Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition By Justin Garvin

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

(A Student's Guide to the Navier–Stokes Equations, 1e Justin Garvin) (Solution Manual all Chapters)

1 Chapter 1 Solutions

Problems 1.1

Find the mass flow rate of a fluid at a constant density of 1.2 kg/m3 passing through a surface whose area is 1 m2 and whose normal is ⃗n = 1 ⃗ = 2ˆi + 3 ˆj + 0kˆ m/s. √ ˆi + 2 ˆj + 3kˆ . The velocity of the flow is: V 14

Solution: The mass passing through a surface is defined as: ρ V⃗ · ⃗ndA

ṁ pass =

A

We can plug our numbers in: ṁ pass = ρ V⃗ · ⃗ndA A

! kg ˆ 1 ˆ ˆ ˆ ˆ 2 i + 3 j m/s · √ i + 2 j + 3k dA = 1.2 3 A m kg 1 14 = 1.2 (2 + 6 + 0) m/sdA √ 3 m 14 A 1 m2

kg 8 = 1.2 2 √ A sm 14 0 9.6 = kg/s √ 14 1.2

Approximate how long will it take, in minutes, to fill up a bathtub with dimensions of 1.3 m x 0.6 m x 0.4 m if water is coming out of a 40 mm diameter faucet at a speed of 1.2 m/s. Assume the density of water is 1000 kg/m3.

3

4

Chapter 1 Solutions Solution: This is a basic mass conservation problem. In words, we have: the change of mass of water in the bathtub in a given unit of time

time rate of mass of water entering the bathtub

=

–

time rate of mass of water leaving the bathtub

There is no mass coming out, which leaves us with: the change of mass of water in the bathtub in a given unit of time

time rate of mass of water entering the bathtub

=

So, the change of the mass in a given unit of time in the tub is just equal to the mass flow rate into the tube: ṁin =

simplifies to

A

ρ V⃗ · ⃗ndA −−−−−−−−→ ρVA

It simplifies to ρVA because ρ and V⃗ are constant and can be pulled out of the area integral. In addition, it is assumed that the velocity and the normal are in the same direction. Plugging in numbers: ṁin = ρVA = (1000) (1.2)

π 0.042 = 0.302kg/s 4

Thus the tube fills at a rate of 0.302 kg/s. The total mass of water that the tube (masstotal) can hold is the volume of the tub multiplied by the density of water: masstube = (1000) (1.3 × 0.6 × 0.4) = 312kg The length of times to fill is: time =

masstotal ṁin

=

312kg

= 1033 s = 17 minutes

0.302kg/s

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems 1.3

5

The divergence of the velocity vector in spherical coordinates can be written as: 2

⃗ ·V ⃗ = ∇

1 ∂Vϕ 1 ∂ r Vr 1 ∂ + (Vθ sin (θ)) + 2 ∂r r sin (θ) ∂θ r sin (θ) ∂ϕ r

where Vr, Vθ, and Vϕ are the velocity coordinates in the r−, θ−, and ϕ− direction, respectively. Determine if a flow with the following flow field velocity is incompressible: Vr = −U cos (θ) 1 –

3R R3 + 2r 2r3

Vθ = U sin (θ) 1 –

3R R3 − 4r 4r3

!

!

Vϕ = 0 where R and U are constants (note, R is not the gas constant in this problem).

Solution: This one is simple enough. We just have to plug in the Vr, Vtheta, and Vz expression into the divergence of velocity equation given for spherical coordinates and see if it is equal to zero, since the divergence of velocity is zero for an incompressible flow. The easiest term in the velocity divergence term is the last one since Vϕ is zero: 1

∂Vphi

r sin θ ∂ϕ

=

1

∂0

r sin θ ∂ϕ

=0

The next term we will look is the first term of the velocity divergence.

Before we do anything, let’s break out the derivative using the product rule: 1 ∂ r2Vr 1 2 ∂Vr 1 ∂r2 + 2 Vr = 2r 2 ∂r ∂r r ∂r r r x˛zx x˛zx =1

=

∂Vr 2 + Vr ∂r r

=2r

We can insert Vr into Equation 1.1 to get:

(product rule) (1.1)

6

Chapter 1 Solutions

1 ∂ r2Vr ∂Vr 2 + Vr = ∂r r 3R r2 ∂r !! 2 U cos ( ) 1 3R !! ∂ U cos ( ) 1 R3 R3 – + θ θ − + = − 2r 2r3 + r − 2r 2r3 ∂r !!

2 3R R3 !!

3R R3 = −U cos θ + 2r2 − 32r4 – U cos θ r − r2 + r4 !

2 3 1 R3 = −U cos θ r − 2r2 − 2 r4 Now onto the second term. We can do the same thing for the second term as we did for the first term. That is, we can expand out the derivative using the product rule: 1 ∂Vθ 1 1 ∂ sin (θ)) = sin θ + V ∂ sin θ (V θ r sin θ ∂θ θ r sin θ ∂θ r sin θ x_˛∂θ z_ x x_______˛ zx 1 r

=

=cos

θ

So now we can plug in Vθ into the above equation to get: 1 ∂Vθ cos θ 1 ∂ sin (θ)) = + V (Vθ r sin (θ) ∂θ r ∂θ r sin θ θ ! !!

3R R3 1∂ 3R R3 cos θ − − = U sin θ 1 − + U sin θ 1 − 4r 4r3 4r 4r3 r ∂θ r sin θ ! !

1 3R R3 U 3R R3 − − 3 + cos θ 1 − = U cos θ 1 − 4r 4r3 4r 4r r r !

3 2 3R R = U cos θ − 2 − 4 r r 2r (1.2)

1.4

As you can see, Equation 1.3 and 1.2 are negatives of each other.

Therefore, adding them up gives us zero and, hence, constitutes an incompressible flow.

An incompressible fluid with density ρ = 1000 kg/m3 travels through a channel with a rectangular cross-section of dimensions 25 mm by 30 mm. The average velocity of the flow in this portion of the channel is 1 m/s. If the flow is at a steady state, what will the velocity be if the rectangular cross-section increases to 50 mm by 50 mm? Start from the integral form of the continuity equation.

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

7

Solution: Starting from the integral form of the continuity equation we have: $ V

∂ρ

∂t

⃗ + ρV · ⃗ndA = 0 A

The slow is at steady state, so the time derivative goes away and we are left with:

A

ρV⃗ · ⃗ndA = 0

We essentially have two sections (areas) that have flow: an inlet and an exit. So, we can break up the area integral into two pieces: „ inlet

ρV⃗ ·⃗ndA +

„ outlet

ρ V⃗ · ⃗ndA = 0

We are going to assume the density and the velocity do not change with the cross-sectional area. In addition, we will assume the inlet normal is −ˆi and the outlet normal is ˆi, giving us: ⃗ inlet · −ˆiAinlet + ρoutlet V ⃗ outlet · ˆiAoutlet = 0 ρinletV If the velocity is only in the x−direction, then we have: – ρinletuinlet Ainlet + ρoutletuoutlet · ˆiAoutlet = 0 – uinlet Ainlet + uoutlet Aoutlet = 0 Ainlet solve for uoutlet −−−−−−−−−→ uoutlet = uinlet Aoutlet Plugging in numbers leads to: (30) (25) uoutlet = 1m/s = 0.3m/s (50) (50) 1.5

Air in a pipe with a diameter of 10 cm starts out at a temperature of 700 K and pressure of 4 × 105 Pa. The initial flow velocity is 10 meters per second. If the pipe diameter contracts to 5 cm with the air speeding up to 115 m/s and the temperature decreases to 500 K, what is the pressure after the contraction? You can assume the molecular weight of air is 28.97 kg/kmol and a steady state. Start from the integral form of the continuity equation.

8

Chapter 1 Solutions Solution: This one is very similar to the previous problem except now we are no longer assuming a constant density. We are still going to start with the intgral form of the continuity equation: $

∂ρ

V

∂t

+ A

ρ V⃗ · ⃗ndA = 0

Again, like last time, we are going to assume steady state:

A

ρV⃗ · ⃗ndA = 0

We have an inlet and an exit section, leaving us with: „ inlet

ρV⃗ ·⃗ndA +

„ outlet

ρ V⃗ · ⃗ndA = 0

We are going to assume the inlet surface has a normal of–ˆi and the outlet surface has a normal of ˆi. In addition, we are going to assume the velocity is only in the x−direction, giving us: ρinletuinletˆi · −ˆi Ainlet + ρoutletuoutletˆi · ˆi Aoutlet = 0 leads to

−−−−→ −ρinletuinlet Ainlet + ρoutletuoutlet Aoutlet = 0

Solve for the density at the exit: ρinletuinlet Ainlet uoutlet Aoutlet pinlet uinlet Ainlet use ρ=RTp −−−−−−→ = RTinlet uoutlet Aoutlet ρoutlet =

(1.3)

Next, utilize the fact that (for an ideal gas), p = ρRT and plug in RTpoutlet outlet for ρoutlet in Equation 1.3 to get: poutlet pinlet uinlet Ainlet = RToutlet RTinlet uoutlet Aoutlet x __ ˛z __ x =

ρoutle

t

Solve for poutlet to get:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

9

Toutlet uinlet Ainlet Tinlet uoutlet Aoutlet 500K 10m/s (10 cm)2 5 = 4 × 10 Pa 700K 115m/s (5 cm)2

poutlet = pinlet

= 99378Pa 1.6

The velocity profile of flow in a pipe in the z−direction (Vz) is given by: ∆pR2 r2 1 – 4µL R2

Vz =

!

where r is the radial component (in cylindrical coordinates), R in this case is the radius of the pipe, L is the length of the pipe, ∆p is considered to be a pressure difference between the entrance and exit of the pipe, and µ is something called the dynamic viscosity. Obtain an expression for the mass flow rate through the pipe.

Solution: The mass flow is given by: „ ṁ =

ρV⃗ · ⃗ndA

A

The velocity in the z−direction is given and it is assumed that the normals of the surface areas of the flow passing through the pipe is also in the z−direction. Therefore, we can replace the dot product of the velocity with the normal with just the velocity in the z−direction: „ ṁ =

A

ρVzdA

We are now dealing with cylindrical coordinates. If you recall, the area, dA in cylindrical coordinates will just be given by rdrdθ, therefore we now have: „ ṁ =

A

ρVzrdrdθ

We can replace Vz with the expression given in the problem: „ ṁ =

A

ρ

∆pR2 r2 1 – 4µL R2

!! rdrdθ

10

Chapter 1 Solutions 2

We can pull out the constant ρ∆pR from the problem and integrate over 4µL r from 0 to R and integrate over θ from 0 to 2π to get: ! r2 1– π rdrdθ ṁ = ρ R2 4µL 0 0 , R ! ∆pR2 r2 =ρ 2π 1 − 2 rdr 4µL R 0 ! , R r3 ∆pR2 =ρ 2π r − 2 dr 4µL R 0 !R 1 ∆ pR2 2π r2 − 1 r4 =ρ 4µL 2 4 R2 0 ! ∆pR2 1 1 =ρ 2π R 2 − R2 4µL 2 4 ∆pR2 ,

2

, R

Therefore, after the algebra: ṁ = ρ 1.7

∆pR4π 8µL

If the velocity profile of flow in a channel is given by: u = UH∞ y, what is the mass flow rate through the channel per length into the page if the height of the channel is H? Solution: Assume that the velocity is only in the x−direction and that the normal is also in the x−direction. Also, assume the density is constant. Therefore: „ „ ρ V⃗ · ⃗ndA = ρ udA A A , H =ρ udy 0

, =ρ =ρ 1.8

0

H

U∞ H

ydy

U∞ 1 2 H U∞H y =ρ H 2 0 2

. per width into the board

Given the following velocity vector: V⃗ = Cy cos(5x)ˆi + D sin(5x)y2 ˆj, what do C and D need to be in order for this vector field to be considered incompressible flow?

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

11

Solution: Take the divergence of velocity: ∂ ⃗ ·V ⃗ = ∂ (Cy cos (5x)) + D sin (5x) y2 ∇ ∂x ∂y = −5Cy sin (5x) + 2Dy sin (5x) In order for the flow to be incompressible, the above expression must equal zero. Thus: – 5Cy sin (5x) + 2Dy sin (5x) = 0 2 ∴C= D 5 1.9

The flux of a quantity given by the vector, f⃗ = 5xyˆi + 10y ˆj + 0kˆ, passes through a cube with dimensions (in x−, y−, and z− directions) of 2 x 1 x 1 (assume the cube “starts” at the origin). Find the value of the area integral, A f⃗ · ⃗ndA. Solution: The geometry for this problem is simple enough: y

x

z

The first thing to notice in this problem is that the flux is two-dimensional since the f⃗ vector does not have a component in the z−direction. Therefore, the front and back faces (i.e. the z−faces) will not be utilized). With this in mind, we can break out the area integral into the following four

12

Chapter 1 Solutions separate integrals corresponding to the left, right, bottom, and top faces, i.e.: f⃗·⃗ndA = A

„ left

f⃗· −ˆi dA +

„ +

bottom

„ right

f⃗ · − ˆj dA +

f⃗ · ˆi dA

„

f⃗ · ˆj dA top

Plugging in the expression for the flux vector and taking the dot products, we get: f⃗ · ⃗ndA = A

„

„ −5xydA +

„

left

+ bottom

5xydA right

–10ydA +

„

top

10ydA

Now we need to perform these integrals. The limits of the integration depend on which face you are at. For example, the left face lies on the x = 0 place with y− and z−varying. Therefore, the left face needs to be integrated with respect to y and z. The right face is at the x = 2 plane with y and z varying, so the right face needs to be integrated over y and z. The top and bottom faces (which are located at y = 1 and y = 0, respectively) need to be integrated over the x and z coordinates. Thus:

A

f⃗· ⃗ndA =

!

, 1, 1 0

0

−5xydydz

! , 1, 1 5xydydz + x=0

! , 1, 2 + −10ydxdz 0

0

0

0

x=2

! , 1, 2 10ydxdz + y=0

0

0

y=1

Notice that, once the integrals are evaluated, the values of x or y need to be set to match whether the surface is the left (x = 0), right (x = 2), bottom (y = 0), or the top (y = 1) surface.

Evaluating the integrals leads to: ! !

1 y=1 z=1 1 y=1 z=1 f⃗ · ⃗ndA = −5x y 2 z + 5x 2 y2 z 2 y=0 z=0 x=0 y=0 z=0 x=2 A x______________________˛z______________________x x____________________˛z____________________x + −10yx

=0 ! x=2 z=1

z

x= 0 z= 0

y=0

+ 10yx

=5 ! x=2 z=1

x____________________˛z____________________x x__ _ __ _ __ _ _ ˛z =0

z

x=0 z=0

y=1

x

=20 __________

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

13

Adding them up thus equals 25!

1.10 Sketch the velocity vector given by: V⃗ = − sin(y)ˆi + sin(x) ˆj.

Solution: We are asked to plot this velocity vector on an x and y grid.

This essentially means calculating the u and v values at different x and y points and plotting the u and v as a vector on an xy grid. The u values are just obtained by calculating − sin(y) with the y− plugged in and the v values are calculated via sin(x) with the x− plugged in. This is done for some x and y values below: x 0 1 1 0 -1 -1 0 2 2 . . .

y 0 0 1 1 -1 0 -1 0 2 . . .

u = − sin(y) 0 0 -0.84 -0.84 0.84 0 0.84 0 -0.909 . . .

v = sin(x) 0 0.84 0.84 0 -0.84 -0.84 0 0.909 0.909 . . .

The values of u and v are made into a “vector” and plotted at the different x and y points. This can all be done relatively easy in either a Matlab or Python script. An example Matlab script is given below along with the corresponding plot: clc clear x = [-2*pi:0.5:2*pi]; %sets up an x-axis y = [-2*pi:0.5:2*pi]; %sets up a y-axis [x,y] = meshgrid(x,y); %takes the axis info and makes a grid %of x and y values u = -sin(y); %calculates the u velocity v = sin(x); %calculates the v velocity quiver(x,y,u,v) %draws the velocity vectors on the grid xlabel(’x-axis’) ylabel(’y-axis’) title(’velocity vector plot’)

14

Chapter 1 Solutions

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

15

2 Chapter 2 Solutions

Problems 2.1

Given an initial temperature distribution of T (x) = sin(x), create a table similar to Table 2.1 for times of 0 s, 0.5 s, 1 s, and 1.5 s when u = 1 m/s.

Plot the final temperature distribution at t = 1.5 s.

Solution: The table is filled out below and the figure at t = 1.5 s is given on the next page.

x (m)

0 1.0 2.0 3.0 4.0 5.0 6.0

T (celsius) (@t = 0) (T=T(x,0)) 0 0.84 0.909 0.141 -0.76 -0.96 -0.28

T (celsius) (@t = 0.5 s) (T=T(x-u*0.5,0)) -0.48 0.48 0.99 0.6 -0.35 -0.98 -0.7

T (celsius) (@t = 1 s) (T=T(x-u*1,0)) -0.84 0 0.84 0.909 0.141 -0.76 -0.96

T (celsius) (@t = 1.5 s) (T=T(x-u*1.5,0)) -0.99 -0.48 0.48 0.99 0.6 -0.35 -0.98

16

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

17

1

T (◦C)

0.5 Advected at 1 m/s

0

−0.5

−1 0

2.2

2.3

1

2

3 x (m)

4

5

6

Notice that the sine plot is just shifted to the right by 1.5*1 in the x−direction.

Explain why the characteristic curves for the advection equation are parallel when the velocity is a constant value.

Solution: The reason why the characteristic curves are parallel for constant velocity is because the characteristic curves provide a way for determining how fast the fluid elements are advecting via the inverse of the slope of the t − x curve. Since the velocity of the fluid elements are the same throughout, the various characteristic curves must also have the same slope.

Given an initial x−velocity distribution of: 1 x ≤ 0.5 u(x, 0) = 2x 0.5 < x < 1 2 x≥1 Does a shock form when the inviscid Burgers’ equation is applied to this initial velocity field? Why or why not? Plot the characteristic curves.

Solution: In order to determine if a shock forms, we can plot the characteristic curves and see if they overlap. The characteristic curves are just the t vs. x curves of various fluid elements as they travel. The velocity of the fluid elements do not change since the inviscid Burgers’ equation, in

18

Chapter 2 Solutions Du = 0. Therefore, we can pick a few points on Lagrangian form, is just Dt our initial velocity distribution, call them fluid elements, and track their position in time via characteristic curves. If the curves overlap, then a shock is considered to have developed since there is an “overtaking” of one fluid element going past another. The initial points (i.e. initial fluid elements) we will pick will be (you can pick whichever points you would like):

x = 0, which has a velocity of u = 1 (Fluid element 1) x = 0.5, which has a velocity of u = 1 (Fluid element 2) x = 0.75, which has a velocity of u = 1.5 (Fluid element 3) x = 1, which has a velocity of u = 2 (Fluid element 4) x = 1.5, which has a velocity of u = 2 (Fluid element 5) We can plot the characteristic curves (the t vs x curves) with the slopes being the inverse of velocity, giving us: 3 Fluid element 1 Fluid element 2 Fluid element 3 Fluid Element 4 Fluid element 5

t

2

1

0

0

1

2

3

4

5

x

2.4

Notice that the characteristic curves appear to be diverging from each other...hence no shock will form as time passes. This is called rarefraction.

Find an expression for the material derivative of temperature if the temperature is given by the following equation T = e−t(sin(2x) + cos(y)) and if the velocity vector is: ⃗V = ˆi + 2 ˆj.

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

19

Solution: The material derivative of the temperature in Cartesian coordinates is given by: DT Dt

=

∂T ∂t

+u

∂T ∂x

+v

∂T ∂y

+w

∂T ∂z

There does not appear to be a z−coordinate, so the z−will go away. The local time derivative of the temperature is given by: ∂T ∂ −t = e (sin(2x) + cos(y)) = −e−t(sin(2x) + cos(y)) ∂t ∂t

(2.1)

The x−derivative is: ∂T ∂ = e−t(sin(2x) + cos(y)) = 2e−t cos(2x) ∂x ∂x

(2.2)

The y−derivative is: ∂T ∂ = e−t(sin(2x) + cos(y)) = −e−t sin(y) ∂y ∂y

(2.3)

The material derivative can now easily be obtained: DT Dt

=

∂T ∂t

+u

∂T ∂x

+v

∂T ∂y

+w

∂T

∂z x˛zx =0

√ , +v . −e−t sin(y) . x_______˛z_______x .̧ x______˛z______x .,

= −e−t(sin(2x) + cos(y)) +u e−t cos(2x) x_______________ _____˛z __ ______________________x from

E quat i o n 2 . 1

from Equation 2.2

from E quati o n 2.3

Plug in 1 for u and 2 for v to get: DT = −e−t(sin(2x) + cos(y)) + 2e−t cos(2x) − 2e−t sin(y) Dt 2.5

For the temperature equation used in Problem 2.4, what is the advective transport term of temperature if the velocity field is given by V⃗ = 2xyˆi − y2 ˆj when x = 2, y = 1, and t = 1?

Solution: The advection transport term is considered the spatial terms in the material derivative (i.e. everything but the local time derivative).

In the previous example, the x−derivative of temperature was:

20

Chapter 2 Solutions ∂T ∂x

= 2e−t cos(2x)

and the y−derivative was: ∂T ∂y

t

= −e− sin(y)

The u−velocity is given by 2xy and the v−velocity is given by −y2.

For x = 2, y = 1, and t = 1, the advection term becomes: u

∂T ∂T = 2xy 2e−t cos(2x) − y2 −e−t sin(y) +v ∂x ∂y = 2 (2) (1) 2e−1 cos(2 · 2) + 12 e−1 sin(1) = −1.614

2.6

Given the velocity field vector: V⃗ = y (A cos(2t) + B sin(3t)) ˆi + 6xyt ˆj m/s, what is the acceleration of the fluid at x = 0.5 m and y = 0.5 m at time, t = 1 s?

Solution: Another material derivative question. In order to find the acceleration, the material derivative of velocity needs to be calculated. Again, it appears there is only an x−velocity and a y−velocity in our velocity vector: u = y (A cos(2t) + B sin(3t)) m/s and v = 6xyt m/s The material derivative of the velocity vector is given by (Equation 2.13 without the z−component): , √ Du , √ ∂u ∂u ∂u . Dt . ..∂t + u ∂x + v ∂y . . . . . .. .= . ¸ Dv , ¸∂v + u ∂v + v ∂v ., Dt ∂t ∂x ∂y The local time derivative is calculated to be:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

21

√ , √ ∂ , √ , . ∂u . . (y (A cos(2t) + B sin(3t))). ..y (−2A sin(2t) + 3B cos(3t)) .. . ∂t . . ∂t . . . = ,. . = ¸. ¸. ∂v ,. ¸. , ∂ (6xyt) 6xy ∂t ∂t (2. 4)

The x−derivative term of advection:

√ , √∂ (y (A cos(2t) + B sin(3t))), .u ∂u . . (y (A cos(2t) B sin(3t))). ..∂x .. ∂x .,. x__________________________ ... x .¸ , = + .¸ ˛z__________________________ ∂v ∂ (6xyt) u ∂x u ∂x √

(2.5) ,

0 . . = . ., ¸ 6yt (y (A cos(2t) + B sin(3t)))

The y−derivative term of advection is:

√ , . v ∂u . √∂ (y (A cos(2t) + B sin(3t))), . .. ∂y ∂y . . . . = 6xyt .. .. x˛ zx .. .., ¸ ∂ ¸v ∂v , v ∂y ∂y (6xyt) (2.6) √ , 6xyt (A cos(2t) + B sin(3t)) .. . = .¸ ., 36x2yt2

Adding up Equations 2.5, 2.6, and 2.4, we get the following expression for the material derivative of velocity:

22

Chapter 2 Solutions √ Du , √ ∂u u ∂u v ∂u , . . .. + ∂x + ∂y . . Dt . . ∂t . = ., ¸.Dv ,. .¸∂v + u ∂v + v ∂v Dt ∂t ∂x ∂y √ , y. (−2A sin(2t) + 3B cos(3t)) + 0 + 6xyt (A cos(2t) + B sin(3t) ) . . =. ., ¸ 6xy + 6yt (y (A cos(2t) + B sin(3t))) + 36x2yt2 √ , 0.5 (−2A sin(2) + 3B cos(3)) + 0 + 6(0.5)(0.5)(1) (A cos(2) + B sin(3)) . . −−−−−−−−−→ = .¸ ., 6(0.5)(0.5) + 6(0.5)(1) ((0.5) (A cos(2) + B sin(3))) + 36(0.5) 2(0.5)(1) 2 plug in values

√−A(0.909) + 1.5B(−0.98999) + 1.5A(−0.416) + 1.5B(0.1411) , . = . .¸ ., 1.5 + 1.5A(−0.416) + 1.5B(0.1411) + 4.5 √ , √ , ..−2.81 .. −1.533A − 1.2733B . . . .. −−−−−−−−−−−−→ ..̧ = .. . , i f A = 1 and B = 1 ¸ 5.59 , 6 − 0.624A + 0.21165B 2.7

The Lagrangian and non-conservation form of the continuity equation can be obtained by applying mass conservation to a moving fluid element, which states that the mass of a moving fluid element does not change in time. Mathematically, this can be written as: d dt

$ V(t)

ρdV = 0

Using the ideas from Section 2.4, where we applied a time derivative to the momentum of a moving fluid element, derive the Lagrangian form of the mass continuity equation as well as the non-conservation form of the continuity equation.

Solution: We can sneak in the time derivative into the volume integral of ρ and utilizing the product rule:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems d dt

23

$

$

ρdV = V(t)

V(⊔)

$ =

V(t)

ρdV Dt Dρ D (dV) dV + ρ Dt Dt

(2.7) !

Recalling the the material derivative of volume is given by: D (dV) ⃗ ⃗ = ∇ · V dV Dt We can plug in the equation above for the material derivative of the infinitesimal volume into Equation 2.7 to get: d dt

$

$ V(t)

Dρ dV + ρ ∇⃗ · V⃗ dV V(t) Dt $ Dρ ⃗ dV = + ρ ∇⃗ · V V(t) Dt x_____________________________˛z_____________________________x

ρd = V

=0

Since the integral needs to equal zero for continuity and since the volume is completely arbitrary (meaning that the limits of the volume integral could be anything), then the integrand must also be zero, therefore: Dρ

⃗ =0 + ρ⃗ V ∇· Dt Which is the Lagrangian form of the continuity equation. We can also expand the material derivative out into the non-conservation form of the continuity equation: ∂ρ ⃗ ⃗ + V · ∇ρ + ρ ∇⃗ · V⃗ = 0 ∂t 2.8

Suppose the density of a fluid element is given by the expression: ρ = e−0.005t + 1. It travels with a velocity of 2 m/s in the x−direction. What is the value of the material derivative after it has gone 60 meters?

Solution: The density of a fluid element as it moves changes in time given the expression in the problem. Assuming the starting time is t = 0 s, the time when the fluid element has gone 60 meters is (60 meters)/(2 meters per second) = 30 seconds. Thus, the material derivative at 30 seconds is nothing but:

24

Chapter 2 Solutions Dρ

2.9

= −0.005e−0.005∗30 = −0.004kg/s Dt What is the summation force vector on a cube fluid element that is 1 meter by 1 meter by 1 meter in size if the density is 1 kg/m3 and ⃗V = 2xˆi + 4y ˆj?

Solution: The force balance on a fluid element is given by: $

ρ V(t)

⃗ DV Dt

d = Σ F⃗ V

Since the velocity given contains spatial coordinates, the material derivative of velocity should be calculated in an Eulerian description. Thus: ⃗ ∂ V⃗ ⃗ ⃗ ⃗ DV = +V V ·∇ Dt ∂t The velocity vector given does not appear to be a function of time, therefore the system must be in a steady state. Thus, only the advective term needs to be calculated: √ ∂u , √. 2x · 2 + 4y · 0,. √ , . . u ∂x + v ∂u . ..4x .. ∂y . . ⃗ V⃗ = V⃗ · ∇ .=. . . = ¸. .¸ ∂v , ¸16y, ∂v , u∂x + v∂y 2x · 0 + 4y · 4 Integrating over volume, we get: $

√ , , 1 , 1 , 1 . 4x .

D V⃗ dxdydz ρ .. ρ dV = ¸ ,.

Dt 0 0 0 V(t) 16y √ , 1 1 1 2 . . √ , 2 kg m/s 2x y z. . . .

0 0 0 = ρ. . = ..¸ .,.

1 8 kg m/s . . . . ¸ 8 2 1 1., y x z 0

0 0

The summation of forces equals the above expression.

2.10 The gradient of a scalar function, f , is defined as: ⃗ f ≡ ˆi ∂ f + ˆj ∂ f + kˆ ∂ f ∇ ∂x ∂y ∂z

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

25

Show that ⃗V · ⃗∇ f = V⃗ · ⃗∇ f Solution: Given how ⃗∇ f is defined above, V⃗ · na⃗bla f is just: ! ⃗V · ⃗∇ f = uˆi + v ˆj + wkˆ · ˆi ∂ f + ˆj ∂ f + kˆ ∂ f ∂x ∂y ∂z ∂f ∂f ∂f =u +v +w ∂x ∂y ∂z This is the same as ⃗V · ⃗∇ f operating on f , observe: ∂ !! f ⃗V · ⃗∇ f = uˆi + v ˆj + wkˆ · ˆi ∂ + ˆj ∂ + w ∂y ∂x ∂z ! ∂ +v ∂ +w ∂ f = u ∂x ∂y ∂z

=u

∂f ∂x

+v

∂f ∂y

+w

∂f ∂z

3 Chapter 3 Solutions

Problems 3.1

Find the stress vector acting on a surface if the stress tensor is: √ , 5 −3 10 . ⃗ .

T⃗ = .−3 2 4. ¸ , 10 4 7 and the normal of the surface is given by: ⃗n =

√1 ˆi + √1 ˆj + 0kˆ.

2 2

Solution: The stress vector can be obtained by simply doing a matrix ⃗ multiplication of T with ⃗n. Before we do that, we should check to en, √ √ 2 1 + 21 = 1. Check. sure ⃗n is a unit normal, i.e. : ||⃗n|| = ⃗n · ⃗n = √ Now let’s perform the matrix multiplication to find the stress vector (i.e. traction vector), τ⃗: √ . 5

,√ 1, −3 10 . √ .

. τ⃗ = .−3 2 . ¸.

10 4 3.2

2

..

√ .

5 √

.

.

2

– , .

√3 2

.

√ . .

, √2 .

2

.

4 . . √1 . = .− √3 + √2 . = .− √1 .

2. .. 2. . 2 . 2.

7

., ..̧ ,.

0

..̧ √10 2

. + √4 , 2

. .̧ √14 ., 2

Do you spot any potential issue with the stress tensor below? √ , 5x 4x 10z ⃗T⃗ . 3y 2xy 4x . = ¸.− ., 2z 4y 7xyz

26

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

27

Solution: Yes, the stress tensor is not symmetric. In general, the stress tensor should be symmetric due to the conservation of angular momentum. We illustrated in the book that not having a symmetric tensor can lead to an imbalance of forces (and hence torque) on an infinitesimal fluid element. Since an infinitesimal fluid element cannot resist motion at all (since there is no moment of inertia), the imbalance of torque would cause the fluid element to infinitely accelerate.

3.3

A stress tensor (in pascals) of a flow is given by: √

, 5x −3y 10z . ⃗​⃗ .

T = −3y 5y 4x . ¸. , 10z 4x 5z What is the acceleration of a fluid particle with density of 1.2 kg3 (asm suming no body force)?

Solution: Here we must use Cauchy’s first law of motion:

ρ

⃗ DV Dt

⃗ · T⃗​⃗ + ρ⃗g =∇

If there is no body force, then:

ρ

⃗ DV Dt

⃗ · T⃗​⃗ =∇

The acceleration of the fluid element is just the material derivative of velocity of the fluid element, thus: acceleration =

D V⃗ Dt

1⃗ ⃗​⃗ = ∇ ·T ρ

Therefore, the acceleration of the fluid element in this case is:

acceleration = =

1

∂

1.2 ∂x

∂ ∂y

√ , . 5x −3y 10z. ∂ .

5y 4x .. ∂z ..−3y ¸ , 10z 4x 5z

1 5 − 3 + 10 0 + 5 + 0 0 + 0 + 5 1.2

28

Chapter 3 Solutions The acceleration is then (written in column vector form): √ 10 , acceleration 4.17 m s2 =.¸ . .., / 4.17

3.4

If the velocity field of a given flow is given by:

⃗V = A cos(x)ˆi + B sin(y) ˆj + C tan(z)kˆ Determine an expression for the total force (per volume) acting on the fluid at a given (x, y, z) point.

Solution: This is another Cauchy first law problem. Again, Cauchy’s first law is:

ρ

⃗ DV Dt

⃗​⃗ = ⃗∇ · T + ρ⃗g

The right-hand side is the total force (per volume). Thus, if we found an expression for the left-hand side by finding the material derivative of the velocity and multiplied by density, we would have an expression for the total force.

The material derivative of velocity can be written in Cartesian coordinates (in an Eulerian description) as:

√ ∂u ∂u ∂u ∂u , .. ∂t + u ∂x + v ∂y + w ∂z .

D V⃗ v ∂v + w ∂v = . ∂v + u ∂v ∂x + ∂y ∂z . ∂t Dt . . .¸ ∂w . ∂w ∂w ∂w , ∂t + u ∂x + v ∂y + w ∂z In our example, u = A cos(x), v = B sin(y), w = C tan(z). In addition, the velocity field appears to be at a steady state, so the time derivatives go away. We can start crossing off terms in the material derivative:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

29

√ ,0 0 . cos(x) ∂(A cos(x)) + B sin(y) ∂(A cos(x))*+ C tan(z) ∂(A cos(x))*.

A ∂x ∂y ∂z . . . . *0 D V⃗ *0 . . ∂(B sin(y)) = + B sin(y) ∂(B sin(y)) + C tan(z) ∂(B sin(y)) ∂y ∂z ..

Dt .. A cos(x) ∂x . ., *0 *0 .̧ (z)) ∂ (C tan (z)) + C tan(z) ∂(C tan(z)) A cos(x) ∂ (C tan + B sin(y) ∂z ∂x ∂y This leads to the following expression for the force per volume after the derivatives have been taken: √ 2 , .−A cos(x) sin(x) .

D V⃗ force per volume = ρ = ρ . B2 sin(y) cos(y) .

Dt .. .. ¸2 , C tan(z) sec2(z) 3.5

For the velocity field given in Problem 3.4, if there are no body forces and if the dynamic viscosity is considered a constant, find an expression for the pressure gradient (given A = 1, B = 1, and C = 0). Be sure to check if this flow is incompressible or not. Is this flow at a steady state?

Solution: For A = 1, B = 1, C = 0, the force per volume (given from the previous problem) is: √– cos(x) sin(x), . .

D V⃗ force per volume = ρ = ρ . sin(y) cos(y) .

Dt .., ..¸ 0

(3.1)

Since there is no time variation in the velocity field, given by: V⃗ = cos(x)ˆi + sin(y) ˆj, the problem is at a steady state. We should also determine if incompressible: ∂ ∂x

∂ ∂y

! cos(x) = − sin(x) + cos(y) ≠ 0 sin(y)

(3.2)

Thus, the flow is not an incompressible flow. Now we need to find the

30

Chapter 3 Solutions pressure gradient. If we assume the compressible Navier-Stokes equations in Lagrangian form with no body force and a constant viscosity, we have:

ρ

⃗ DV

† 2 = − ∇⃗ p − µ ∇⃗ ⃗∇ · V⃗ + µ ∇⃗ · ⃗∇V⃗ + ⃗∇V⃗ 3 Dt

Solving for the pressure gradient gives us: √ , † .. DV⃗ 2 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ + ∇V ⃗ . ∇p = − ρ + µ ∇ ∇ · V − µ ∇ · ∇V ¸ , Dt 3 We already obtained ρ DV⃗ from Equation 3.1. We can now find an Dt

⃗ · V⃗ . From Equation 3.39, we have: expression for 23 µ ∇⃗ ∇ √ ∂ ∇⃗·V⃗ , . . . ∂x .

2 ⃗ ⃗ ⃗ 2 . ∂ ∇⃗ ·V⃗ .

V = µ µ∇ ∇ · 3 . ∂y .

3 .¸∂ ∇⃗·V⃗ ,. ∂z

We can plug our velocity divergence from Equation 3.2 into the above equation to get: √ , .− cos(x).

2 ⃗ ⃗ V⃗ = 2 µ . − sin(y) .

(3.3) µ∇ ∇ · 3 3 . . .¸ ., 0 ⃗ ⃗ ⃗ V⃗ † term. To calculate Next up we need to calculate the µ ∇ · ∇ V⃗ + ∇ this term, we need to first find the velocity gradient. The velocity gradient is just determined by the following expression: ∂u

∂v ∂x

∂w , ∂x .

∂u ∂y

∂v ∂y

∂w ∂y

∂z

∂v ∂z

∂w , ∂z

. ∂x ⃗V ⃗ = ∇

. .¸∂u

. .

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

31

Plugging in our velocity field of V⃗ = cos(x)ˆi + sin(y) ˆj leads to: √ – sin(x) . ⃗∇V⃗ = . .¸

0

, 0 .

0

cos(y)

0.

0

0

., 0

† Thus, ⃗∇⃗V + ⃗∇V⃗ becomes:

⃗∇⃗V + ⃗∇V⃗

†

√– sin(x) .

0

0

cos(y)

0

0

= . .̧.

0, √– sin(x) . .

0. + . .. ..̧ , 0

√ .−2 sin(x)

0

, 0.

0

2 cos(y)

0

0

0. . ., 0

= . .¸

0

0., .

0

cos(y)

0.

0

0

., 0

The diffusive transport term then becomes:

⃗ ⃗ ⃗V + ∇ ⃗V ⃗ † =µ µ∇ · ∇

∂ ∂x

∂ ∂y

= µ −2 cos(x)

∂ ∂z

√ .−2 sin(x) 0 . . .¸

2 cos(y)

, 0.

0.

0

. ., 0

0

0

−2 sin(y)

0 (3.4)

Putting together Equations 3.1, 3.3, and 3.4 (written as a column vector) leads to the following for the pressure gradient:

32

Chapter 3 Solutions √ , † 2 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ . ∇ = − .ρDV ⃗ + µ ∇ ∇ · V − µ ∇ · ∇V⃗ + ∇V⃗ ⃗p ¸ , Dt 3 √ , √ , , √ √ , .− cos(x). .−2 cos(x) .. .. . .. − cos(x) sin(x) . . −2 sin(y) .. . . sin(y) cos(y) . 2 . − sin(y) . = − .ρ . . + 3µ . . − µ. ... . .. ..¸ .¸ ¸. , ., .̧ ., ., 0 0 0

3.6

Determine the Newtonian stress tensor if the pressure is given by 10x pascals and a velocity field of V⃗ = 2x sin(2y)ˆi + x cos(2y) ˆj m/s at location x = 1 m, y = 2 m. Assume a dynamic viscosity of 10−4 Pa·s.

Solution: The Newtonian stress tensor is given by: ⃗​⃗ ⃗​⃗ 2 ⃗ ⃗ V⃗ + ⃗∇V⃗ † = T = − p I − µ ⃗∇ · V⃗ I+ µ ∇ 3 √ 2 ⃗ · V⃗ + 2µ∂u . −p − µ ∇ 3 ∂x . µ ∂u + ∂v . ∂y ∂x . .¸ ∂w µ ∂u ∂z + ∂x

µ ∂v + ∂u ∂x ∂y

∂u µ ∂w ∂x + ∂z

⃗ · V⃗ + 2µ∂v −p − 2 µ ∇ 3

∂y

∂w µ ∂v ∂z + ∂y

, .. .

∂v µ ∂w ∂y + ∂z

. . . , −p − 32 µ ⃗∇ · V⃗ + 2µ∂w ∂z

The velocity field has no z−component, so we can just write the stress tensor in two-dimensions: ⃗​⃗ ⃗​⃗ 2 ⃗ ⃗ V⃗ + ⃗∇V⃗ † = T = − p I − µ ⃗∇ · V⃗ I+ µ ∇ 3 √ . −p − 2 µ ⃗∇ · V⃗ + 2µ∂u 3 ∂x . .¸ ∂v µ ∂u ∂y + ∂x

, . . . ∂v ,

µ ∂v + ∂u ∂x ∂y

−p − 32 µ ⃗∇ · ⃗V + 2µ ∂y

The pressure term is simple: 0 ⃗​⃗ −10x I −p = 0 −10x

! = x=1

−10 0 0 −10

!

(3.5)

Next we need to find the divergence of velocity:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

⃗ · V⃗ = ∇

∂ ∂x

33

√ , 2x . sin(2y) . ∂ ∂y . .¸ .., = 2 sin(2y) − 2x sin(2y) x cos(2y)

⃗ Now that we have the divergence of velocity, we can find −32 µ ⃗∇ · ⃗V I at x = 1 m, y = 2 m and a dynamic viscosity of 10−4 Pa via:

2 ⃗ ⃗ ⃗ – µ ∇ ·V I 3 √ 2 . −3 µ (2 sin(2y) − 2x sin(2y)) = .. ¸ 0

, .. . ., − 32 µ (2 sin(2y) − 2x sin(2y)) x=1,y=2,µ=10−4

√ 2 −4 . 310 (2 sin(4) − 2 sin(4)) .− = .. ¸ 0

, . . ., − 32 10−4 (2 sin(4) − 2 sin(4))

0

0

√ , 0 0 . . =. . ¸ , 0 0 (3.6) ⃗ V⃗ + ∇ ⃗ V⃗ Next up is the µ ∇ √ ∂u . ⃗V ⃗ .∂x ∇ = ..̧ u ∂ ∂y

†

term, The velocity gradient in 2D is:

√ ∂(2x sin(2y)) ∂x .. = ., .̧ ∂(2x sin(2y)) ∂v

∂v ∂x ..

∂(x cos(2y)), . ∂x .

∂y

∂(x cos(2y)) . , ∂y

∂y

√ 2 sin(2y) = .. ¸4x cos(2y)

.

cos(2y) , . ., −2x sin(2y)

⃗ ⃗V + ∇ ⃗ V⃗ † turns out to be: Using the velocity gradient above, the µ ∇

34

Chapter 3 Solutions ⃗V ⃗ ⃗V † ⃗+ ∇ µ ∇

=µ

.

, cos(2y) . √2 sin(2y) . . + ., ¸. −2x sin(2y) cos(2y)

2 sin(2y)

¸.

4x cos(2y) √

4 sin(2y)

= µ .̧ cos(2y) + 4x cos(2y) √

4 sin(4) .. = . ¸ 5 cos(4)

, 4x cos(2y) . . ,. −2x sin(2y)

4x cos(2y) + cos(2y),. . . , x=1,y=2,µ=10−4 −4x sin(2y)

, 5 cos(4) . √−3.027 . . ., = ¸. −4 sin(4) −3.27

−3.27., . ., 3.027 (3.7)

Adding up Equations 3.5, 3.6, and 3.7, the stress tensor becomes: √ , √ , ! .. ..−3.027 −3.27 −13.027 −3.27 ! .

10 0 ⃗ .

0 0 − T⃗ = . . . . + + = 0 −10 ¸ −3.27 −6.973 , ¸ , 0 0 −3.27 3.027 3.7

What is the pressure, in pascals, a person experiences when they have dived 5 meters below the water surface? Assume atmospheric pressure is 105 pascals.

Solution: This problem deals with hydrostatic pressure. The equation of which is just: p = ρ|gy|depth + patm Plugging in the numbers from the problem gives us (assume water density is 1000 kg/m3 and the gravitational acceleration is 9.8 m/s2: p = ρ|gy|depth + patm = 1000 ∗ 9.8 ∗ 5 + 105 = 149000Pa

3.8

Estimate the shear stress (i.e. viscous force) on a surface defined by ⃗n = ˆi where the velocity distribution is given by:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

u=

U H

35

y

v=w=0 where U is a velocity parameter and H is a length scale parameter. ⃗ V⃗ † term in the Solution: The shear stress comes from the µ ⃗∇V⃗ + ∇ stress tensor, which is given by: √ ⃗ V⃗ + ∇ ⃗ V⃗ µ ∇

†

2µ∂u ∂x

. = .µ ∂u + ∂v ∂y ∂x . .¸ ∂u ∂w µ ∂z + ∂x 0 µU H . = .µ U 0 H . .¸ 0 0

µ ∂v + ∂u ∂x ∂y 2µ∂v

∂y

∂w µ ∂v ∂z + ∂y

, µ ∂w + ∂u ∂x ∂z . . ∂v µ ∂w ∂y + ∂z . . . , 2µ∂w ∂z

0 .

0. . ., 0

The shear stress vector (τ⃗shear) can be written as: √ .. 0 † ⃗V ⃗V ⃗+ ∇ ⃗ · n = .µ U τ⃗shear = µ ∇ ⃗ . H .. ¸ 0

3.9

µ HU 0

,√ , 0 . .1 .

0. .0. = µ .. .

0

U ˆj H

,. ¸. ,.

0 0

The shear stress on the ⃗n = ˆi surface is µHU in the y−direction. ⃗ ⃗V = ⃗V · ∇ ⃗ ⃗V in two dimensional Cartesian coordinates.

Show that ⃗V · ∇ ⃗ operator in 2D Cartesian coordindates is given by: Solution: The ⃗V · ∇ ⃗V ⃗ = u ∂ + v ∂ ·∇ ∂x ∂y Applying this operator to V⃗ = uˆi + v ˆj yields:

36

Chapter 3 Solutions ∂ ⃗ ⃗V = u ∂ + v uˆi + v ˆj ⃗V · ∇ ∂y ∂x ! ! ∂u ∂u ˆ ∂v ∂v ˆ = u +v i+ u +v j ∂x ∂y ∂x ∂y ⃗ V⃗ is calculating the dot product of velocity with the velocity The V⃗ · ∇ gradient. This is given by:

⃗ V⃗ = u V⃗ · ∇

√ ∂u . ∂x v . .¸∂u ∂y

∂v ∂x .

∂u

∂u

. = u ∂x + v ∂y

∂v ., ∂y

∂v + v ∂v u ∂x ∂y

The above result, while written as a row vector, yields the same result ⃗ V⃗ . Thus, the calculation order turns out to as when we calculated V⃗ · ∇ not matter in this instance.

3.10 Please explain, in your own words, why an equation of state is not necessarily needed for an incompressible flow.

Solution: The reason for not needing an equation of state for an incompressible flow is simply because the incompressible Navier-Stokes equations are closed by including the divergence-free velocity condition.

Thus, no thermodynamics is needed to be able to solve for the flow field of an incompressible flow under usual circumstances. Even more to the point, often incompressible flows assume a constant density throughout the flow.

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

4 Chapter 4 Solutions

4.1 Chapter 4 Solutions Problems 4.1

Find ⃗A ⊗ ⃗B if ⃗A = 3ˆi + 5 ˆj − 10kˆ and ⃗B = 5ˆi − 2 ˆj − kˆ.

Solution: Matrix multiply the column vector form of ⃗A with the row vector form of B⃗ : √ , √3 , . 15 −6 −3 . .. . ⃗⊗B ⃗ =.

A 5 . 5 −2 −1 = . 25 −10 −5. .¸ ., .¸ ., −10 −50 20 10

4.2

Determine A ρ V⃗ ⊗ V⃗ ·⃗ndA for a 1 cm x 1 cm x 1 cm Cartesian element √ if V⃗ = 5 x ˆi m/s and the density is 1000 kg/m3.

Solution: First thing first, let’s determine V⃗ ⊗ V⃗ : √ √ , .5 x. √25x 0 0, √ .0 0 0 . = ⃗ ⊗V ⃗ =. 0 . 5 x 0 0 V ., .¸ .¸ ., 0 0 0 0 Thus: √ , 25000x 0 0 . ⃗ ⃗ ρ V⊗ V = 0 0 0. ., .¸ 0 0 0 Next, instead of using the form given in the problem, we can use the divergence theorem to get: 37

38

Chapter 38 Solutions ρ V⃗ ⊗ V⃗ · ⃗ndA = A

$ ⃗ · ρ V⃗ ⊗ ⃗V dV ∇ V

Performing the divergence gives:

⃗∇ · ρ V⃗ ⊗ V⃗ =

∂ ∂x

∂ ∂y

√ , 25000x 0 0 . . ∂ 0 0 0 = 25000 0 0 ., ∂z ..¸ 0 0 0

# ⃗ Thus, the integral, ∇ · ρ V⃗ ⊗ V⃗ dV becomes (written in column V vector form): $

⃗ dV = ⃗ ρ V⃗ ⊗ V V ∇ ·

=

$

√25000, . . 0 .. dV . ., V ¸ 0

√ , 25000

, 0.01 m , 0.01 m , 0.01 m

. 0 . dxdydz .¸ ., 0 √ , 25000 (0.01) (0.01) (0.01) . . = .¸ 0 ., 0 √0.025, . .

2 = .. 0 . kg m /s ., ¸ 0 x=0

4.3

y=0

z=0

Consider incompressible flow between two parallel plates a distance, H, apart. If the bottom plate moves with a velocity of UB and the top is fixed, find an expression of the velocity profile assuming there is no pressure gradient.

Solution: We will make the following assumptions: steady state, onedimensional flow, no gravity, and no pressure difference. The continuity equation is: ∂u ∂v + =0 ∂x ∂y Setting v = 0 leads us to:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

39

∂u =0 ∂x Next we look the Navier-Stokes equations. We can start with the incompressible Navier-Stokes equations in the x−direction: √ !

2 , ∂/u/ ∂/u/ ∂u/ ∂/ p / ..∂2// u ∂ u. u v/ .,. ρ + + = − + µ .̧ + /∂t /∂x /∂y /∂x /∂ x 2 ∂y2 After our assumptions we are left with:

0=

d2u dy2

This is the governing equation for Couette flow. The general solution to this equation is: u = C1y + C2 where C1 and C2 are constants of integration. In order to find values for C1 and C2, boundary conditions need to be applied. The no-slip boundary condition for u is utilized at the top and bottom plates. Recall that the noslip boundary condition implies that the velocity of the fluid is the same as the velocity of the adjacent solid surface. Thus, in mathematical form, the boundary conditions are: at y = 0, u = UB at y = H, u = 0 Applying the first boundary condition (i.e. at y = 0, u = UB) to our general solution, we get: UB = C10 + C2 ∴ C2 = UB Next apply the second boundary condition (i.e. at y = H, u = 0) to our general solution, i.e.:

40

Chapter 40 Solutions 0 = C1H + UB −UB ∴ C1 = H The solution for the x-velocity profile is now obtained by plugging in the expressions for C1 and C2 into the general solution: u=

4.4

−UB H

y + UB

Find an expression for the shear stress on the bottom and top plates from Problem 4.3.

Solution: We can obtain the shear stress on the top and bottom plates by ⃗ V⃗ + ∇ ⃗ V⃗ † , recognizing that the shear term of the stress tensor is given by µ ∇ i.e.: term in the stress tensor, which is given by: √

2µ∂u ∂x

µ ∂v + ∂u ∂x ∂y

. † ⃗ ⃗ ⃗ ⃗ µ ∇V + ∇V = .µ ∂u + ∂v 2µ∂v ∂y ∂x ∂y . .¸ ∂u ∂w ∂w µ ∂z + ∂x µ ∂v ∂z + ∂y √ 0,. −µ UHB . 0 B 0 0. = .−µ UH . . .¸ ., 0 0 0

, µ ∂w + ∂u ∂x ∂z . . ∂v µ ∂w ∂y + ∂z . . . , 2µ∂w ∂z

Therefore, the shear stress (i.e. the stress vector, τ⃗shear) on the bottom (given by normal, ⃗nb = − ˆjcan be written as: √ .. 0 † ⃗ ⃗V + ⃗∇⃗V ·⃗n = .−µ UB τ⃗shear,bottom = µ ∇ b . H ..̧ 0

0

,√ , 0 . .0 .

0. .−1. = µ

0

.. . ., .¸ ., 0 0

−µ UHB

U B ˆi H

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

41

Likewise, the shear stress at the top is given by: √ .. 0 −µ UHB † ⃗V ⃗ ⃗ + ∇V ⃗ · ⃗n = .−µ 0 UB τ⃗shear,top = µ ∇ t H . ..̧ 0 0 4.5

,√ , 0 . .0 .

0. .1. = −µU B i ˆ .. .

H ., .¸ ., 0 0

Consider incompressible flow between two parallel plates a distance, H = 0.5 cm, apart. The top plate is moving with a velocity to the right of 1 m/s. There is also a constant pressure gradient that is resisting the flow in the x−direction with a magnitude of 2 pascals per meter. The dynamics viscosity of the fluid is 2x10 −5 Pa · s. Determine the velocity at the midpoint of the parallel plates.

Solution: Like before, we will make the following assumptions: incompressible flow, steady state, one-dimensional flow, and no gravity. The continuity equation is (after setting v = 0): ∂u =0 ∂x The incompressible Navier-Stokes equations in the x−direction is: √ !

2 , u ∂p ∂/u/ u ∂/u/ v ∂// ..∂2// u ∂ u. ρ + + = − + µ .̧ 2 + 2 .,. ∂x ∂y /∂t /∂x /∂y /∂ x After our assumptions we are left with: 0=−

∂p ∂x

+µ

d2u dy2

This is the governing equation for Couette flow. The general solution to this equation is: u=

1 ∂p 2 y + C1y + C2 2µ ∂x

where C1 and C2 are constants of integration. In order to find values for C1 and C2, boundary conditions need to be applied. The boundary conditions are:

42

Chapter 42 Solutions at y = 0, u = 0 at y = H, u = UT Applying the first boundary condition (i.e. at y = 0, u = 0) to our general solution, we get: 1 ∂p 2 0 + C10 + C2 2µ ∂x ∴ C2 = 0

0=

Next apply the second boundary condition (i.e. at y = H, u = UT ) to our general solution, i.e.: UT =

1 ∂p

H2 + C1H 2µ ∂x UT 1 ∂p ∴ C1 = − H H 2µ ∂x

Thus the solution for the x-velocity profile is: !

1 ∂p 2 UT 1 ∂p u= y + − H y 2µ ∂x 2µ ∂x H To find the value of the velocity at the midpoint, we plug in y = 2H to get: ! !

1 ∂p H2 1 ∂p 2 1 ∂p 2 UT UT − − H = H umidpoint = + 2 2 8µ ∂x 4µ ∂x 2µ ∂x 4 ∂p = 2 Pa/m, µ = 2x10−5Pa · s, and UB = 1 m/s, we get: Plugging in ∂x

!

UT 1 ∂p 2 1 H2 − umidpoint = H = − m/s 2 8µ ∂x 2 8x10−5 If H = 0.005 meters (or 0.5 centimeters), then: umidpoint = 0.1875 m/s

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems 4.6

43

In your own words, describe the various forms of the Navier-Stokes equations discussed in this book and how they were obtained.

Solution: A quick discussion of this is on the video podcasts.

4.7

Show that the conservation form and the non-conservation form of the Navier-Stokes equations are equivalent. You may use Cartesian coordinates.

Solution: The non-conservation form of the Navier-Stokes equations is: √ , ! ∂ V⃗ ⃗ ⃗ ⃗ ⃗ 2µ ∇ ⃗ · V⃗ + ∇ ⃗·µ∇ ⃗ V⃗ + ∇ ⃗ V⃗ † + ρ⃗g ρ¸ . + V · ∇V, . = − ∇⃗ p − ∇ ∂t 3 The conservation form of the Navier-Stokes equations is: ∂ ρ V⃗ ∂t

! † 2 ⃗ ⃗ ⃗ ⃗ ⃗ + ρ⃗g ⃗ ⃗ · µ∇V ⃗ ⃗ + µ ∇V ⃗ ⃗ + ∇ · ρV ⊗ V = −∇p − ∇ µ∇ · V + ∇ 3

As you can tell, the main difference between the two versions is the left-hand side. Thus, we need to show that: , √ ⃗ ⃗V ⃗ . ∂ ρ V⃗ .. ∂ V⃗ V ⃗ ⃗ ⃗ · ρV ⊗ V ρ ¸. + · ∇ ,. = +∇ ∂t ∂t The easiest way to show this equivalency is by using Cartesian coordinates. For example, recall that ρ V⃗ ⊗ V⃗ equals: √ , ρuu ρuv ρuw . ⃗ ⊗ ⃗V = . ρvu ρvv ρvw . ρV ., ¸. ρwu ρwv ρww Taking the divergence we have:

⃗ · ρ V⃗ ⊗ V⃗ = ∇

∂ ∂x

∂ ∂y

∂ ∂z

This leads to (in column vector form):

√ρuu ρuv ρuw , .ρvu ρvv ρvw .,. . .¸ ρwu ρwv ρww

44

Chapter 44 Solutions √ ∂(ρuu) ∂(ρvu) ∂(ρwu) , . ∂x + ∂y + ∂z . . . ∂(ρvv) ∂(ρwv) ⃗∇ · ρ V⃗ ⊗ V⃗ = . ∂(ρuv) ∂x + ∂y + ∂z . . . .¸ ∂(ρuw) ∂(ρvw) ∂(ρww) ., + ∂y + ∂z ∂x

Adding the time derivative: √ ∂(ρuu) ∂(ρvu) ∂(ρwu) , . ∂x + ∂y + ∂z . . . ∂ ρ V⃗ ⃗ ∂(ρuv) + ∂(ρvv) ∂(ρwv) ∂ρv ⃗ ⃗ + ∇ · ρV ⊗ V = . ∂t . + . ∂x ∂y + ∂z . . ∂t . . . . .¸ ., .¸ . ∂w ∂(ρuw) + ∂(ρvw) + ∂(ρww) , √ , . ∂ρu . . ∂t .

∂t

∂x

∂y

∂z

We can take the result from above (for simplicity, let’s just take the x−direction) and expand it out via product rule: ∂ (ρu) ∂ (ρuu) ∂ (ρvu) ∂ (ρwu) + + + ∂t ∂x ∂y ∂z ∂u ∂u ∂u ∂u ∂ρ ∂ (ρu) ∂ (ρv) ∂ (ρw) + ρv + u + ρw + u =ρ + u + ρu ∂x + u ∂y ∂z ∂t ∂t ∂x ∂y ∂z x__ _ _ _˛ z___ _ __x x__________ ___ _˛ z _ _____________x x__________ ___ _˛ z_ _____________x x_______________˛ z_ x =

∂(ρu) ∂t

=

∂ (ρ u u ) ∂x

=

∂ (ρ v u ) ∂y

=

∂ (ρ w u _______ ) ∂z

! ! ∂u + u ∂u + v ∂u + w ∂u ∂ρ ∂ (ρu) ∂ (ρv) (ρw) + + + +u −−−−−−−→= ρ ∂t ∂t ∂y ∂z ∂t ∂x ∂y ∂z x_______________________________˛ z_______________________________x x_________________________ _________ _˛z_ _ ________ x group terms

n o n - c o n s erva t iv e f o r m

0 = f r om conit n u i y t______

! ∂u ∂u ∂u ∂u =ρ +u +v +w ∂t ∂x ∂y ∂z Thus, the conservation and the non-conservation forms of the equations are equivalent. The same procedure can be done for the other directions.

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

5 Chapter 5 Solutions

Problems 5.1

Evaluate the viscous dissipation term at an (x, y) location of (1, 5) meters with a dynamic viscosity of 10−3 Pa· s if velocity field is given by: V⃗ = 2y cos(2x)ˆi + y2 sin(2x) ˆj m/s.

Solution: The viscous dissipation term is given by: 2 ⃗ ⃗ 2 ⃗ ⃗V + ∇ ⃗V ⃗ ⃗V ⃗ † :∇ − µ∇ ·V +µ ∇ 3 ⃗ V⃗ + ∇ ⃗ V⃗ For an incompressible flow, it is just µ ∇

†

⃗ V⃗ :∇

Let’s first see if the flow is incompressible by calculating the divergence of velocity (with u = 2y cos(2x), v = y2 sin(2x), and w = 0: ⃗ · V⃗ = ∂u + ∂v + ∂w ∇ ∂x ∂y ∂z =

∂ (2y cos(2x))

∂ y2 sin(2x)

+ ∂y ∂x = −4y sin(2x) + 2y sin(2x) = −2y sin(2x) ⃗ · V⃗ Therefore, the −3 2 µ ∇

2

portion is:

2 ⃗ ⃗ 2 2 8 − µ ∇ · V = − µ (−2y sin(2x))2 = − µy2 sin2(2x) 3 3 3 Plugging in the numbers for the problem yields: 45

46

Chapter 46 Solutions 8 8 J − µy2 sin2(2x) = − 10−3 (5)2 (0.909)2 = −0.055 3 3 3 ms

(5.1)

† Next up we need to calculate µ ⃗∇⃗V + ⃗∇V⃗ : ⃗∇⃗V . This is essentially a 2D scenario, so the velocity gradient is:

√ ∂u . ∂x ⃗V ⃗ = ∇ .¸ ∂u

∂y

⃗ V⃗ The ⃗∇V⃗ + ∇

†

∂v ∂x .

! −4y sin(2x) y2 cos(2x) = 2 cos(2x) 2y sin(2x) ∂v ., ∂y

portion turns out to be:

! !

4y sin(2x) y2 cos(2x) −4y sin(2x) 2 cos(2x) ⃗ V⃗ + ∇ ⃗V ⃗ †= – + 2 ∇ 2 cos(2x) 2y sin(2x) y cos(2x) 2y sin(2x)

=

y2 cos(2x) + 2 cos(2x) ! −8y sin(2x) 2 cos(2x) + y2 cos(2x) 4y sin(2x)

⃗ V⃗ † : ⃗∇⃗V : Next we calculate the ⃗∇⃗V + ∇ ⃗V ⃗V ⃗V ⃗ + ∇ ⃗ † :∇ ⃗ ∇

=

y2 cos(2x) + 2 cos(2x) ! −4y sin(2x) y2 cos(2x) −8y sin(2x) : 2 cos(2x) + y2 cos(2x) 4y sin(2x) 2 cos(2x) 2y sin(2x)

!

=64y2 sin2(2x) + y4 cos2(2x) + 2y2 cos2(2x) + 4 cos2(2x) + 2y2 cos2(2x) + 16y2 sin2(2x) = 1322.9 + 108.234 + 8.658 + 0.693 + 8.659 + 330.729 = 1779.87

5.2

Multiplying this result by mu gives 1.77987 and adding to Equation 5.1 leads to the value for the viscous dissipation: 1.72 mJ3 s.

Calculate the power (per volume) if the stress tensor is: √10x 5y 6x , . ⃗ ⃗ T = .5y 2y 25y . . ¸. , 6x 25y 3z

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

47

with a velocity vector is ⃗V = ˆi + 2 ˆj − 2kˆ and no body force.

Solution: The power (per volume) without body forces can be written as: ⃗ · ⃗V · ⃗​⃗ Ẇ = − ∇ T The velocity dotted with the stress tensor leads to: √10x . ⃗ ⃗V · T⃗ = 1 2 −2 . 5y ¸ 6x

5y 6x , 2y 25y. , 25y 3z

= 10x + 10y − 12x 5y + 4y − 50y 6x + 50y − 6z = −2x + 10y −41y 6x + 50y − 6z The power per volume is therefore: ! ∂ Ẇ = – ˆi + ˆj ∂ + kˆ ∂ · (−2x + 10y) ˆi + −41y ˆj + (6x + 50y − 6z) kˆ ∂y ∂z ∂x = − (−2 − 41 − 6) = 49

5.3

watts

m3 Does the follow expression satisfy the Laplace equation?

T = T0 + T1

sin(πx) sinh(πy) sinh(πL)

You may assume T0, T1, and L are constants.

Solution: The Laplace equation is: 2 ∂2T ∂2T ∂2T ⃗ ⃗ ∇ T = ∇ · ∇T = 2 + 2 + 2 = 0 ∂x ∂y ∂z

Taking the derivative of T with respect to x gives: ∂T ∂x

= T1

cos(πx) sinh(πy) π sinh(πL)

48

Chapter 48 Solutions Taking the derivative with respect to x again gives: ∂2T

= −T1 ∂x2

sin(πx) sinh(πy) 2 π sinh(πL)

Taking the derivative of T with respect to y gives (the derivative of hyperbolic sine is hyperbolic cosine): ∂T ∂y

= T1

sin(πx) cosh(πy) π sinh(πL)

The second derivative is (the derivative of hyperbolic cosine is hyperbolic sine): ∂2T

= T1 ∂y2

5.4

sin(πx) sinh(πy) 2 π sinh(πL)

which is the negative of the second derivative with respect to x. Thus, addding them up leads to zero and this solution for temperature is a solution to Laplace’s equation.

Consider a 2D flow whose velocity vector is given by V⃗ = −10xˆi + 10y ˆj m/s and whose temperature is T = 50e−0.03t sin(2x) cosh(5y) kelvin. The W , a density of 1000 kg/m3, and fluid has a thermal conductivity of 0.6 mK a specific heat of 4180 J kgK . What is the value of the heat generation at a time of 10 seconds and (x, y) coordinates of (1, 1) meters?

Solution: Let’s first check to see if the flow in incompressible or not: ! ⃗ ·V ⃗ = ˆi ∂ + ˆj ∂ + kˆ ∂ · −10xˆi + 10y ˆj = −10 + 10 = 0 ∇ ∂z ∂y ∂x Yes, the flow is incompressible. So, we need to use the 2D incompressible energy equation: ! ! ∂T + u ∂T + v ∂T ∂2T ∂2T + ρcp =k + Φ + q˙gen ∂x2 ! ∂y2 ∂t ∂x ∂y 2 √ !2 , ∂u ∂v . ∂u!2 ∂v . . where Φ = µ .¸2 + + +2 . ∂y ∂x ∂y , ∂x With u = −10x and v = 10y, the viscous dissipation becomes:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems !2 √ !2 , ∂u ∂v .. ∂u!2 ∂v . + + Φ = µ .¸2 ∂y ∂x + 2 ∂y ., ∂x = µ 2 (−10)2 + (0 + 0)2 + 2 (10)2

49

(5.2)

= 400µ The x− and y−derivatives of T are: ∂T ∂x

= 100e−0.03t cos(2x) cosh(5y) = 100e−0.3 cos(2) cosh(5)

(5.3)

= 5467 kelvin/m and ∂T ∂y

= 250e−0.03t sin(2x) sinh(5y) = 250e−0.3 sin(2) sinh(5)

(5.4)

= 12496 kelvin/m The second-order derivatives are: ∂2T ∂x2

= −200e −0.03t sin(2x) cosh(5y) = −200e−0.3 sin(2) cosh(5)

(5.5)

= −9997.9 kelvin/m2 and ∂T ∂y

= 1250e−0.03t sin(2x) cosh(5y) = 1250e−0.3 sin(2) cosh(5)

(5.6)

= 62487 kelvin/m2 The time derivative is: ∂T = –0.03 (50) e−0.03t sin(2x) cosh(5y) ∂t = −1.5e−0.3 sin(2) cosh(5) = −75 kelvin/s Putting it all together yields:

(5.7)

50

Chapter 50 Solutions ! ! ∂T + u ∂T + v ∂T ∂2T ∂2T + =k + Φ + q˙gen ∂x2 ∂y2 ∂t ∂x ∂y leads to (1000)(4180) (−75 + (−10)(5467) + (10)(12496)) = 0.6 (−9997.9 + 62487) + 400µ + q˙gen −−−−−→ q˙gen = 2.9e11 − 400µ watts per meter cubed ρcp

5.5

Find the steady state temperature in the middle of a 10 cm bar whose sides in the y− and z−directions can be ignored and the temperature of the right (at x = 10 cm) is fixed at 0 degrees Celsius and the flux on the left is 1 W/m2. The thermal conductivity is 0.6mKW .

Solution: The temperature distribution for this problem can be obtained by simply solving the following equation: ∂2T ∂x2

=0

This is just the equation for diffusion in one dimension at steady state.

Simple enough. The general solution for this problem is just a straight line: T = C1 x + C2 We now need to apply boundary conditions: at x = 0, ⃗q′′ · ⃗n = 1 watt per meter squared at x = 10 cm, T = 0◦C Applying the first boundary condition (at x = 10 cm, T = 0◦C) leads to:

0 = C10.1 + C2

(notice we switched to using meters)

∴ C2 = −0.1C1 Applying the next boundary condition (i.e. at x = 0, ⃗q ′ ′ ·⃗n = 1 watt per meter squared) yields (with ⃗n = −ˆi since it is considered the left face):

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

⃗q′′ · ⃗n = − k

51

! ∂T ˆ ∂T ˆ ∂T ˆ ∂T i+ j + k · −ˆi = −k ∂x ∂y ∂z ∂x

leads to

−−−−→ − kC1 = 1

W m2

◦ 1 1 C ∴C1 = − = − = −1.67 k 0.6 m

The temperature in the middle of the bar (i.e. at x = 5 cm or x = 0.05 m) leads to: !

1 −1 T = − (0.05) + −0.1 = 0.0833◦C 0.6 0.6 5.6

Does the following function for temperature satisfy the heat equation when there is no heat generation (α is thermal diffusivity): ! −2π2α πx πy T = T0 + Tm exp t sin sin L L L2 You may assume L, T0, and Tm are also constant.

Solution: The heat equation is given by (in 2D): ∂T ∂2T ∂2T =α + 2 ∂x2 ∂y ∂t

!

The time derivative of the given temperature function is: ∂T ∂t

= Tm

! ! −2π2α πx πy −2π2α t exp sin sin 2 L L L L2

The second derivative of T with respect to x is: ∂2T ∂x2

= Tm

−π2 L2

!

! −2π2α πx πy t exp sin sin L2 L L

The second derivative of T with respect to y is: ∂2T ∂y2

= Tm

! ! −2π2α πx πy −π2 t exp sin sin 2 L L L 2 L

52

Chapter 52 Solutions Adding the second derivative with respect to x with the second derivative with respect to y and multiplying with α gives: ! ! ! ! −π2 πy −2π2α ∂2T ∂2T πx + exp sin t sin = α 2Tm α ∂x2 ∂y2 L2 L L L2

5.7

Upon inspection, the above equation is equal to the time derivative.

Hence, this equation satisfies the heat equation.

Consider a system of water with a pinch of salt centered in the middle. If we were to model the “stirring” of the system of water using the advection equation, what would the result be? What would we need to do in order to more accurately model the physical reality of the situation?

Solution: If the pinch of salt was very small so that the velocity at the different parts of the salt would be essentially the same, then the salt would stay relatively the same shape and size and just move with the stirring of the water. If the pinch of salt was somewhat big to begin with, then the various areas of the salt pinch might experience differing velocities, and hence cause the salt to get stretched. Either way, the salt will just move with the velocity vectors of the flow but will not spread out and diffuse into the water like you would expect. In order to more accurately model the situation, a diffusion term would need to be added. In other words, the convection-diffusion equation should be used.

5.8

Consider incompressible flow between two parallel plates. The flow is driven by the bottom plate moving to the right with a velocity of 100 m/s. The viscosity of the fluid is given by 0.001 Pa ·s and the thermal conductivity is 0.6 WmK. If both the bottom and the top temperatures are fixed at 0◦C, what is the maximum temperature of the flow? You may assume the flow is in a steady-state and is only in the x−direction. In addition, you may ignore any pressure gradient as well as any heat generation. How is the maximum temperature of the flow different than the plate temperatures? What accounts for the difference?

Solution: This problem starts out like the other cases between two plates.

In particular, the continuity equation simplifies to: du dx

=0

And the Navier-Stokes equations (in the x−direction) simplifies to:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

53

d2u

=0 dy2 The general solution of the velocity profile is: u = C1y + C2 We now need to apply the following velocity boundary conditions: at y = 0, u = UB at y = H, u = 0 Applying the first boundary condition leads to: UB = C10 + C2 ∴ C2 = UB The next velocity boundary condition is: 0 = C1H + UB −UB ∴ C1 = H Thus, the velocity profile is: u=

−UB

y + UB H Next, we need to find the temperature profile. The temperature profile is obtained from the energy equation. Making the assumption that the flow is only in the x−direction, steady state, the derivative of temperature with respect to x is zero, there is no heat generation, and an incompressible flow, we have: ! du 2 0=k +µ dy dy2 d2T

B Solving for the general solution for temperature, withdydu = −U H , we get:

T =−

µ UB 2 2 y + C 3y + C 4 k H

54

Chapter 54 Solutions We now need to apply the following velocity boundary conditions: at y = 0, T = 0 at y = H, T = 0 Applying the first boundary condition leads to: 0

µ UB 2 =−

k ∴ C4 = 0

02

H

+

C 0 3

+

C 4

The next temperature boundary condition is: µ UB 2 2 H + C 3H k H µ UB 2 ∴ C3 = H k H The temperature profile is thus: 0=−

µ UB 2 2 µ UB 2 y + Hy k H k H The maximum temperature of the flow is located at the point where the derivative of the temperature with y is zero, thus: T =−

dT µ UB 2 µ UB 2 H=0 y+ = −2 k H dy k H leads to H y = −−−−→ 2 Thus, the maximum occurs (as expected) at y = H/2. Plugging in y = H/2 into the temperature equation leads to: T

max = −

Plugging in values gives: Tmax =

µ

4k

µ UB 2 k

2

U2 = B

+

µ 2k

U2

µ

B = 4k

U2

B

0.001

1002 = 4.16◦C 4(0.6)

The maximum temperature is greater than the sides of the wall because viscous dissipation (i.e. frictional heating due to shearing) has increased the flow temperature.

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems 5.9

55

In your own words, what is the definition of a boundary layer?

Solution: There is a short discussion on boundary layers in the video podcasts.

6 Chapter 6 Solutions

Problems 6.1

A metal block, whose dimensions in the x−, y−, and z−directions are respectively L x H x W, is in a steady state with each of its sides all held fixed at various temperatures. If L is much smaller than both H and W, show, by scaling, that the governing equation for this problem is simply the conduction term in the x−direction is equal to zero.

Solution: The first step in this problem is to simplify and write down the governing equation. This problem deals with steady state conduction, thus the governing equation is: ∂2T

∂2T ∂2T + + 2 =0 ∂x2 ∂y2 ∂z

The second step is to introduce scaled variables into the equation. The scaling we will use is the following: θ∗ =

T − Tshi f t

,

Ts

x y z x∗ = , y ∗= , z∗ = L H W

where the * superscript indicates the non-dimensional version of the variable in question, Tshi f t and Ts are general shift and scale factors, respectively, for the temperature.

We can introduce this scaling into the governing equation to get: Ts ∂2θ∗ L2 ∂x∗2

+

Ts ∂2θ∗ H2 ∂y∗2

+

Ts ∂2θ∗ W2 ∂z∗2

=0

Step 3 would just be to divide out by a dimensional coefficient in front 56

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

57

of one of the terms. Typically, the largest term is the one that is advised to be divided out. In this case, since L is less than both H and W, we can divide out by 2Ts to get: L

∂2θ∗ L2 ∂2θ∗ L2 ∂2θ∗ ∂x∗2 + H2 ∂y∗2 + W2 ∂z∗2 = 0 We now have a non-dimensional equation.

In the2fourth step we can simplify this equation by recognizing that L2 and L is really small and can thus be ignored, i.e.: H2

W2

∂2θ∗ =0 ∂x∗2

6.2

This is the simplified scaled equation, which is just the conduction term in x equals zero.

Step 1: Simplify the problem and equations (and boundary/initial conditions) as much as possible.

Step 2: Introduce scaled (i.e. non-dimensional) variables and plug them into the simplified equations and conditions.

Step 3: Divide the whole equation by a dimensional “coefficient.” This dimensional coefficient is usually determined to be a coefficient in front of one of the terms in the equation. If there are multiple terms in the equation and each one has a coefficient in front, it is usually advised to divide by the biggest coefficient.

Step 4: Make any additional simplifications with the new scaled variables if necessary.

Step 5: Set any characteristic scale factors not previously defined such that the coefficient terms are of order one (this is the hardest part and will be explained as we go). Note, this may not show up in all problems.

Step 6: Solve the resulting equation if applicable.

Scale the one dimensional heat equation with the heat generation term included. Write the non-dimensional heat equation in terms of Fourier number (Fo), a non-dimensional time parameter defined as: αt Fo = L2 where α is the thermal diffusivity and L is the length scale.

Solution: The one-dimensional heat equation is (first step):

58

Chapter 6 Solutions ∂2T q˙gen =α 2 + ∂t ∂x ρcp

∂T

Introducing scaled variables (second step): θ∗ =

T − Tshi f t Ts

,

x t z x∗ = , t ∗= , z∗ = L ts W

where Tscale and tscale and the temperature and time characteristic scales, respectively. Their values (or expression) has not yet been defined and may change depending on the specifics of the problem. In addition, we are not going to scale the density, specific heat, and heat generation term as we will just assume those to be a constant. With the scaled variables, the heat equation becomes: 2 ∗ q˙ Ts ∂θ∗ = α Ts ∂ θ + gen L2 ∂x∗2 ts ∂t∗ ρcp

We now need to divide out by a dimensional coefficient (third step).

The term we need to divide out by is not as obvious because Ts and ts have still not been defined. At this point, we will just pick a term to divide out by and so we will just divide out by thet s Ts : ts q˙gen ts ∂2θ∗ ∂θ∗ + ∗ = α ∂t L2 ∂x∗2 Ts ρcp We have a scaled equation. Now the question is can we make any simplifications (step 4) and/or define any characteristics scale not previously defined (step 5)? The answer is yes. For one thing, we have not defined a characteristic time scale. To do so, we will set the coefficient in front of the diffusion term to be one, thus: ts

gives us

L2

α 2 = 1 −−−−−→ ts = L α We now have a scale factor for time.

Introducing this scale factor for time into our scaled equation we get: ∂θ∗ = ∂2θ∗ + L2 q˙gen ∂x∗2 ∂t∗ αTs ρcp We can simplify further by setting the last term to one such that:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

59

L2 q˙gen L2q˙gen gives us = 1 −−−−−→ Ts = αTs ρcp αρcp Introducing the temperature scale into the scaled equation gives us: ∂θ∗ ∂t∗

=

∂2θ∗ +1 ∂x∗2

We have one more thing to do. We still need to put this equation in terms of an often used non-dimensional parameter, the Fourier number.

If we plug the time scale, ts, expresssion that we obtained earlier into the scaling the time, notice that we get: t∗ =

t αt t = L2 = 2 ts L α

Comparing non-dimensional time, ts, to the Fourier number, we notice that they are the same thing. In fact, the Fourier number is nothing but a non-dimensional time parameter for conduction problems. So, we can replace the non-dimensional time with Fourier number (Fo) in our final expression to get: ∂θ∗ ∂2θ∗ = ∂Fo ∂x∗2 + 1

6.3

Note, depending on the problem, it may have been more fruitful to use different scale factors for either temperature or time. However, with such little information, the scaling done here was the easiest.

Consider a conduction problem similar to the one given in Figure 5.5, except the boundary condition of the right side (at x = L) is no longer held at a fixed temperature and is instead in contact with a fluid at temperature T∞. The boundary condition on the right side is given by a convection boundary condition, i.e.: ⃗q′′ · ⃗n = h (T − T∞) where h is a parameter known as the heat transfer coefficient. Obtain an expression for the non-dimensional steady state temperature in terms of a non-dimensional parameter known as the Biot number (Bi), given by: Bi = hLk .

60

Chapter 6 Solutions Solution: The first thing to do is to write down the simplified governing equation and boundary conditions: d2T dx2

=0

(6.1)

Notice the boundary condition on the right side (i.e. at x = L) contains a heat flux. The heat flux vector dotted with the normal on the right side (i.e. ⃗n = ˆi becomes:

⃗q′′ · ⃗n = −k

! ∂T ∂T ∂T ˆ ∂T + + · i = −k ∂x ∂y ∂z ∂x

Thus, the boundary conditions become (the partial is replaced by an ordinary derivative):

at x = 0, T = Tle f t dT at x = L, – k = h (T − T∞ ) dx

(6.2)

We now want to introduced scaled variables for temperature and x (step 2): x T − T∞ x∗ = , θ∗ = L Ts Notice that we are shifting by T∞ instead of Tle f t. We could have shifted

by Tle f t, however, as you will see, shifting by T∞ will make the right boundary condition a little easier to handle. Shifting by Tle f t would make the left boundary condition a little easier. So, you will need to make a decision. In addition, the characteristic scale factor for temperature, Ts, will be determined later.

Introducing the scaling into the governing equation yields: Ts d2θ∗ L2 dx∗2

=0

Introducing the scaling into boundary conditions yields:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

61

at Lx∗ = 0, Tsθ∗ + T∞ = Tle f t x˛ zx x______˛ z______x x

T

√ , Ts dθ∗ ∗ = L, −k at x˛Lxzx .Tsθ∗ + Tz∞x−T∞.,.

L dx∗ = h .¸x______˛ x_____ ˛ zx x =T dT −k dx

The governing equation and boundary conditions still are not nondimensional. To do that we need to divide out by a dimensional coefficient (step 3). For the governing equation, this is easy, we just divide out

by T2s to get: L

d2θ∗ dx∗2

=0

Likewise, the boundary conditions are also not yet non-dimensional.

We can first consider the boundary condition at x = 0 (or Lx∗ = 0): at Lx∗ = 0, Tsθ∗ + T∞ = Tle f t x__˛z_ _x x___ _____ _______˛ z _________ _ _____x divide

by L subt ra c t by

→

x∗ = 0, θ∗ =

T ∞ a nd d i iv d e b y Ts

Tle f t − T∞

Ts x_______˛z_______x =1

If we set the boundary condition on the right to 1, we get an expression now for the characteristic scale scale, namely: Ts = Tle f t − T∞ Scaling the next boundary condition looks like this :

d ivi bydLe

→x∗ = 1,

Ts dθ∗

−k

at Lx∗ = L, __x_˛ z___x

= h (Tsθ∗ + T − T ) ∞

L dx∗

∞

______ ___ ____˛z __ __ ____ ___ ____ ___ ____ _x x _________ divide out by −k Ts and subtra c t L o u t t h e

dθ∗ dx∗

=−

hL

θ∗ k x˛ z x =B i

Thus the scaled boundary conditions are:

T

∞

o n t h e right

62

Chapter 6 Solutions

at x∗ = 0, θ∗ = 1 dθ∗ at x∗ = 1, = –Biθ∗ dx∗ Notice that we introduced a new parameter, the Biot number (Bi).

Solving our scaled governing equation (step 6) leads to the general solution of: θ∗ = C1 x∗ + C2 Applying our first scaled boundary condition (i.e. at x = 0, θ∗ = 1) to our scaled general solution leads to: 1 = C10 + C2 ∴ C2 = 1 Applying the next boundary condition gives us: dθ∗ dx

∗

= −Biθ∗ x∗=1

∗

x =1 √ , →C1 = −Bi .C11 + x˛1zx,. ¸ =C2 Bi ∴ C1 = – 1 + Bi

So our final expression for the non-dimensional temperature is: −Bi ∗ x +1 1 + Bi The book provided the boundary layer equations in scaled form. Rescale the boundary layer equations in dimensional form. θ∗ =

6.4

Solution: The scaled boundary layer equations are: Continuity for an incompressible flow past a flat plate: ∂u∗

∂v∗ + =0 ∂x∗ ∂y∗

x−momentum for a steady incompressible flow past a flat plate (with a constant free stream velocity):

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

u∗

63

∂u∗

∗ ∂2u∗ ∗ ∂u = + v ∂x∗ ∂y∗ ∂y∗2

y−momentum for a steady incompressible flow past a flat plate:

∂p∗ ∂y∗

=0

To rescale, all that we have to do is reintroduce the expressions for the scaled variables into the equations:

u∗ =

u

v

, v∗ =

U∞

,

p∗ =

p

,

x∗ =

ps

Vs

x

, y∗ =

X

where:

Vs =

δU∞

, ,

ps = ρU2 ,

X

δ=

∞

Thus, the continuity equation becomes:

√ v, ! u ∂ ∂ ..¸δU∞ .., U∞ X + =0 y x ∂ ∂ X δ ∂u ∂v → + =0 ∂x ∂y The Navier-Stokes in x−becomes:

µX ρU∞

y δ

64

Chapter 6 Solutions ! ! ! u u 2 u ∂ ∂ ∂ U∞ U∞ U∞ u v + = y 2 U∞ ∂ x δU∞ ∂ y ∂ X δ X δ divide out by ∞X2

−−−−−−−−−−U−→ u ∂u + v ∂u = U∞δ2 ∂ u2 ∂x ∂y X ∂y 2

µX

∂u ∂u U∞ ∂2u ρU∞ −−−−−−−→ u + v = X ∂y2 ∂x ∂y

introduce δ

∂u −−−−−−−→ u

with ν=µ/ρ

∂u ∂2u +v = ν 2 ∂x ∂y ∂y

The momentum equation in y− is simply: ∂

p ps

∂

y δ

→ 6.5

!

∂p ∂y

=0

=0

Scale Cauchy’s momentum equation.

Solution: The Cauchy momentum equation (i.e. Cauchy’s first law) is: √ , ∂ V⃗ ⃗ ⃗ ⃗ ⃗ · T⃗​⃗ + ρ ..¸ + V · ∇V, . = ∇ ∂t Introduce the following ρ⃗g scaled variables (we will assume density is a constant): ⃗ V∗ =

⃗ V U∞

,

t∗ =

⃗ ∇ ⃗∗ , ∇ = , ts 1/L t

⃗ T ⃗​⃗ ⃗g T ∗ = σ , ⃗g∗ = g s

0

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

65

where U∞ is a velocity scale (we will just assume is known), g0 is a gravity scale, ts is a time scale to be determined, and σs is a stress scale to be determined. Plugging this scaling into Cauchy’s equation leads to: √ , ⃗∗ σs ⃗ ⃗​⃗ ⃗∗ g g∗ .. U∞ ∂ V U2∞ V⃗ ∗ ⃗ ∗ V T ∗ + ρ .¸ .,. = ∇ · + ρ 0⃗ · ∇ ∗ ts ∂t L L We can pull out the to get:

U2 L

∞

term from the parentheses on the left-hand side

√ , U2 . U∞ ∂ V⃗ ∗ ⃗ ∗ ⃗ ∗ V σs ⃗∗ ⃗​⃗ ⃗ g g∗ T ∗ + V · ∇ ∗.,. = ρ ∞ .¸ + ρ 0⃗ ∇ · L tsL ∂t∗ L U2

We can divide out by ρ L∞ to get: U∞ ∂ V⃗ ∗ tsL ∂t∗

⃗ ∗ V⃗ ∗ = + V⃗ ∗ · ∇

σs ⃗ ∗ ⃗ ⃗ ∗ g0 L ∗ ∇ ·T + ⃗g 2 ρU ∞2 U∞

We can now set the first term on the left to be one (making ts = L ) U∞ and the first term on the right to be one (making σ = ρU2 s ∞). In addition, the gravity term coefficient ( gU02L) is just the inverse of the Froude number ∞ squared. Thus, the scaled Cau chy equation is: ∂ V⃗ ∗ ⃗ ∗ ⃗ ∗ ⃗ ⃗ ∗ ⃗ ⃗ ∗ 1 ∗ + V · ∇ V = ∇ · T + ⃗g ∂t∗ Fr2 Note that there are other ways of generally scaling Cauchy’s equation and the exact scaling will depend on the problem at hand. For instance, some may scale the stress tensor with a general known pressure scale, p0, and also include a scaling for density of ρ∗ = ρρ , where ρ0 is a known 0 density scale, leading to a Cauchy first law equation of: ∂ V⃗ ∗ ⃗ ∗ ⃗ ∗ ⃗ p0 1 ⃗ ∗ ⃗ ⃗ ∗ 1 ⃗g∗ +V ·∇ V = ∇ ·T + ∂t∗ Fr2 ρ∗ ρ0U2∞ ρ∗ The

p0

2

ρ0U∞

is often given the name, Euler number, Eu, thus: ∂ V⃗ ∗

Eu 1 ⃗g∗ ⃗​⃗ + V⃗ ∗ · ⃗∇∗ V⃗ = ⃗∇∗ · T ∗ + ∂t∗ ρ∗ Fr2 ρ∗

66 6.6

Chapter 6 Solutions Consider Couette flow between two plates a distance of 2 millimeters apart whose top plate moves with a velocity of 1 m/s and the bottom plate is held fixed. The temperature of both plates are held fixed at zero W and the dynamic degrees Celsius. The thermal conductivity is 0.6 mK −3 viscosity is 10 Pa · s. Find a value for the non-dimensional temperature in the middle in the flow. Unscale the result and obtain a value for the temperature in the middle of the flow in Celsius.

Solution: The first step is to write down the x−momentum and energy equations for this problem, which was obtained in Chapter 5: d2u

=0 dy2 !2 d2T µ du =− dy2 k dy

(x−momentum) (energy)

There is no y−momentum equation for this problem because we are going to assume only a one-dimensional flow.

The boundary conditions are: at y = 0, u = 0 and T = TB (0◦C) at y = H (2 mm), u = UT (1 m/s) and T = TT (0◦C) For the next step, we can introduce non-dimensional variables: y∗ =

y

, u∗ =

H

u

, θ∗ =

UT

T − TB Ts

Notice that we are not defined the temperature scale as TT − TB. The reason for not doing so is that, for this particular problem, it would result in dividing by zero. Plugging these scaled variables into our simplified governing equations give: UT ∂2u∗ H2 ∂y∗2

=0

(x−momentum) ∂u∗ !2

2 Ts ∂2θ∗ = −µU T H2 ∂y∗2 kH2 ∂y∗

(energy)

When we plug the scaled variables into the boundary conditions we get:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

67

at Hy∗ = 0, UT u∗ = 0 and Tsθ∗ + TB = TB at Hy∗ = H, UT u∗ = UT and Tsθ∗ + TB = TT Our third step is to properly scale the equations by dividing out by a coefficient in front of one of the terms. In the case of the x–momentum equation, the only coefficient is the U2T . In the case of the temperature H equation, there are two terms. The term on the left (the diffusion term) has a coefficient of Ts 2 in front of it and the term on the right (the viscous 2 H dissipation term) has the coefficient µUT . We will divide out by Ts . Thus, kH2

H2

Ts dividing out the x−momentum equation by UHT2 and the energy by H2 yields the following result for the scaled governing equations:

d2u∗ =0 dy∗2 ! d2θ∗ = −µU2T du∗ 2 dy∗2 kTs dy∗

(scaled x−momentum) (scaled energy)

The boundary conditions from Equation also need to be scaled by dividing out by a coefficient. Since the temperatures at both the top and the bottom are zero, the scaled temperature at the top and bottom will also be zero. The scaled velocity at the top will be 1 and the scaled velocity at the bottom will be zero. Thus: at y∗ = 0, u∗ = 0 and θ∗ = 0 at y∗ = 1, u∗ = 1 and θ∗ = 0 Note now that we can find an expression for Ts (step 5) by setting the coefficient in front of the viscous dissipation term on the right hand side of the energy equation to be zero, i.e.: µU2 T

kTs

implies

= 1 −−−−→ Ts =

µU2 k

T

Thus, our new energy equation is: ! d2θ∗ du∗ 2 =− dy∗ dy∗2 So, the final scaled equations and boundary conditions we are to solve are:

68

Chapter 6 Solutions

∂2u∗ =0 ∂y∗2 ! ∂2θ∗ ∂u∗ 2 =− ∂y∗ ∂y∗2

(scaled x−momentum) (scaled energy)

With boundary conditions: at y∗ = 0 : u∗ = 0 and θ∗ = 0 at y∗ = 1 : u∗ = 1 and θ∗ = 0 We can now solve this problem (step 6). The x−momentum has a general solution that is linear. That is: u∗ = C1y∗ + C2 where C1 and C2 are the unknowns. Applying the boundary conditions for u∗ leads to: at y∗ = 0, u∗ = 0 → 0 = C10 + C2 ∴ C2 = 0 at y∗ = 1, u∗ = 1 → 1 = C11 + 0 ∴ C1 = 1 So the final x−velocity profile is just a line given by: u∗ = y∗ Next up is the energy equation. The energy equation requires the derivative of the x−velocity with respect to y, which is nothing but 1. Thus, the general solution to the energy equation is: 1 ∗2 du∗!2 solve d2θ∗ ∗ y + C3y∗ + C4 = − −−−→ θ = − ∗ dy∗2 2 dy x_˛ z _x =1

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Problems

69

Applying the boundary conditions for the energy equation leads to: 1 at y∗ = 0, θ∗ = 0 → 0 = − 02 + C30 + C4 2 ∴ C4 = 0 1 at y∗ = 1, θ∗ = 0 → 0 = − 12 + C31 2 1 ∴ C3 = 2 So, our final energy equation is (with constants C3 and C4 included): 1 1 θ∗ = − y∗2 + y∗ 2 2 The non-dimensional temperature in the middle of the flow is obtained by setting y∗ = 1/2 to get: 1 1 θm∗ idpoint = − 2 2

!2

+

11 1 = 22 8

The unscaled temperature in the middle of the flow is obtained simply by multiplying by the temperature scale, i.e.:

6.7

2 −3 2 Tmidpoint = θm∗ idpointTs = 1 µU T = 1 (10 )(1) = 0.0002◦C 8 k 8 0.6 Consider flow past a flat plate. At what point downstream from the leading edge does the boundary layer double in size compared to the size of the boundary layer at x = 1 meter? You may assume the Reynolds number never reaches 105, which is the transition to turbulence.

Solution: The boundary layer thickness is given by: X δBL = 5 √ Rex Which just equals: X

δBL = 5 , U X = 5 ∞

ν

,

νX U∞

Thus the expression for the boundary layer thickness when x = X = 1 m is given by:

70

Chapter 6 Solutions , δBL = 5

ν U∞

The distance downstream when the boundary layer thickness is twice as thick can be obtained by: , , ν ! νX = 2 5 5 U∞ U ∞

Solving for X yields: X = 4 meters

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

1 A Student’s Guide to the Navier-Stokes Equations: A Supplement on an Alternative Calculation for the Velocity Gradient in Cartesian Coordinates

Let’s write out the velocity gradient in the following manner: ! ∂ ˆ ∂ ∂ ˆ ⃗ ⃗ ˆ ˆ ∇V = i + j + k u i + v ˆj + wkˆ ∂x ∂y ∂z ∂ uˆi ∂ uˆi ∂ uˆi = ˆi + ˆj + kˆ ∂x ∂y ∂z ˆ ˆ ∂ vj ∂ vj ∂ v ˆj + ˆi + ˆj + kˆ ∂x ∂y ∂z ˆ ˆ ∂ wk ∂ wk ∂ wkˆ + ˆi + jˆ + kˆ ∂x ∂y ∂z

(1.1)

What have we done here? What we have done is “foiled” the derivatives from the del operation so that every derivative is applied to every component of velocity. Notice that the derivatives technically are also applied to the base vectors. For Cartesian coordinates, the base vectors are not affected by the derivatives. Thus, you can pull them out of the derivatives, leaving: ⃗V ⃗ = ˆiˆi∂u + ˆjˆi∂u + kˆˆi∂u ∇ ∂x ∂y ∂z ∂v ∂v ∂v + ˆiˆj + ˆjˆj + kˆ ˆj ∂x ∂y ∂z ∂w ˆ ˆ ∂w ˆ ˆ ∂w ˆ ˆ + ik + jk +kk ∂x ∂y ∂z

(1.2)

Note that the base vectors are not “dotted” with each other. Instead, they appear to be multiplied by each other in some fashion. The base vectors are multiplied by what is called an outer product, sometimes called a dyadic prod1

2

Velocity Gradient

uct or tensor product, and is often given by the symbol ⊗ in between the two vectors. From a matrix algebra perspective, the outer product is similar to the inner product (i.e. dot product) except now the left vector is a column vector and the right vector is the transpose of a column vector (i.e. a row vector).

Thus: 1 1 0 0 ˆˆ ˆ ˆ ˆˆ ii = i ⊗ i = ii† = 0 1 0 0 = 0 0 0 0 0 0 0 1 0 1 0 ˆˆ ˆ ˆ ˆˆ i j = i ⊗ j = i j† = 0 0 1 0 = 0 0 0 0 0 0 0 1 0 0 1 ˆˆ ˆ ˆ ˆˆ ik = i ⊗ k = ik† = 0 0 0 1 = 0 0 0 0 0 0 0 and so on...

Note that the base vectors (ˆi for x, ˆj for y, and kˆ for z) are represented in column vector form.

Using the outer product, the velocity gradient given in Equation 1.2 can be expanded out via: 1 0 0 0 0 0 0 0 0 ∂u ∂u ∂u ∇⃗ V⃗ = 0 0 0 + 1 0 0 + 0 0 0 0 0 0 ∂x 0 0 0 ∂y 1 0 0 ∂z 0 0 0 0 0 0 0 1 0 ∂v ∂v ∂v + 0 0 0 + 0 1 0 + 0 0 0 0 0 0 ∂x 0 0 0 ∂y 0 1 0 ∂z 0 0 0 0 0 0 0 0 1 ∂w ∂w ∂w + 0 0 0 + 0 0 1 + 0 0 0 ∂x ∂y 0 0 0 0 0 0 0 0 1 ∂z Leaving us with the following expression for the velocity gradient (which is what we obtained in the textbook): ∂u ∂x

⃗​⃗ = ∇V

∂v ∂x

∂w ∂x

∂u

∂v

∂w

∂y

∂y

∂y

∂u ∂z

∂v ∂z

∂w ∂z

(1.3)

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

1 A Student’s Guide to the Navier-Stokes Equations: A Supplement on the Bernoulli Equation

The Bernoulli equation is another important equation in fluid mechanics. It actually predates the Navier-Stokes equations by about a hundred years. It is very useful for engineering purposes but it does require some fairly significant assumptions. The most significant assumptions are that the flow is inviscid as well as something called irrotational.

There are a number of ways to derive Bernoulli’s equation. It can be derived from energy conservation principles or from Newton’s second law. The energy conservation approach is usually the easiest to grasp and understand. In particular, the mechanical energy equation, Equation 5.48 from the textbook, is the most useful starting point: ⃗ ⃗ · DV = V ρV ⃗ · T⃗​⃗ + ρ V⃗ · ⃗g ⃗·∇ Dt We can write the left hand side as (from Equation 5.46): ⃗ ·V ⃗ ρD V ⃗ ·⃗​⃗ T + ρV⃗· ⃗g 2________ D_t_______ = V⃗· ∇ x ˛z x =ρ V⃗ ·DtDV⃗

We can write out the material derivative in the above equation in an Eulerian description to get: ρ

⃗ ·V ⃗ ∂V

2

∂t

⃗ ⃗ ⃗ ⃗ +V·∇ V·V

⃗ ⃗ ⃗​⃗ ⃗ = V · ∇ · T + ρV · ⃗g

At steady state the time derivative goes away to give us: 1

2

Bernoulli Equation

2

ρ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ V · ∇ V · V = V · ∇ · T⃗​⃗ + ρ V⃗ · ⃗g

⃗ · to get: We can move every term to the left hand side and factor out the V ⃗ · T⃗​⃗ − ρ⃗g = 0 ⃗V · ρ ∇ ⃗V · ⃗V − ∇ 2 ⃗ = 0 (which is the The expression above holds under two circumstances: if V ⃗ · T⃗​⃗ − ρ⃗g = 0. The interesting case is when: ⃗ ·V ⃗ −∇ trivial case) or if ρ ∇ V 2

ρ ⃗ ⃗ ⃗ ∇ V · V − ∇ · T⃗​⃗ − ρ⃗g = 0 2

(1.1)

The stress tensor for an inviscid flow (where µ = 0) just contains the pressure term, thus: ⃗ T⃗​⃗ = − p I

(1.2)

⃗​⃗ where I is the identity matrix. Plugging Equation 1.2 into Equation 1.1 leads to: ρ 2

⃗ · − p ⃗I −ρ⃗g = 0 ⃗ ·V ⃗ −∇ ∇V x_____˛ z_____x ⃗p =− ∇

⃗ · − p ⃗I term becomes a gradient of pressure.

Recall from the textbook that the ∇ Now we have: ρ 2

⃗ p − ρ⃗g = 0 ⃗ ·V ⃗ +∇ ∇V

For a constant density flow (incompressible), we can sneak the ρ2 in the first term into the gradient to get: ∇ρ

⃗V · V ⃗ 2

⃗ − ρ⃗g = 0 + ∇p

(1.3)

The gravity term, i.e. the ρ⃗g term, is usually considered just a constant value.

In addition, the gravity is usually assumed to be in either the z−direction or the y−direction. As such, we can write the gravity term in the following manner (with z being the coordinate direction of gravity):

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Bernoulli Equation

3

⃗ (−ρgzz) ρ⃗g = ∇

(1.4)

where gz is the magnitude of the gravitational acceleration in the z−direction and z is the coordinate direction pointing in the opposite direction as gravity (i.e. pointing up instead of down, hence the negative sign). Writing the force as a gradient of some function is indicative of what is called a conservative force and is a requirement for the Bernoulli equation. To see how why this relationship works out, just perform the gradient operation from the right hand side of Equation 1.4: ! ⃗ (−ρg z) = ˆi ∂ + ˆj ∂ + kˆ ∂ (−ρg z) ∇ z z ∂y ∂x ∂z ∂ (−ρgzz) ˆ ∂ (−ρgzz) ˆ ∂ (−ρgzz) + j +k ∂x ∂y ∂z = 0ˆi + 0 ˆj − ρgzkˆ = ρ⃗g = ˆi

!

Note that −ρgz is considered a constant and the ⃗g is defined as ⃗g = −gzkˆ.

Plugging Equation 1.4 into Equation 1.3 leads to: ∇ρ

⃗ ·V ⃗ V 2

⃗ +∇ ⃗ (ρg z) = 0 + ∇p z

(1.5)

Notice every term in Equation 1.5 is a gradient term. We can factor out the gradient to get: ∇ ρ

⃗ ·V ⃗ V 2

+ p + ρg zz = 0

Since the gradient is equal to zero, the term inside the gradient must be a constant, thus we now have the Bernoulli equation:

ρ

2

+ p + ρgzz =

(1.6)

Equation 1.6 is the most common form of the Bernoulli equation (and the original version). It assumes an inviscid, incompressible flow with a conservative body force. In addition, although this is not obvious, it also assumes an irrotational flow. An irrotational flow is a flow that has zero vorticity, ω⃗.

Vorticity is defined as the curl of velocity, written as:

4

Bernoulli Equation ⃗ ×V ⃗ ω⃗ = ∇

(1.7)

To see how the Bernoulli equation also assumes an irrotational flow, consider Cauchy’s first law: ⃗ DV ⃗ · T⃗​⃗ + ρ⃗g =∇ Dt We know from earlier that the right hand side can be written as the negative of the gradient of pressure and the gradient of −ρgzz, thus: ρ

⃗ DV

⃗ p+∇ ⃗ (−ρgzz) = −∇ Dt Writing the above equation in non-conservative form by expanding out the material derivative and assuming a steady state, we get: ρ

0 ⃗​⃗ ⃗ ⃗ ⃗ /‘ ∂V ρ / + V⃗ · ∇V = −∇p + ∇ (−ρgzz) /∂t

(1.8)

⃗V ⃗ ) can be written using the following vector idenThe advective term ( V⃗ · ∇ tity: Lamb vector ______________ ______________ z xx ˛ ⃗V ⃗ V ⃗ ·∇ ⃗ = 1∇ ⃗ ·V ⃗ −V ⃗ × ∇⃗ × V ⃗ V 2 x___˛z___x

(1.9)

=0 if irrotational

The last term on the right is called the Lamb vector, named after Horace Lamb. The curl of velocity is contained in the last term, if the case of irrota⃗ ×V ⃗ = 0. Plugging Equation 1.9 into Equation 1.8 gives us: tional flow, ∇ ρ⃗ ⃗ ⃗ ⃗ (−ρgzz) ∇ V · V = − ∇⃗ p + ∇ 2 This is essentially the same expression as we had earlier. We can move all of the terms to the left-hand side and factor out the gradient to get: ⃗ ∇

V⃗ · V⃗ ρ + p + ρg z z = 0 2

Thus leading to, again, the Bernoulli equation:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Bernoulli Equation

ρ

2

+ p + ρgzz =

5

(1.10)

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

1 A Student’s Guide to the Navier-Stokes Equations: A Supplement on the Use of the Del Operator in Non-Cartesian Coordinates

The book used mostly used Cartesian coordinates when discussing the equations. In doing so, we performed the various operations using a “Cartesianfriendly” approach. Such an approach is fine to get your feet wet but the vector operations get somewhat more involved when non-Cartesian coordinates are used. This document serves as an introduction for applying some of the del operations we learned about in the book to a non-Cartesian coordinate system.

Seeing these operations in detail may help you later on as you continue in your studies of fluid mechanics. One reason for this is that many times the fluid mechanics equations are written in what is called tensor notation. Tensor notation is a nice, succinct way to deal with the operations using the del operation (as well as other vector and tensor operations) without having to perform lines and lines of computations.

To start understanding how the operations can sometimes be different for non-Cartesian coordinate systems, it is best to first consider an example using Cartesian coordinates. For example, when taking the divergence of two vectors in Cartesian coordinates, we simply treated the operation in some ways like ⃗ = matrix multiplication. Thus, taking the divergence of the velocity vector, V ˆ ˆ ˆ u i + v j + wk , in Cartesian coordinates was simply the following operation:

⃗ ·V ⃗ = ∂x∂ ∇

∂ ∂y

∂ ∂z

√u , ∂u ∂v ∂w + + v = ∂x ∂y ∂z .¸. .,. w

You would perform this operation much like you would matrix multiplication except that you would be applying the derivatives of the row vector to the column vector as opposed to multiplying. This approach worked well in Cartesian coordinates. However, the correct way to perform the divergence opera1

2

Del Operations

tion is to “foil” through the derivatives to all of the velocity terms. Therefore, written in full, the divergence operation should be written as:

! ⃗ ·V ⃗ = ˆi ∂ + ˆj ∂ + kˆ ∂ · uˆi + v ˆj + wkˆ ∇ ∂y ∂z ∂x ∂ ∂ ∂ = ˆi · uˆi + ˆi · v ˆj + ˆi · ∂x ∂x ∂x wkˆ ∂ ∂ ∂ + ˆj · uˆi + ˆj · v ˆj + ˆj ∂y ∂y ∂y · ∂ ∂ wkˆ ∂ + kˆ · uˆi + kˆ · v ˆj + kˆ ∂z ∂z ∂z · wkˆ Notice how there are nine terms (as opposed to three). The derivatives of the individual velocity components multiplied by their base vectors need to be taken. We can do that for the first three terms here (utilizing the product rule from calculus):

∂ˆi uˆi = uˆi · + ˆi · ˆi ∂u = ∂u x ˛ zx ∂x ∂x x ˛ zx e qua ls 1 ∂x e qua ls 0 ˆ ˆi ∂ · v ˆj = vˆi · ∂ j + ˆi · ˆj ∂v = 0 x˛ zx ∂x ∂x x ˛ ∂x z x e qua ls 0 ˆi ∂ ∂x

·

(1.1)

e q ua ls 0

ˆ ˆi ∂ · wkˆ = wˆi · ∂k + ˆi · kˆ ∂w = 0 x ˛ zx ∂x ∂x x ˛ zx e qua ls 0 ∂x e qua ls 0

All we are doing here is utilizing the product rule with an added wrinkle, the dot product between base vectors (i.e. i, j, and k) is taken in each term. Since a Cartesian coordinate system is an orthogonal coordinate system (i.e. all of the base vectors are perpendicular to each other), then the base vectors dotted with each other are zero if we are dotting two different base vectors and one if the base vectors being dotted are the same. We can continue with this trend for the next six terms in the divergence:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Del Operations

3

ˆ ˆj ∂ · uˆi = u ˆj · ∂ i + ˆj · ˆi ∂u = 0 x˛ zx ∂y ∂y ∂y x ˛ z x e qua ls 0 ·

ˆj ∂ ∂y

e q ua ls 0

∂ ˆj v ˆj = v ˆj · + ˆj · ˆj ∂v = ∂v ∂y x ˛ z x x ˛ zx ∂y ∂y e q ua ls 0

e qua ls 1

ˆ ˆj ∂ · wkˆ = w ˆj · ∂k + ˆj · kˆ ∂w = 0 x˛ zx ∂y ∂y ∂y x ˛ z x e qua sl 0 e q ua ls 0

∂ ∂ˆi ∂u kˆ · uˆi = ukˆ · + kˆ · ˆi =0 x˛ zx ∂z ∂z ∂z x ˛ z x e qua ls 0

(1.2)

e q ua ls 0

∂ ∂ ˆj ∂v kˆ · v ˆj = vkˆ · + ˆj · kˆ =0 x˛ zx ∂z ∂z ∂z x ˛ z x e qua ls 0 ∂ kˆ ∂z

·

e q ua ls 0

∂kˆ wkˆ = wkˆ · + kˆ · kˆ ∂w = ∂w x ˛ zx ∂z ∂z x ˛ z x e qua ls 1 ∂z e q ua ls 0

We can add Equations 1.1 and 1.2 to get the divergence in Cartesian coordinates: ⃗ ·V ⃗ = ∂u + ∂v + ∂w ∇ ∂x ∂y ∂z That is a lot of computations just to get a result we already knew. However, for non-Cartesian coordinate systems, this more extensive operation is usually neeeded. We can see this by performing the divergence in cylindrical coordinates using the del operator. In cylindrical coordinates, the velocity vector is defined as: ⃗ = Vreˆr + +Vθeˆθ + Vzeˆz V

(1.3)

The del in cylindrical coordinates is defined as the following: ⃗ = eˆr ∂ + eˆ 1 ∂ + eˆ ∂ ∇ θ r ∂θ z ∂z ∂r

(1.4)

⃗ in cylindrical coordinates, you should avoid To take the the divergence of V

4

Del Operations

taking the “matrix algebra” approach and instead write it out with the base vectors included, i.e.:

! ⃗ ·V ⃗ = eˆr ∂ + eˆ 1 ∂ + eˆz ∂ · (Vreˆr + +V eˆ ∇ θ θ + Vzeˆz) θ r ∂θ ∂r ∂z = eˆ ∂ · (V eˆ ) + eˆ ∂ · (V eˆ ) + eˆ ∂ · (V eˆ ) r

∂r 1∂

r r

r

r

z z

∂r1 ∂ θ θ ∂r 1 ∂ + eˆ · (V eˆ ) + eˆ · (V eˆ ) + eˆ · (V eˆ ) θ r r θ θ θ θ z z r ∂θ r ∂θ r ∂θ ∂ ∂ ∂ + eˆz · (Vreˆr) + eˆz · (V eˆ ) + eˆz · (Vzeˆz) θ θ ∂z ∂z ∂z

Now, the key thing here is that the derivatives of the base vectors are not going to be zero in all cases, unlike Cartesian coordinates. In particular, we have the following relationships:

∂eˆr ∂θ

= eˆθ and

∂eˆθ ∂θ

= −eˆr

All other derivatives of the base vectors are zero. With this information, the nine terms can be written out in full (using the product rule of calculus):

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Del Operations

5

∂ ∂Vr ∂eˆr ∂Vr eˆr · (Vreˆr) = eˆr · eˆr + Vreˆr · = x ˛ z x ∂r ∂r x∂r ˛ z x ∂r e qua ls 1 e q ua ls 0

∂ eˆr · (V eˆ ) = eˆ · eˆ ∂Vθ + V eˆ · ∂eˆθ = 0 θ r θ θ xr˛ zxθ ∂r ∂r x∂r ˛ zx e qua ls 0 ∂

e q ua ls 0 ∂eˆz

∂Vz

eˆr · (Vzeˆz) = eˆr · eˆz + Vzeˆr · =0 ∂r x ˛ zx ∂r ∂r x˛ zx e qua ls 0

e q ua ls 0

1 ∂ · (Vreˆr ) = eˆθ · eˆr 1 ∂Vr + Vr eˆθ ∂eˆr Vr eˆθ · = x ˛ z x r ∂θ r ∂θ r ∂θ r e q ua ls 0 x˛zx equals eˆ xθ x _ ˛z______ equals 1

∂eˆθ = 1 ∂Vθ 1 ∂ · (Vθeˆθ ) = eˆθ · eˆθ 1 ∂Vθ + Vθ eˆθ · eˆθ x ˛ z x r ∂θ r x˛∂θ zx r ∂θ r ∂θ eq ua ls 1 s − êr _______equal_______ x ˛ z x eq ua ls 0

∂ 1 ∂Vz Vz eˆθ 1 · (Vzeˆz) = eˆθ · eˆz + eˆθ · ∂eˆz = 0 r ∂θ x ˛ z x r ∂θ r x∂θ ˛ zx e q ua ls 0

∂

e q ua ls 0

∂Vr

∂eˆr

eˆz · (Vreˆr) = eˆz · eˆr + Vreˆz · =0 ∂z x ˛ zx ∂z ∂z x˛ zx eˆz

∂

∂z

e qua ls 0

e q ua ls 0

· (V eˆ ) = eˆ · eˆ ∂Vθ + V eˆ θ θ

z

θ

x ˛ zx ∂z

e qua ls 0

∂

∂Vz

∂eˆθ =0 ∂z x˛ zx

θ z·

e q ua ls 0

∂eˆz ∂Vz eˆz · (Vzeˆz) = eˆz · eˆz + Vzeˆz · = ∂z ∂z x ˛ zx ∂z ∂z x˛ zx e qua ls 1

e q ua ls 0

We can plug all of these values into our divergence calculation to get:

(1.5)

6

Del Operations

∂ 1∂ ∂ r eˆ r + +Vθeˆθ + V zeˆz) θ r ∂θ + eˆz ∂z · (V ∂r ∂ ∂ ∂ = eˆr · (Vreˆr) + eˆr · (V θeˆ θ) + eˆr · (Vzeˆz)

∇ · V = eˆr

∂Vr ∂r

1 ∂

1 ∂

r r

θ r ∂θ

∂ ∂z

r r

z

=0

=

∂ ∂z

θ θ =0

z z

θ r ∂θ

0

1 ∂Vθ r ∂θ

r r

+ eˆz

1 ∂

θ θ

θ r ∂θ

z

∂ ∂z

z z ∂Vz ∂z

∂Vr Vr 1 ∂Vθ ∂Vz + + + ∂r r r ∂θ ∂z

This was a considerable amount of calculations for the cylindrical coordinates. However, this is what is necessary in order to fully calculate the divergence using the del operator.

We can do a similar operation for the advective term in cylindrical coordinates. For starters, the advective operator is: ∂! ⃗ = (Vreˆr + Vθeˆθ + Vzeˆz) · eˆr ∂ + eˆ 1 ∂ + eˆ ⃗ ·∇ V z ∂z θ r ∂θ ∂r ∂ 1 ∂ ∂ = Vr eˆr · eˆr + Vr eˆr · eˆθ Vθ + Vr eˆr · eˆz x˛zx ∂r x˛zx ∂z x˛zx r ∂θ =1

=0

=0

∂ + Vθ eˆθ · eˆr + Vθ eˆθ · eˆθ + Vθ eˆθ · eˆz x˛zx ∂r x˛zx r ∂θ x˛zx ∂z =0

∂

1∂

=1

∂

1∂

(1.6)

=0

∂ + Vz eˆz · eˆr + Vz eˆz · eˆθ + Vz eˆz · eˆz x˛zx ∂r x˛zx r ∂θ x˛zx ∂z =0

= Vr

∂

∂r

+ Vθ

1∂ r ∂θ

=0

+ Vz

∂

=1

∂z

Foiling through for this case is relatively straightforward since all of the dot products are zero except when the base vectors are the same. This is always the case for orthogonal coordinate systems (such as a cylindrical coordinate system). Applying Equation 1.6 to the velocity vector gives us:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Del Operations

7

! ∂ 1∂ ∂ ⃗ ⃗ ⃗ V · ∇V = Vr + Vθ + Vz (Vreˆr + V eˆθ θ+ Vzeˆz) r ∂θ ∂r ∂z We can apply the advective term operator to each velocity coordinate term separately (again, taking advantage of the product rule): For Vr: Vr

!

1∂ ∂ ∂ + Vθ + Vz Vreˆr r ∂θ ∂r ∂z

∂ (Vreˆr) 1 ∂ (Vreˆr) ∂ (Vreˆr) , √ , = Vr√ ∂r + Vθ r ∂θ + Vz ∂z √ , . . ∂V . ∂V . .. = V V ∂eˆ +eˆ + V V +eˆ . ∂eˆ ∂eˆ ∂V 1 V r r r r +eˆr r r + Vθ r r z r r ∂z r r r ∂θ ∂θ ∂z . . ..¸ x˛∂rzx . ., ∂r ., , ¸. x˛zx ¸ x˛zx =0

= Vr

∂Vr eˆr + ∂r

=êθ

=0

1 1 ∂Vr ∂Vr V____ eˆr + Vz eˆr θ V____ reˆθ +Vθ r ∂θ ∂z x ˛r z x goe s to θ-direction

(1.7) For Vθ: ! ∂ 1 ∂ ∂ Vr + V θ + Vz V θeˆθ r ∂θ ∂r ∂z ∂ (Vθeˆθ) 1 ∂ (Vθeˆθ) ∂ (Vθeˆθ) , √ = Vr√ ∂r + Vθ r ∂θ + V√z ∂z , , . . . .. ∂V = V V ∂eˆ +eˆ ∂V .

1 + V V ∂eˆ +eˆ ∂V . ∂eˆ +eˆ + V V θ θ θ θ θ θ θ θ r . θ z θ ∂z θ θ θ r . ∂θ ∂θ ., ∂z ., .¸ x˛∂rzx ∂r .., ¸ x ˛ zx ¸ x˛zx =0

= Vr ∂Vθeˆθ ∂r

1 − V2eˆ r __ _ θ___r x ˛z x

=− êr

=0

+V θ 1 ∂Vθ eˆθ + V ∂Vθ eˆ z θ r ∂θ ∂z

goes to r − d ir e c tio n

(1.8)

8

Del Operations For Vz:

Vr

!

1∂ ∂ ∂ + Vθ + Vz Vzeˆz r ∂θ ∂r ∂z

∂ (Vzeˆz) 1 ∂ (Vzeˆz) ∂ (Vzeˆz) + Vθ + Vz ∂r r ,∂θ √ ∂z .√ √ , , . . . ∂eˆz . . ∂V ∂eˆ z z ∂V ∂eˆ ∂V z z z .. ..+ V ..V . + Vθ 1 .Vz = Vr ..Vz +eˆz +eˆz +eˆ z z z . . . .. . ∂z ∂z . r ¸.. x˛∂θzx ∂θ ,. ∂r ., .¸ ¸ x˛∂rzx x˛zx , = Vr

=0

= Vr

=0

=0

1 ∂Vz ∂Vz ∂Vz eˆz + Vθ eˆz + Vz eˆz ∂r r ∂θ ∂z

(1.9)

Notice that two of the terms “switch” to another direction, namely the Vθ 1r Vreˆθ from the Vr advective term and the − 1 V2eˆr from the Vθ advective term. Adding r

θ

up Equations 1.7, 1.8, and 1.9 leads to the following for the material derivative in cylindrical coordinates:

∂V r

Dt

=

∂V

∂V

∂Vθ r ∂r

1 ∂Vθ r ∂θ

+ V r ∂rr + V 1r ∂θr + V

∂V

∂V

∂V r ∂r

∂V r 1 2 ∂z − r V ∂Vθ ∂z

1 ∂Vz r ∂θ

1 θr r ∂V ∂z

The first row of Equation 1 is the r−direction, the second row is the θdirection, and the third row is the z−direction.

The same procedure can be done for the Laplacian operator in cylindrical coordinates:

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Del Operations

9

! ! ⃗ ·∇ ⃗ = eˆr ∂ + eˆ 1 ∂ + eˆz ∂ · eˆr ∂ + eˆ 1 ∂ + eˆz ∂ ∇2 = ∇ θ r !∂θ θ r ∂θ ! ∂r ∂z ∂r ∂z ! ∂ ∂ ∂ ∂ · eˆ + eˆ · eˆ r r z 1 ∂ ∂ · eˆ = eˆr + eˆr θ r ∂θ ∂r ∂r ∂r ∂r ∂z ! ! ! ∂ ∂ 1∂ 1∂ 1 ∂ + eˆ 1 ∂ + eˆθ + eˆθ r ∂θ · eˆr θ r ∂θ · eˆz r ∂θ · eˆθ r ∂θ ∂r ∂z ! ! ! ∂ ∂ ∂ ∂ + eˆz · eˆθ 1 ∂ + eˆz ∂ · eˆz + eˆz · eˆr r ∂θ ∂z ∂r ∂z ∂z ∂z

(1.10)

We can go term by term now with the reminder that the only base vectors affected by the derivatives are the θ-derivatives of eˆθ and eˆr: ∂

∂

!

∂ ∂ eˆr · eˆr = eˆr · eˆr 2 = 2 x˛zx ∂r2 ∂r2 ∂r ∂r ! =1 ! ∂ 1∂ ∂ 1∂ eˆr · eˆθ = eˆr · eˆ =0 x˛zxθ ∂r r ∂θ ∂r r ∂θ ! ! =0 ∂ ∂ ∂ ∂ eˆr · eˆz = eˆr · eˆz =0 x˛zx ∂r ∂z ∂r √∂z , =0 . . 1 ∂2 1 ! eˆ 1 eˆ ∂2 eˆ eˆ 1∂ ∂eˆr ∂ eˆ ∂ 1∂ ∂ eˆ eˆθ · · = · + · = eˆ + = r θ r θ r ∂θ r ∂r θ θ ∂θ ∂r . r x˛ r ..¸ ∂θ∂r zx ∂θ∂r r x˛ zx ∂r r ∂r x˛zx ., =0 =1 =êθ √ , ! .

2 1∂ 1∂ 1 ∂ eˆθ ∂ . 1 ∂2 1 ∂ 1 ∂2 · eˆ + eˆθ · (−eˆr ) = 2 2 eˆθ eˆθ 2 · eˆθ 2 + eˆ eˆ = = · θ θ 2 2 θ r ∂θ . r ∂θ r ..¸ ∂θ x ∂θ r2 x ____˛z ____x ∂θ r ∂θ ˛ z ∂θ x., r x˛zx ∂θ ! = − eˆr

eˆθ · eˆz = eˆθ · eˆz 2 = 0 1 ∂ ∂ 1 ∂ r ∂θ ∂z r x˛zx ∂θ∂z ! =0 ∂ ∂ ∂ eˆz · eˆr = eˆz · er 2 = 0 x˛zx ∂r∂z ∂z ∂r =0 ! ∂ 1 ∂ 1 ∂2 = 0 eˆz · eˆθ = eˆz · eˆθ x˛zx r ∂θ∂z ∂z r ∂θ ! =0 ∂ ∂ ∂ ∂ eˆz · eˆz = eˆz · eˆz 2 = 2 x˛zx ∂z2 ∂z2 ∂z ∂z =1

=1

=0

10

Del Operations

We can add up the terms to get the Laplacian in cylindrical coordinates: ∂2 1∂ 1 ∂2 + + 2 2 2 ∂r r ∂r r ∂θ

2

∂2 ∂z2

(1.11)

Sometimes the r− derivatives are “grouped” into a single term to get an alternative for the Laplacian in cylindrical coordinates:

2

=

=

1∂

∂

+

∂2 1 ∂2 + r2 ∂θ2 ∂z2

(1.12)

Going over these mathematical manipulations for some sample operations (divergence, material derivative, and the Laplacian) in cylindrical coordinates hopefully gives you a taste of how to use to del operator in its various forms for non-Cartesian coordinate systems. The examples used in this supplemental document used cylindrical coordinates, which is still an orthogonal coordinate system. Even greater computations are required for non-orthogonal coordinate systems. Luckily, much of these computations can be encapsulated and written more succinctly with the help of tensor notation. A great book on tensors can be found in the collection of Cambridge’s students guides, namely: A Student’s Guide to Vectors and Tensors by Daniel Fleisch.

In general, however, when using cylindrical or spherical coordinates, you normally can just “look” up the operations involving the del symbol.

The governing equations are given below in cylindrical coordinate systems.

Continuity (conservation form): ∂ρ ∂t

+

∂ (ρVr) ∂r

V 1 ∂ (ρVθ) ∂ (ρVz) +ρ r + + =0 r r ∂θ ∂z

Compressible Navier-Stokes (non-conservation form): r−coordinate: ρ

, ! ∂Vr ∂Vr + Vθ ∂Vr Vθ 2 ∂V r . ∂p ∂ 2⃗ ⃗ − + Vr + Vz – µ ∇ · V . =− r ∂t ∂r ∂z , ∂r ∂r 3 r ∂θ ! !! ∂ ∂Vr 1 ∂ 1 ∂Vr ∂Vθ + 2µ + µ + ∂r ∂r r ∂θ r ∂θ ∂r !! ! ∂ ∂Vr ∂Vz 2µ ∂Vr 1 ∂Vr Vr + µ + + − − + ρgr ∂z ∂z ∂r r ∂r r ∂θ r

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Del Operations θ-coordinate:

11

! ! ⃗· ⃗ ∂Vθ ∂Vθ Vθ ∂Vθ VθVr ∂Vθ 1 ∂p 1 ∂ µ 2 ∇ ρ + Vr + + + Vz =− − V ∂t ∂r r ∂θ r ∂z r ∂θ r ∂θ 3 !! !! ∂ ∂Vθ Vθ 1 ∂Vr 1 ∂ 2 ∂Vθ 2Vr + µ − + + µ + ∂r ∂r r r ∂θ r ∂θ r ∂θ r !! ! ∂ ∂Vθ 1 ∂Vz 2µ 1 ∂Vr ∂Vθ Vθ + µ + + + − + ρg θ ∂z ∂z r ∂θ r r ∂θ ∂r r z−coordinate: ρ

! ! ∂Vz ∂Vz Vθ ∂Vz ∂Vz ∂p ∂ 2⃗ ⃗ + Vr + + Vz =− − µ ∇· V ∂t ∂r r ∂θ ∂z ∂z ∂z 3 !! !! ∂ ∂Vz ∂Vr 1 ∂ 1 ∂Vz ∂Vθ + µ + + µ + ∂r ∂r ∂z r ∂θ r ∂θ ∂z ! ! ∂ ∂Vz µ ∂Vr ∂Vz + 2µ + + + ρgz ∂z ∂z r ∂z ∂r

Energy Equation (non-conservation form): ! ! ! ∂T ∂T Vθ ∂T ∂T 1∂ ∂T 1 ∂ k ∂T ρcp + Vr + + Vz = kr + ∂t ∂r ! r ∂θ ∂z r ∂r ∂r r ∂θ ! r ∂θ ∂ ∂T ∂p ∂p Vθ ∂p ∂p + k + βT T + Vr + + Vz + q˙gen + Φ ∂z ∂z ∂t ∂r r ∂θ ∂z Where:

√ !2 !2 !2 , 1 ∂Vθ Vr ∂Vz . .. ∂Vr + + + Φ = 2µ .¸ . r ∂θ r ∂z , ∂r √ !2 !2 !2, .. 1 ∂Vr ∂Vθ Vθ ∂vθ 1 ∂Vz ∂Vz ∂Vr .. + − + + + + +µ .¸ . r ∂θ ∂r r ∂z r ∂θ ∂r ∂z , !2 2 1 ∂ (rVr) 1 ∂Vθ ∂Vz – µ + + 3 r ∂r r ∂θ ∂z

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

1 A Student’s Guide to the Navier-Stokes Equations: A Supplement on the Reynolds Transport Theorem

A very common, and somewhat quick, way of deriving the integral form of the governing equations involves what is known as the Reynolds transport theorem. The Reynolds transport theorem involves taking the derivative of an integral whose limits are not constants but are instead functions of the variable of the derivative. The Reynolds transport theorem is essentially an extension of the Leibniz integration rule, also known as differentiation under the integral sign rule. You may see a number of forms for the Reynolds transport theorem but the most common is:

d V(t)

f V(t) ∂t

(1.1) A(t)

where f is some function (which does not have to be a scalar but also could be a higher order tensor). The velocity in this case is the velocity of an small differential area segment on a moving fluid element. Notice on the left-hand side of the equation that we are attempting to take the time derivative of an integral whose limits are also functions of time (i.e., the V(t) is a function of time). When we came across this type of situation before, we cheated a little and attempted to “sneak” the derivative into integral with a trick where the derivative of the differential volume was related to the divergence of velocity (see Section 2.4). This trick can be avoided by simply using the Reynolds transport theorem.

Instead of deriving the Reynolds transport theorem, we are going to look at some examples of its use. The simplest example of the use of the Reynolds transport theorem is if we set f = 1, giving us: 1

2

Reynolds Transport Theorem

d dt

0 / ‘ ∂1 / dV + V(t) /∂t

$

$ dV =

V(t)

x________ ˛ z x

V⃗ · ⃗ndA

x__# _________˛ z ___ ___ x

V ( t)

DV(t) → = Dt

A(t)

$

V(t )

∇⃗ · V⃗ dV

⃗ ·V ⃗ d V ∇

V(t)

shrink D (dV(t)) ⃗ ⃗ −−−−→ = ∇ · VdV Dt

Notice that the result of setting f = 1 leads us right to the idea that the material derivative of volume of a moving fluid element is proportional to the divergence of velocity, which is something we found out in Chapter 2.

The next example we can look at is to set f = ρ, which is just mass per volume (or density). This leads us to: $ $ d ∂ρ ρdV = dV + ρV⃗ · ⃗ndA (1.2) dt ∂t A(t) V(t) V(t) By setting the right-hand side of Equation 1.2 to zero, we obtain the integral form of the continuity equation: $

∂ρ V(t)

∂t

dV + A(t)

ρV⃗ · ⃗ndA = 0

We can do the same thing with momentum as we did with mass. By setting f = ρ V⃗ (i.e., momentum per volume as opposed to mass per volume), the Reynolds transport theorem now states: d dt

$ V(t)

ρ V⃗ dV =

∂ ρ V⃗

$ V(t)

∂t

dV + A(t)

ρV⃗ ⃗V · ⃗ndA

(1.3)

The key thing here is recognizing that the V⃗ V⃗ term on the right-hand side of Equation 1.3 is actually the outer product of the velocity vector with itself, i.e. ⃗V ⊗ V⃗ . Therefore: d dt

$ V(t)

⃗ ρVdV =

$

∂ ρ V⃗ V(t)

∂t

dV + A(t)

⃗ ⃗ ρV ⊗ V · ⃗ndA

(1.4)

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Reynolds Transport Theorem

3

The right-hand side of Equation 1.4, when set equal to the summation of forces, will lead to the integral form of the Navier-Stokes equations.

If we set f = ρe (or energy per volume), we get: d dt

$

$ V(t)

ρedV =

∂ (ρe) dV + ∂t V(t)

⃗ · ⃗ndA ρeV A(t)

Setting the right-hand side of the above equation equal to Q˙ − Ẇ +S ource gives the starting point for our energy equation in integral form.

As you can probably tell, we could have started with the Reynolds transport theorem and derived all of the equations from this theorem. However, the approach taken in the book, in the author’s opinion, ends up offering a more intuitive approach to the governing equations.

Solution Manual for A Student's Guide to the Navier–Stokes Equations 1st Edition by Justin Garvin

Solutions to A Student’s Guide to the Navier-Stokes Equations Justin W. Garvin