International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 4, Oct 2013, 1-8 ÂŠ TJPRC Pvt. Ltd.

SEPARATION OF A BINARY FLUID MIXTURE DUE TO SORET EFFECT IN UNSTEADY MHD CONVECTION FLOW IN A POROUS MEDIUM B. R SHARMA & KABITA NATH Department of Mathematics, Dibrugarh University, Dibrugarh, Assam, India

ABSTRACT Effects of heat source, magnetic field, and temperature gradient on separation of a binary mixture of incompressible viscous thermally and electrically conducting, and heat absorbing fluids in two dimensional boundary layer laminar unsteady flow through a semi-infinite porous medium bounded one side by a permeable plate in presence of a weak transverse magnetic field are investigated. The momentum, energy and concentration equations are reduced to nonlinear coupled ordinary differential equations by using separation of variables and series expansion methods. These coupled ordinary nonlinear differential equations are solved numerically by using MAT LABâ€™s built in solver bvp4c. These numerical results are exhibited graphically from which it has been found that the effects of various parameters are to separate the components of the binary mixture by collecting the lighter and rarer component near the plate and throwing the heavier one away from it.

KEYWORDS: Soret Effect, Binary Fluid Mixture, Porous Medium, Magnetic Field INTRODUCTION Separation processes of a thermally and electrically conducting binary mixture of incompressible viscous fluids under the influence of magnetic field are considered to be of significant importance due to their applications in many engineering problems such as nuclear reactors and those dealing with liquid metals. Mixed convection flow with simultaneous heat and mass transfer from different geometries embedded in porous media has many engineering and geophysical applications such as geothermal reservoirs, drying of porous solids, thermal insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling of nuclear reactors and underground energy transport. Buoyancy is also of importance in an environment where differences between land and air temperatures can give rise to complicated flow patterns (Chandrasekhara et al., [1]). Furthermore, magnetohydrodynamic (MHD) has attracted the attention of a large number of scholars due to its diverse applications (Geindreau and Auriault, [2]). In astrophysics and geophysics, it is applied to study the stellar and solar structures, interstellar matter, radio propagation through the ionosphere, etc. In engineering it finds its application in MHD pumps, MHD bearings, etc. Workers like Chamkha [3] , Chamkha et al., [4] and Hossain and Mandal [5] have investigated the effects of magnetic field on natural convection flow past a vertical surface. Mass diffusion effects on natural convection flow past a flat plate were studied by researchers like (Makinde, [6]; Makinde et al., [7]; Martynenko et al., [8]; Muthukumaraswamy et al., [9]). A comprehensive account of the boundary layers flow over a vertical flat plate embedded in a porous medium can be found in Kim and Vafai [10] and Liao and Pop [11]. Sharma and Singh ([12], [13], [14]), Sharma and Nath [15] and Sharma et al. ([16], [17]) studied the effect of magnetic field on demixing of a binary fluid mixture. Sharma and Singh ([18], [19]) studied the effect of temperature gradient on demixing of species in hydromagnetic flow of a binary mixture of incompressible viscous fluids between two parallel plates, first taking the plates horizontal and in the second case by taking the plates to be vertical. They found that the effect of temperature gradient is to separate the components of the binary mixture and the magnetic field increases the effect of species demixing.

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B. R Sharma & Kabita Nath

The aim of the research paper is to investigate analytically/numerically the effects of heat source, magnetic field and temperature gradient on separation of a binary mixture of incompressible viscous thermally and electrically conducting, and heat absorbing fluids flowing through a semi-infinite porous medium bounded one side by a permeable plate.

MATHEMATICAL FORMULATION We consider unsteady two-dimensional laminar boundary flow of an incompressible, viscous, thermally and electrically conducting; and heat-absorbing binary mixture of fluid past a semi-infinite vertical permeable moving plate embedded in a uniform porous medium and subjected to a weak uniform transverse magnetic field in presence of thermal and concentration buoyancy effects. It is assumed that there is no applied voltage which implies the absence of an electrical field. The transversely applied magnetic field is assumed to be very small so that the induced magnetic field and the Hall effect are negligible. A consequence of the small magnetic Reynolds number is the uncoupling of the Navier-Stokes equations from Maxwell’s equation. The governing equations for this investigation are based on the balances of mass, linear momentum, energy and concentration of species. Taking into consideration the assumptions made above, these equations can be written in Cartesian frame of reference, as follows: (1) (2) (3) and , where

,

, and

(4)

are the distances along and perpendicular to the plate and time, respectively.

the components of velocitity along

and

and

are

directions, respectively, ρ is the fluid density, ν is the kinematic viscosity, c p

is the specific heat at constant pressure, σ is the fluid electrical conductivity, B0 is the magnetic induction, K* is the permeability of the porous medium, T is the temperature, Q 0 is the heat absorption coefficient, c is the concentration, α is the fluid thermal diffusivity, D is the mass diffusivity, g is the gravitational acceleration,

and

are the thermal and

concentration expansion coefficients, respectively and ST is the Soret number. The magnetic and viscous dissipations are small and hence neglected in this study. The third and fourth terms on the RHS of the momentum equation (2) denote the thermal and concentration buoyancy effects, respectively. Also, the last term of the energy equation (3) represents the heat absorption effects. It is assumed that the permeable plate moves with a constant velocity in the direction of fluid flow, and the free stream velocity follows the exponentially increasing small perturbation law. In addition, it is assumed that the temperature and the concentration at the wall as well as the suction velocity are exponentially varying with time. Under these assumptions, the appropriate boundary conditions for the velocity, temperature and concentration fields are , T= Tw + ε

at

,

(5)

as where

,

and

are the wall dimensional velocity, concentration and temperature respectively.

(6) ,

and

3

Separation of a Binary Fluid Mixture Due to Soret Effect in Unsteady MHD Convection Flow in a Porous Medium

are the free stream dimensional velocity, concentration and temperature, respectively.

and

are constants.

It is clear from equation (1) that the suction velocity at the plate surface is a function of time only. We assume that it takes the following exponential form: (7) where A is a real positive constant, ε and εA are small (less than unity), and

is a scale of suction velocity which

is non-zero positive constant. Outside the boundary layer equation (2) gives .

(8)

It is convenient to employ the following dimensionless variables:

(9) Equations (2)-(4) are coupled non-linear partial differential equations. Introducing the relation (7)-(9) into the equations (2)-(4) we obtain the following coupled non-linear ordinary differential equations: ,

(10)

,

(11) ,

where medium,

,

is the magnetic field parameter,

is the permeability of the porous

is the thermal Grashof number, is the Prandtl number,

number ,

(12)

is the solutal Grashof number,

is the dimensionless heat absorption coefficient,

is the dimensionless exponential index and

=

is the Schmidt

is the thermal diffusion number.

The dimensionless form of the boundary conditions (5) and (6) become at η = 0

(13)

as η → ∞.

(14)

SOLUTION OF THE PROBLEM Equations (10)-(12) represent a set of partial differential equations that cannot be solved in closed form. However, it can be reduced to a set of coupled non linear ordinary differential equations in dimensionless form by using series expansion method. This can be done by representing the velocity, temperature and concentration as (15) ,

(16) (17)

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B. R Sharma & Kabita Nath

Substituting equations (15)-(17) into equations (10)-(12), equating the harmonic and non-harmonic terms, and neglecting the terms of

and higher than that, one obtains the following pairs of equations for

and

. ,

(18) (19) (20) ,

(21) (22) (23)

where a prime denotes differentiation with respect to η. The corresponding boundary conditions can be written as at η = 0

(24)

as η → ∞ .

(25)

Still the equations (18) - (23) are highly coupled and nonlinear and so cannot be solved analytically. Hence we solved the equations numerically by using MAT LAB’s built in solver bvp4c.

RESULTS AND DISCUSSIONS Numerical calculations have been carried out for concentration of the rarer and lighter component of the binary fluid mixture for various values of the parameters Sc, n,

, ϕ, Pr and A, and are plotted against η in Figures 1-6. Six cases

are considered: Case I: M=0.1, K=0.5,

= 2.0,

= 1.0, n =0.1, A = 0.5, Pr = 0.7, ϕ = 1,

=0.0001,

Up = 1 and Sc = (0.3, 0.5, 0.7). Case II: M=0.1, K=0.5,

= 2.0,

= 1.0, Sc =0.3, A = 0.5, Pr = 0.7, ϕ = 1,

=0.0001,

Up = 1 and n = (0.1, 0.9, 1.5). Case III: M=0.1, K=0.5, Up = 1 and

= 2.0,

= 1.0, n =0.1, A = 0.5, Pr = 0.7, ϕ = 1, Sc=0.3,

= (0.001, 0.045, 0.094).

Case IV: M=0.1, K=0.5,

= 2.0,

= 1.0, n =0.1, A = 0.5, Pr = 0.7, Sc = 0.3,

=0.0001,

Up = 1 and ϕ = (1, 4, 6). Case V: M=0.1, K=0.5,

= 2.0,

= 1.0, n =0.1, A = 0.5, Sc = 0.3, ϕ = 1,

=0.0001,

Up = 1 and Pr = (0.7, 1, 7). Case VI: M=0.1, K=0.5,

= 2.0,

= 1.0, n =0.1, Sc = 0.3, Pr = 0.7, ϕ = 1,

=0.0001,

Up = 1 and A = (0.3, 0.5, 0.7). Figure 1, 2, 4, 5 and 6 depict that the concentration of the rarer and lighter component of the binary mixture decreases with increase in the values of the parameters Sc, n, ϕ, Pr and A. It is found that concentration of the rarer and

Separation of a Binary Fluid Mixture Due to Soret Effect in Unsteady MHD Convection Flow in a Porous Medium

5

lighter component of the binary mixture is more near the surface of the plate and decreases exponentially as η increases to 14. Thereafter in the region

14 no variation in C is observed. Thus we conclude that the separation of the binary

mixture takes place mostly in the region

and thereafter separation is found to be negligible. It is evident from

Figure 1, 2, 4, 5 and 6 that the rate of separation can be enhanced by increasing the values of Sc, n, ϕ, Pr and A; and from Figure 3 reverse effect is observed with increase in the values of parameter

.

In the present investigation the effects of all these parameters are to demix the binary mixture by collecting the rarer and lighter component of the binary fluid mixture near the surface of the plate and throwing the heavier component away from it. Taking into account the conclusion derived in this paper, gas separating instruments can be installed in big cities where harmful gases are present in very small quantities that can be sucked after separating them and thus pollutants can be removed.

Figure 1: The Graph of C against η M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Pr = 0.7, ϕ = 1, =0.0001 and Up = 1 for Various Values of Sc

Figure 2: The Graph of C against η M=0.1, K=0.5, = 2.0, = 1.0, Sc =0.3, A = 0.5, Pr = 0.7, ϕ = 1, =0.0001 and Up = 1 for Various Values of n

Figure 3: The Graph of C against η M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Pr = 0.7, ϕ = 1, Sc=0.3 and Up = 1 for Various Values of

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B. R Sharma & Kabita Nath

Figure 4: The Graph of C against η M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Pr = 0.7, Sc = 0.3, =0.0001 and Up = 1 for various Values of ϕ

Figure 5: The Graph of C against η M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Sc = 0.3, ϕ = 1, =0.0001 and Up = 1 for Various Values of Pr

Figure 6: The Graph of C against η M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, Sc = 0.3, Pr = 0.7, ϕ = 1, =0.0001 and Up = 1 for various Values of A Finally, the effects of the rate of heat and mass transfer are shown in Table 1. The behaviour of these parameters is self – evident from the Table 1 and hence any further discussion about them seems to be redundant. Table 1: Numerical Values of M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Pr = 0.7, ϕ = 1, =0.0001 and Up = 1 Sc 0.3 0.5 0.7 1 1 1 3.5256 3.4512 3.3879 0 0 0 4.8264 4.6559 4.5307 1 1 1 -1.2569 -1.2569 -1.2569 1 1 1

M=0.1, K=0.5, = 2.0, = 1.0, Sc =0.3, A = 0.5, Pr = 0.7, ϕ = 1, =0.0001 and Up = 1 n 0.1 0.9 1.5 1 1 1 3.5256 3.5256 3.5256 0 0 0 4.8264 4.5742 4.5030 1 1 1 -1.2569 -1.2569 -1.2569 1 1 1

M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Pr = 0.7, ϕ = 1, Sc=0.3 and Up = 1 0.001 1 3.5258 0 4.8271 1 -1.2569 1

0.045 1 3.5369 0 4.8627 1 -1.2569 1

0.094 1 3.5498 0 4.9039 1 -1.2569 1

7

Separation of a Binary Fluid Mixture Due to Soret Effect in Unsteady MHD Convection Flow in a Porous Medium

-1.5323 1 -0.1301 1 -0.4935

-1.5323 1 -0.2351 1 -0.7921

-1.5323 1 -0.3530 1 -1.0907

M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Pr = 0.7, Sc = 0.3, =0.0001 and Up = 1 ϕ 1 4 6 1 1 1 3.5256 3.3001 3.2302 0 0 0 4.8264 4.5698 4.4903 1 1 1 -1.2569 -2.0595 -2.4291 1 1 1 -1.5323 -2.2895 -2.6495 1 1 1 -0.1301 -0.1228 -0.1210 1 1 1 -0.4935 -0.4946 -0.4949

Table 1: Contd., -1.5323 -1.7635 -1.9117 1 1 1 -0.1301 -0.1301 -0.1301 1 1 1 -0.4935 -0.7468 -0.8815

-1.5323 1 -0.1294 1 -0.4911

M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, A = 0.5, Sc = 0.3, ϕ = 1, =0.0001 and Up = 1 Pr 0.7 1 7 1 1 1 3.5256 3.4080 2.8633 0 0 0 4.8264 4.6525 4.0178 1 1 1 -1.2569 -1.6180 -7.8875 1 1 1 -1.5323 -2.0167 -11.0845 1 1 1 -0.1301 -0.1260 -0.1135 1 1 1 -0.4935 -0.4941 -0.4951

-1.5323 1 -0.0958 1 -0.3730

-1.5323 1 -0.0573 1 -0.2418

M=0.1, K=0.5, = 2.0, = 1.0, n =0.1, Sc = 0.3, Pr = 0.7, ϕ = 1, =0.0001 and Up = 1 A 0.5 0.7 0.9 1 1 1 3.5256 3.5256 3.5256 0 0 0 4.8264 5.3149 5.8035 1 1 1 -1.2569 -1.2569 -1.2569 1 1 1 -1.5323 -1.6273 -1.7224 1 1 1 -0.1301 -0.1301 -0.1301 1 1 1 -0.4935 -0.5394 -0.5852

ACKNOWLEDGEMENTS This research work is funded by grants from the UGC, New Delhi, India (File No. 39-43/2010 (SR)) as a Major Research Project awarded to Dr. B. R. Sharma. Kabita Nath is associated with the project as a Project Fellow. The authors are grateful to UGC for providing financial support during the research work and Dibrugarh University for providing other facilities.

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2.

Geindreau, C & Auriault, J.L. (2002). Magnetohydrodynamics flows in porous media. Journal of Fluid Mechanics, 466, 343-363.

3.

Chamkha, A.J. (2004). Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption. Int. Journal of Engg. Sci., 42, 217-230.

4.

Chamkha, A.J & Abdul-Rahim Khaled, A. (2000). Hydromagnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid saturated porous medium. Int. Journal of Numerical Methods Heat and Fluid Flow, 10(5), 455-476.

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Hossain, M.A & Mandal, A.C. (1985). Mass transfer effects on the unsteady hydromagnetic free convection flow past an accelerated vertical porous plate. Journal of physics D: Applied physics, 18, 163-169.

6.

Makinde, O.D. (2005). Free-convection flow with thermal radiation and mass transfer past a moving vertical porous plate. Int. Communications in Heat and Mass Transfer, 32, 1411-1419.

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Makinde, O.D, Mango, J.M & Theuri, D.M. (2003). Unsteady free convection with suction on an accelerating porous plate. A.M.S.E., Modelling, Measurement and Control, 72(3-4), 39-46.

8.

Martynenko, O.G, Berezovsky, A.A & Yu and Sokovishin, A. (1984). Laminar free convection from a vertical plate. Int. Journal of Heat and Mass Transfer, 27, 869-881.

9.

Muthukumaraswamy, R, Ganesun, P & Souldalgekar, V. M. (2001). The study of the flow past an impulsively started isothermal vertical plate with variable mass diffusion. Journal of Energy, Heat and Mass Transfer, 23, 6372.

10. Kim, S.J & Vafai, K. (1989). Analysis of natural convection about a vertical plate embedded in porous medium. Int. Journal of Heat and Mass Transfer, 32, 665-677. 11. Liao, S.J & Pop, I. (2004). Explicit analytic solution for similarity boundary-layer equations. Int. Journal of Heat and Mass Transfer, 47, 75-85. 12. Sharma, B.R & Singh, R.N. (2008). Barodiffusion and thermal diffusion a binary fluid mixture confined between two parallel discs in presence of a small axial magnetic field. Latin American Applied Research, 38, 313-320. 13. Sharma, B.R & Singh, R.N. (2009). Thermal diffusion in a binary fluid mixture confined between two concentric circular cylinders in presence of radial magnetic field. J. Energy Heat Mass Transfer, 31, 27-38. 14. Sharma, B.R & Singh, R.N. (2010). Separation of species of a binary fluid mixture confined between two concentric rotating circular cylinders in presence of a strong radial magnetic field. Heat Mass Transfer, 46, 769777. 15. Sharma, B.R & Nath, Kabita. (2012). The effect of magnetic field on separation of binary mixture of viscous fluids by barodiffusion and thermal diffusion near a stagnation point- a numerical study. Int. Jour. Mathematical Archive, 3(3), 1118-1124. 16. Sharma, B.R, Singh, R.N, Gogoi, Rupam Kr. (2011). Effect of a Strong Transverse Magnetic Field on Separation of Species of a Binary Fluid Mixture in Generalized Couette Flow. Int. Journal of Applied Engineering Research, 6, 2223-2235. 17. Sharma, B.R, Singh, R.N , Gogoi , Rupam Kr. & Nath, Kabita. (2012). Separation of species of a binary fluid mixture confined in a channel in presence of a strong transverse magnetic field. Hem. Ind. 66(2), 171-180. 18. Sharma, B.R & Singh, R.N. (2004). Soret effect in generalized MHD Couette flow of a binary mixture. Bull Cal Math Soc., 96, 367-374. 19. Sharma, B.R & Singh, R.N. (2007). Soret effect due to natural convection between heated vertical plates in a horizontal small magnetic field. Ultra Science, 19, 97-106.

2 common fixed point

Published on Sep 4, 2013

Effects of heat source, magnetic field, and temperature gradient on separation of a binary mixture of incompressible viscous thermally and e...