Feb14

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

UK: Managing Editor International Journal of Innovative Technology and Creative Engineering 1a park lane, Cranford London TW59WA UK E-Mail: editor@ijitce.co.uk Phone: +44-773-043-0249 USA: Editor International Journal of Innovative Technology and Creative Engineering Dr. Arumugam Department of Chemistry University of Georgia GA-30602, USA. Phone: 001-706-206-0812 Fax:001-706-542-2626 India: Editor International Journal of Innovative Technology & Creative Engineering Dr. Arthanariee. A. M Finance Tracking Center India 17/14 Ganapathy Nagar 2nd Street Ekkattuthangal Chennai -600032 Mobile: 91-7598208700

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

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IJITCE PUBLICATION

International Journal of Innovative Technology & Creative Engineering Vol.4 No.2 February 2014

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

From Editor's Desk Dear Researcher, Greetings!

Research article in this issue discusses about Peristaltic MHD Flow of a Jeffrey.

Let us review research around the world this month; The virtual currency is about more than money – the real innovation is what people are doing with the technology it is based on. BITCOIN has been called many things, from the future of money to a drug dealer's dream and everything else in between. But beyond creating the web's first native currency, the true innovation of Bitcoin's mysterious designer, Satoshi Nakamoto, is its underlying technology, the "block chain". That fundamental concept is being used to transform Bitcoin – and could even replace it altogether. So what is the block chain? It is a ledger of transactions that keeps Bitcoin secure and allows all users to agree on exactly who owns how many bitcoins. Each new block requires a record of recent transactions along with a string of letters and numbers, known as a hash, which is based on the previous block and produced using a cryptographic algorithm.

The floating school built for Nigeria's water world.In Makoko, Nigeria, a water world of homes above the sea is no futuristic fantasy. For generations, people have lived in this sprawling slum on stilts above the lagoon that surrounds the country's largest city, Lagos. This floating school aims to bring much-needed education to the children of Makoko. At the same time, its architects say it is an exploration of how African cities can cope with the breakneck pace of urbanisation, and the flooding and sea level rise brought on by climate change. The school was built by local builders out of reclaimed materials from the area: it uses the empty plastic barrels abundant in Lagos as floats, for example. Its three levels include an open play space on the bottom, an enclosed classroom area in the middle and workshop space on the top.

The school opened in spring 2013 and can accommodate up to 100 students.

CLEAN energy inspired by the stars is the dream of scientists pursuing nuclear fusion, in which atomic nuclei fuse together and release energy. In a first for laser-driven fusion, scientists at a US lab say they have reached a key milestone called fuel gain: they are producing more energy than the fuel absorbed to start the reaction. But the National Ignition Facility (NIF) in Livermore, California, is still a long way from sparking a self-sustaining fusion reaction with an overall gain in energy – a process called ignition. Currently, the reactor as a whole needs more energy to operate than the amount that is produced.

India unveils its first home-grown astronaut capsule. INDIA is about to take one small step towards human space flight.

Last week the country's space agency unveiled a prototype of its first crew capsule, a 4-metre-high module designed to carry two people into low Earth orbit.Built by Hindustan Aeronautics in Bangalore, the prototype capsule cannot be hermetically sealed and so cannot take people into space. But if the rocket launch is a success, ISRO should be able to remotely test some in-flight controls and see how the module survives the stresses of re-entry and landing at sea.

It has been an absolute pleasure to present you articles that you wish to read. We look forward to many more new technologies related research articles from you and your friends. We are anxiously awaiting the rich and thorough research papers that have been prepared by our authors for the next issue.

Thanks, Editorial Team IJITCE

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

Editorial Members Dr. Chee Kyun Ng Ph.D Department of Computer and Communication Systems, Faculty of Engineering, Universiti Putra Malaysia,UPM Serdang, 43400 Selangor,Malaysia.

Dr. Simon SEE Ph.D Chief Technologist and Technical Director at Oracle Corporation, Associate Professor (Adjunct) at Nanyang Technological University Professor (Adjunct) at Shangai Jiaotong University, 27 West Coast Rise #08-12,Singapore 127470 Dr. sc.agr. Horst Juergen SCHWARTZ Ph.D, Humboldt-University of Berlin, Faculty of Agriculture and Horticulture, Asternplatz 2a, D-12203 Berlin, Germany Dr. Marco L. Bianchini Ph.D Italian National Research Council; IBAF-CNR, Via Salaria km 29.300, 00015 Monterotondo Scalo (RM), Italy Dr. Nijad Kabbara Ph.D Marine Research Centre / Remote Sensing Centre/ National Council for Scientific Research, P. O. Box: 189 Jounieh, Lebanon Dr. Aaron Solomon Ph.D Department of Computer Science, National Chi Nan University, No. 303, University Road, Puli Town, Nantou County 54561, Taiwan Dr. Arthanariee. A. M M.Sc.,M.Phil.,M.S.,Ph.D Director - Bharathidasan School of Computer Applications, Ellispettai, Erode, Tamil Nadu,India Dr. Takaharu KAMEOKA, Ph.D Professor, Laboratory of Food, Environmental & Cultural Informatics Division of Sustainable Resource Sciences, Graduate School of Bioresources, Mie University, 1577 Kurimamachiya-cho, Tsu, Mie, 514-8507, Japan Mr. M. Sivakumar M.C.A.,ITIL.,PRINCE2.,ISTQB.,OCP.,ICP Project Manager - Software, Applied Materials, 1a park lane, cranford, UK Dr. Bulent Acma Ph.D Anadolu University, Department of Economics, Unit of Southeastern Anatolia Project(GAP), 26470 Eskisehir, TURKEY Dr. Selvanathan Arumugam Ph.D Research Scientist, Department of Chemistry, University of Georgia, GA-30602, USA.

Review Board Members Dr. Paul Koltun Senior Research ScientistLCA and Industrial Ecology Group,Metallic & Ceramic Materials,CSIRO Process Science & Engineering Private Bag 33, Clayton South MDC 3169,Gate 5 Normanby Rd., Clayton Vic. 3168, Australia Dr. Zhiming Yang MD., Ph. D. Department of Radiation Oncology and Molecular Radiation Science,1550 Orleans Street Rm 441, Baltimore MD, 21231,USA Dr. Jifeng Wang Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign Urbana, Illinois, 61801, USA Dr. Giuseppe Baldacchini ENEA - Frascati Research Center, Via Enrico Fermi 45 - P.O. Box 65,00044 Frascati, Roma, ITALY. Dr. Mutamed Turki Nayef Khatib Assistant Professor of Telecommunication Engineering,Head of Telecommunication Engineering Department,Palestine Technical University (Kadoorie), Tul Karm, PALESTINE.

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Dr. T. Christopher, Ph.D., Assistant Professor & Head,Department of Computer Science,Government Arts College(Autonomous),Udumalpet, India.

Dr. T. DEVI Ph.D. Engg. (Warwick, UK), Head,Department of Computer Applications,Bharathiar University,Coimbatore-641 046, India.

Dr. Renato J. orsato Professor at FGV-EAESP,Getulio Vargas Foundation,São Paulo Business School,Rua Itapeva, 474 (8° andar),01332-000, São Paulo (SP), Brazil Visiting Scholar at INSEAD,INSEAD Social Innovation Centre,Boulevard de Constance,77305 Fontainebleau - France Y. Benal Yurtlu Assist. Prof. Ondokuz Mayis University Dr.Sumeer Gul Assistant Professor,Department of Library and Information Science,University of Kashmir,India Dr. Chutima Boonthum-Denecke, Ph.D Department of Computer Science,Science & Technology Bldg., Rm 120,Hampton University,Hampton, VA 23688 Dr. Renato J. Orsato Professor at FGV-EAESP,Getulio Vargas Foundation,São Paulo Business SchoolRua Itapeva, 474 (8° andar),01332-000, São Paulo (SP), Brazil Dr. Lucy M. Brown, Ph.D. Texas State University,601 University Drive,School of Journalism and Mass Communication,OM330B,San Marcos, TX 78666 Javad Robati Crop Production Departement,University of Maragheh,Golshahr,Maragheh,Iran Vinesh Sukumar (PhD, MBA) Product Engineering Segment Manager, Imaging Products, Aptina Imaging Inc. Dr. Binod Kumar PhD(CS), M.Phil.(CS), MIAENG,MIEEE HOD & Associate Professor, IT Dept, Medi-Caps Inst. of Science & Tech.(MIST),Indore, India Dr. S. B. Warkad Associate Professor, Department of Electrical Engineering, Priyadarshini College of Engineering, Nagpur, India Dr. doc. Ing. Rostislav Choteborský, Ph.D. Katedra materiálu a strojírenské technologie Technická fakulta,Ceská zemedelská univerzita v Praze,Kamýcká 129, Praha 6, 165 21 Dr. Paul Koltun Senior Research ScientistLCA and Industrial Ecology Group,Metallic & Ceramic Materials,CSIRO Process Science & Engineering Private Bag 33, Clayton South MDC 3169,Gate 5 Normanby Rd., Clayton Vic. 3168 DR.Chutima Boonthum-Denecke, Ph.D Department of Computer Science,Science & Technology Bldg.,Hampton University,Hampton, VA 23688 Mr. Abhishek Taneja B.sc(Electronics),M.B.E,M.C.A.,M.Phil., Assistant Professor in the Department of Computer Science & Applications, at Dronacharya Institute of Management and Technology, Kurukshetra.

(India).

Dr. Ing. Rostislav Chotěborský,ph.d, Katedra materiálu a strojírenské technologie, Technická fakulta,Česká zemědělská univerzita v Praze,Kamýcká 129, Praha 6, 165 21

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Naik Nitin Ashokrao B.sc,M.Sc Lecturer in Yeshwant Mahavidyalaya Nanded University Dr.A.Kathirvell, B.E, M.E, Ph.D,MISTE, MIACSIT, MENGG Professor - Department of Computer Science and Engineering,Tagore Engineering College, Chennai Dr. H. S. Fadewar B.sc,M.sc,M.Phil.,ph.d,PGDBM,B.Ed. Associate Professor - Sinhgad Institute of Management & Computer Application, Mumbai-Banglore Westernly Express Way Narhe, Pune - 41 Dr. David Batten Leader, Algal Pre-Feasibility Study,Transport Technologies and Sustainable Fuels,CSIRO Energy Transformed Flagship Private Bag 1,Aspendale, Vic. 3195,AUSTRALIA Dr R C Panda (MTech & PhD(IITM);Ex-Faculty (Curtin Univ Tech, Perth, Australia))Scientist CLRI (CSIR), Adyar, Chennai - 600 020,India Miss Jing He PH.D. Candidate of Georgia State University,1450 Willow Lake Dr. NE,Atlanta, GA, 30329 Jeremiah Neubert Assistant Professor,Mechanical Engineering,University of North Dakota Hui Shen Mechanical Engineering Dept,Ohio Northern Univ.

Dr. Xiangfa Wu, Ph.D. Assistant Professor / Mechanical Engineering,NORTH DAKOTA STATE UNIVERSITY Seraphin Chally Abou Professor,Mechanical & Industrial Engineering Depart,MEHS Program, 235 Voss-Kovach Hall,1305 Ordean Court,Duluth, Minnesota 55812-3042 Dr. Qiang Cheng, Ph.D. Assistant Professor,Computer Science Department Southern Illinois University CarbondaleFaner Hall, Room 2140-Mail Code 45111000 Faner Drive, Carbondale, IL 62901 Dr. Carlos Barrios, PhD Assistant Professor of Architecture,School of Architecture and Planning,The Catholic University of America Y. Benal Yurtlu Assist. Prof. Ondokuz Mayis University Dr. Lucy M. Brown, Ph.D. Texas State University,601 University Drive,School of Journalism and Mass Communication,OM330B,San Marcos, TX 78666 Dr. Paul Koltun Senior Research ScientistLCA and Industrial Ecology Group,Metallic & Ceramic Materials CSIRO Process Science & Engineering Dr.Sumeer Gul Assistant Professor,Department of Library and Information Science,University of Kashmir,India Dr. Chutima Boonthum-Denecke, Ph.D Department of Computer Science,Science & Technology Bldg., Rm 120,Hampton University,Hampton, VA 23688

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014 Dr. Renato J. Orsato Professor at FGV-EAESP,Getulio Vargas Foundation,S찾o Paulo Business School,Rua Itapeva, 474 (8째 andar)01332-000, S찾o Paulo (SP), Brazil Dr. Wael M. G. Ibrahim Department Head-Electronics Engineering Technology Dept.School of Engineering Technology ECPI College of Technology 5501 Greenwich Road Suite 100,Virginia Beach, VA 23462 Dr. Messaoud Jake Bahoura Associate Professor-Engineering Department and Center for Materials Research Norfolk State University,700 Park avenue,Norfolk, VA 23504 Dr. V. P. Eswaramurthy M.C.A., M.Phil., Ph.D., Assistant Professor of Computer Science, Government Arts College(Autonomous), Salem-636 007, India.

Dr. P. Kamakkannan,M.C.A., Ph.D ., Assistant Professor of Computer Science, Government Arts College(Autonomous), Salem-636 007, India.

Dr. V. Karthikeyani Ph.D., Assistant Professor of Computer Science, Government Arts College(Autonomous), Salem-636 008, India.

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Mr. Md. Musfique Anwar B.Sc(Engg.)

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Mrs. Smitha Ramachandran M.Sc(CS)., SAP Analyst, Akzonobel, Slough, United Kingdom.

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Dr. Pradeep H Pendse B.E.,M.M.S.,Ph.d Dean - IT,Welingkar Institute of Management Development and Research, Mumbai, India Muhammad Javed Centre for Next Generation Localisation, School of Computing, Dublin City University, Dublin 9, Ireland Dr. G. GOBI Assistant Professor-Department of Physics,Government Arts College,Salem - 636 007 Dr.S.Senthilkumar Post Doctoral Research Fellow, (Mathematics and Computer Science & Applications),Universiti Sains Malaysia,School of Mathematical Sciences, Pulau Pinang-11800,[PENANG],MALAYSIA.

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014 Manoj Sharma Associate Professor Deptt. of ECE, Prannath Parnami Institute of Management & Technology, Hissar, Haryana, India RAMKUMAR JAGANATHAN Asst-Professor,Dept of Computer Science, V.L.B Janakiammal college of Arts & Science, Coimbatore,Tamilnadu, India Dr. S. B. Warkad Assoc. Professor, Priyadarshini College of Engineering, Nagpur, Maharashtra State, India Dr. Saurabh Pal Associate Professor, UNS Institute of Engg. & Tech., VBS Purvanchal University, Jaunpur, India Manimala Assistant Professor, Department of Applied Electronics and Instrumentation, St Joseph’s College of Engineering & Technology, Choondacherry Post, Kottayam Dt. Kerala -686579 Dr. Qazi S. M. Zia-ul-Haque Control Engineer Synchrotron-light for Experimental Sciences and Applications in the Middle East (SESAME),P. O. Box 7, Allan 19252, Jordan Dr. A. Subramani, M.C.A.,M.Phil.,Ph.D. Professor,Department of Computer Applications, K.S.R. College of Engineering, Tiruchengode - 637215 Dr. Seraphin Chally Abou Professor, Mechanical & Industrial Engineering Depart. MEHS Program, 235 Voss-Kovach Hall, 1305 Ordean Court Duluth, Minnesota 55812-3042 Dr. K. Kousalya Professor, Department of CSE,Kongu Engineering College,Perundurai-638 052 Dr. (Mrs.) R. Uma Rani Asso.Prof., Department of Computer Science, Sri Sarada College For Women, Salem-16, Tamil Nadu, India.

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

Contents Effect of Heat Transfer on the Peristaltic MHD Flow of a Jeffrey Fluid in An Inclined Channel by V.P. Rathod and Mahadev. M................................................................................................................................................................[180]

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

Effect of Heat Transfer on the Peristaltic MHD Flow of a Jeffrey Fluid in An Inclined Channel V.P. Rathod* and Mahadev. M. Department of studies and Research in Mathematics, Gulbarga University, Gulbarga-585106, Karanataka, India Corresponding Author E-mail address: rathod_vijaykumar@rediffmail.com

Abstract— Peristaltic transport of a conducting Jeffrey fluid in an inclined channel with heat transfer is studied under long wave length and low Reynolds number approximations. An exact solution is presented for the temperature field. The expressions for axial velocity, axial pressure gradient, Stream line and coefficient of heat transfer have been obtained analytically. The effect of various parameters on the flow characteristics are discussed with the help of graph.

Key words — Jeffrey fluid, MHD flow, heat transfer,

inclined channel.

I. INTRODUCTION

Peristaltic pumping is a form of fluid transport which occurs in a biological system. The study of peristaltic motion has gained considerable interest because of its extensive role in transporting many physiological fluid in the body in various situations such as urine transport from the kidney to bladder, transport of the spermatozoa in the ducts afferents of male reproductive tract, the moment of chyme in the gastrointestinal tract, swallowing of food through esophagus and vasomotion of small blood vessels. Many mechanical devices have been designed on the peristaltic pumping to transport fluids without internal moving parts, for example, the transport of slurries, sensitive or corrosive fluids, sanitary fluid and noxious fluids in the nuclear industries.

The literature on this topic is quite extensive. Mention may be made to some recent theoretical and experimental contributions Shapiro and Jaffrin [1], Hayat and Sajid [2], El Misery et al.[3],Yin and Fung[4,5], Pzrikidis [6], Shukla and gupta[7], Vjravelu et al.[8,9], Raju and Devanathan[10] in the field for Newtonian fluids. Such approximation is true in the ureter but it fails to give an adequate understanding of food mixing and chyme moment in the intestine, fuel slurries, flow of plasma, flow of mercury amalgams, and lubrication with heavy oils and greases etc. Also, the assumptions that most of the physiological fluid behave like Newtonian fluid is not true in reality. With all these facts in mind, it is clear that non Newtonian fluid plays an indispensable role in peristaltic flow problem.

A number of researchers have been discussed the effect of magnetic field on peristaltic flow Mekheimer [11], Harikrishana and Subbareddyy [12], Suryanarayana Reddy et al. [13], Subba Reddy and Gangadhar [14], Elshahed and Haroun [15], Kothadapani [16, 17], Affi and Gad[118], Krishana kumari and Raman Murthy [19], Nadeem and Akbar [21], due to its applications in bio engineering and medical devices. Specially, magnetic wound or cancer treatment causes hyperthermia, bleeding reduction during surgeries, and targeted transport of drugs using magnetic particles as drug carries are few examples.

The interaction of peristalsis in connection with heat transfer has also received some attention Nadeem and Akbar [21], Srinivas and Gayatri [22], Hayat and Heena [23], Hayat and Ali [24], Hayat et al.[25] as it might be relevant in processes like hemodialysis and oxygenation. Rathod and Mahadev [26] have studied the effect of thickness of porous material on the peristaltic pumping of Jeffrey fluid with non-erodible porous ling wall. Effects of magnetic field and an endoscope on peristaltic motion is studied by Rathod and Asha [27].

Rathod and Pallavi [28] have discussed the influence of wall properties on MHD peristaltic transport of dusty fluid.

The present article studies the effect of heat transfer on peristaltic MHD flow of Jeffrey fluid in an inclined channel. The governing equations of Jeffrey fluid in Cartesian co-ordinate have been modeled. The equations are simplified using long wave length and low Reynolds number approximations. The closed form of solution for velocity field and temperature are obtained.

The influence of various parameters on the flow characteristics, the temperature and heat transfer coefficient are discussed through graph.

2. FORMULATION OF THE PROBLEM Consider the peristaltic pumping of a Jeffrey fluid with heat transfer in an inclined channel of halfwidth ‘a’. A longitudinal train of progressive sinusoidal wave takes place on the upper and lower wall of the channel. We further assume that the fluid is electrically conducting. A uniform magnetic field B0 is applied in the transverse direction to the flow. The Reynolds number is taken small so that the induced magnetic field is neglected. For simplicity, we restrict our discussion to the half width of the channel as shown in figure .(1)

180

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

Figure.1. Physical model The wall deformation is given by

H ( x, t )  a  bCos

2



( X  ct )

(1)

where b is amplitude of the waves and λ is the wave length.

The constitutive equations for an incompressible Jeffrey fluid are

T   PI  S

(2)

S

(3)

 (  2  ) 1  1

T and S are Cauchy stress tensor and extra stress tensor, p is the pressure, I is the identity tensor,  is the dynamic viscosity, 1 is the ratio of relaxation to retardation times, 2 is the retardation time,  is the shear where

rate and dots over the quantities denote differentiation.

Under the assumptions that the channel length is an integral multiple of the wave length  and the pressure difference across the ends of the channel is a constant, the flow is inherently unsteady in the laboratory frame

( X , Y ) and become steady in the wave frame ( x , y ) which is moving with velocity ‘c’ along the wave. The transformation between these two frames is given by x  X  ct , y  Y , u  U , v  V

(4) where U and V are velocity components in the laboratory frame and u and v are the velocity components in the wave frame. In the many physiological situations it is proved experimentally that the Reynolds number of the flow is very small. So, we assume that the wavelength is infinite, the flow is of Poiseuille type at each local cross-section.

Introducing the non-dimensional quantities

x t

2 x

 2 a 2



,y

y u v 2 a 2 a 2 p ,u  ,v  ,  ,p , a c c   c

,S

a b  ga 2  ca c S ,   ,G  ,   , R  , Pr  , c a c   k

c2   , T   (T1  T0 )  T0 , u  ,v c p (T1  T0 ) y x where R is the Reynolds number, G is the gravitational parameter, Ec is the Eckert number and Pr is the Prandtl Ec 

number.

The equations governing the flow become

u u   0, x y

(5)

181

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.2 FEBRUARY 2014

S xy S u u  p  v      xx    2 (u  1)   g sin  x y  x x y  S S  v v  p  3 R (u  1)  v      2 xy   yy   g cos  x y  x x y  2  u 2  u 2    2 2  2    2 v u     2  PrR (u  1)  v     2   2   2 N        N     x y  y   x   y     x y   x 2  2c  v   u where S xx  1 (u  ) , 1  1  a x  y  x 

 R (u  1)

2  2c  v   u 1 (u  ) , 1  1  a x  y  y 1  2c   v     u v  S xy  1  u       , 1  1  a  x  y    y x 

(6)

(7)

(8)

S yy 

and

 S xy  1  2u    2  y  0 1  1 y

The non dimensional boundary conditions are

u  0 ;  0 at y  0 y y u  1 ;   1 at y  h

(9) (10)

Using long wavelength approximation and dropping terms of order Eqs. (6) - (8) reduces to



and higher.

  1 u  p 2    M (u  1)  GSin  y  1   y  x p 0 y

(11) (12)

2

 u  1  2  Ec    0 2 Pr y  y  where

(13)

 B0 a 2 M   2

3. SOLUTION OF THE FROBLEM Solving the Eq. (11) with the boundary conditions (9), we get

u where

P

  P  Gsin  M 2  P  Gsin cosh (( M 1   ) y )    M2 M 2cosh( Mh 1   )  

p . x

Integrating the equation (14) and using the conditions 



  at y = 0 , we get the stream function as

 ( P  G sin  )sinh hM 1    { y cosh[hM 1   ] ( M 2  P  G sin  )     M 1  







(14)



  * 

(15)

(cosh hM 1    sinh h M 2 (1   ) 1  Tanh M 2 (1   ) } Substituting Eq. (14) into Eq. (13) subject to the boundary conditions (10), the temperature is

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

1 {2M 2 (2M 2  Br ( p  G sin  )2 (h2  y 2 )(1   ))  (4M 4  Br ( p  G sin  ) 2 ) 8M 4

(16)

 cosh(2hM 1   )  Br ( p  G sin  ) cosh(2My 1   )sech( Mh 1   ) } where Br  EcPr is the Brinkman number.

2

2

The coefficient of heat transfer is given by Z  hx y .

(17)

The volume flux ‘q’ through each cross section in the wave frame is given by h

q   udy

(18)

0



1 ( P  Gsin )Tanh(M 1   )h   P  Gsin  M 2  h 2 M M 1 

(19)

The expression for pressure gradient from Eq. (19) is given by 2 2 p M 1   ((q) M  ( M  Gsin )h)  Gsin .Tanh( M 1   h)  x Tanh( M 1   h)  Mh 1  

The instantaneous volume flow rate

(20)

Q( x, t ) in the laboratory frame between the central line and the wall is

h

Q( x, t )   (u  1)dy  q  h

(21)

0

Averaging the Eq. (21) over one period yields the time mean flow rate (time averaged flow rate)

Q

1 Q dt  q  1 T 0

Q as

T

(22)

The pressure difference and frictional force across one wave length in an inclined channel is given by

p dx 0 x 1 p F   h  dx 0 x

p  

1

(23) (24)

4. Results and Discussion To Study the behavior of the distribution of the axial velocity u, numerical calculations for several values of Jeffrey fluid parameter  , Hartman number M , angle of inclination  , gravitational parameter G are carried out. Fig. 2(a) shows that an increase in  results in increase of velocity distribution. The effect of  on the velocity distribution can be seen through Fig. 2(b). It reveals that the axial velocity decreases with increasing  . Fig. 2(c) displays the effect of G on the velocity distribution for fixed values of other parameters. It is observed that the velocity increases with increase of G. The effect of the M on the velocity distribution is illustrated in Fig. 2(d). It is evident that, increase the value of M has a tendency to slow down the fluid motion and fluid moves like a block, which shows some sort of rigidity. This is because of the presence of the transverse magnetic field creates a resistive force similar to the drag force that acts in the opposite direction of the fluid motion, thus causing the velocity of the fluid to decrease.

The effect of heat transfer on peristalsis is illustrated in Fig.3. Fig. 3(a) is made to see the variation temperature  for various values of Brinkman number Br . It is observed that the temperature profiles are almost parabolic and the increase of Br , the temperature distribution increases. From Fig. 3(b), it can be noticed that the temperature decreases with an increase of M. Figs. 3(c) and 3(d) are plotted to see the influence of Jeffrey fluid parameter and angle of inclination on the temperature distribution. It is observe that an increase in Jeffrey fluid parameter  affects the temperature profile in an opposite way to that of angle of inclination.

Fig. 4 is plotted to study the effect of Jeffrey fluid parameter  , Hartman number M , angle of inclination  , gravitational parameter G on the pressure gradient Figs. 4(a) and 4(c) indicate that the pressure gradient increases with increase of M and G. From Figs. 4(b) and 4(d), it is observe that pressure gradient decreases with increase of  and  .

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Fig.5 shows the behavior of heat transfer coefficient Z. From this figure it is observed that due to peristalsis the heat transfer coefficient is in oscillatory behavior. The absolute value of heat transfer coefficient decreases with increasing  and M while it increases with increase of Br and  .

Figure. 2. The velocity distribution for ( 

5. TRAPPING Trapping is an interesting phenomenon in peristaltic motion. It is basically the formation of an internally circulating bolus of the fluid by closed stream lines. The effect of M on trapping can be seen in Fig. 6.

We observed that the size of the bolus reduces with an increase in M. The effects of  and G on the stream lines are plotted in Figs.7-8. It is observed that the size of trapping bolus increases with increasing  and G.

 0.4 , x  0.25 ); other parameters are (a)  

M  2, G  0.5,   1 , (c)  

 4

, M  2,   1 , (d)  

184

 4

 4

, G  0.5, M  2 , (b)

, G  0.5,   1 .

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Figure. 3.Temperature distribution for (  (a)  (c) 

 

 4



4

 0.4 , x  0.25 ); other parameters are

, G  0.5, M  2,   1 . (b)  

 4

, G  0.5,   1, Br  3 .

, M  2, G  0.5, Br  3 . (d) M  2, G  0.5,   1, Br  3.

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Figure. 4. Pressure gradient versus x for (a)  (b)

G  2,   2, M  1,   0.4 . (c)  

 3

 0.4 ,   2,  

,   2, M  1,   0.4 .(d)  

186



 3

3

,G  3 .

, G  2, M  1,   0.4 .

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Figure. 5. Coefficient of heat transfer for (a)  (b)



 3

 0.6, M  5, G  0.2,   2, Br  3.

, G  0.2,   2, Br  3,   0.6 . (c)   (d)



 3

 3

, G  0.2,   2, M  5,   0.6

, G  0.2, Br  3, M  5,   0.6 .

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Figure. 6. Streamlines for (a) M  0.1. (b) M  0.2. (c) M  0.3. (d) M  0.4 ; other parameters are

  0.4,  300 ,   4, G  0.8 .

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  2 (b)   3 (c)   4 (d)   5 ; other parameters are   0.4,  300 , M  0.4, G  0.8

Figure. 7. Streamlines for (a)

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Figure.8. Streamlines for different values of (a) G  4 (b) G  5 (c) G  6 (d) G  7 ; Other parameters are 

 0.4,  300 , M  0.4,   2 .

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