INFLUENCE OF HEAT TRANSFER ON PERISTALTIC FLOW OF JEFFREY FLUID THROUGH A POROUS MEDIUM IN AN INCLINED ASYMMETRIC CHANNEL

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1 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved INFLUENCE OF HEAT TRANSFER ON PERISTALTIC FLOW OF JEFFREY FLUID THROUGH A POROUS MEDIUM IN AN INCLINED ASYMMETRIC CHANNEL G Ravindranath Reddy, S Ravikumar and G S S Raju 3 Department of Mathematics, Tadipatri Engineering College, Veerapuram, Tadipatri, India Department of Mathematics, N B K R Institute of Technology, Nellore, (AP), India 3 Department of Mathematics, JNTUA College of Engineering Pulivendula, Pulivendula, (AP), India ravindranathreddy98@gmailcom ABSTRACT In this paper, we studied the peristaltic flow of a Jeffrey fluid through a porous medium in an inclined asymmetric channel under the assumptions of long wavelength The expressions for the velocity and pressure gradient are obtained analytically The effects of various pertinent parameters on the pumping characteristics and temperature field are studied in detail with the aid of graphs Keywords: asymmetric channel, darcy number, froude number, jeffrey fluid, prandtl number INTRODUCTION A lot of researchers imagined the fluid to behave similar to a Newtonian fluid for physiological peristalsis including the flow of blood in arterioles But such a model cannot be suitable for blood flow unless the non - Newtonian nature of the fluid is incorporated in it Chyme is undoubtedly a non-newtonian fluid Provost and Schwarz (99) have explained a theoretical study of viscous effects in peristaltic pumping and assumed that the flow is free of inertial effects and that non-newtonian normal stresses are negligible In addition, the Jeffrey model is quite simpler linear model using time derivatives instead of convected derivatives, for example the Oldroyd- B model does, it represents rheology different from the Newtonian In spite of its relative simplicity, the Jeffery model is able to specify the changes of the rheology on the peristalsis even under the assumption of long wavelength, low Reynolds number and small or large amplitude ratio Hayat et al (006) investigated the effect of endoscope on the peristaltic flow of a Jeffrey fluid in a tube Nagendra et al (008) have studied the peristaltic flow of a Jeffrey fluid in a tube A large amount attention has been curbed to symmetric channels or tubes, but there exist also flows which could not be symmetric An example for a peristaltic type motion is the intra-uterine fluid flow due to momentarily contraction, where the myometrial contractions may occur in both symmetric and asymmetric directions An interesting study was made by Eytan and Elad (999), whose results have been used to analyze the fluid flow pattern in a non-pregnant uterus In another paper, Eytan et al (00) discussed the characterization of non-pregnant women uterine contractions as they are composed of variable amplitudes and a range of different wave lengths Mishra and Ramachandra Rao (003) have investigated the peristaltic flow of a Newtonian fluid in an asymmetric channel Elshewey et al (006) have analyzed the peristaltic flow of a Newtonian fluid through a porous medium in an asymmetric channel Peristaltic transport of a power law fluid in an asymmetric channel was investigated by Subba Reddy et al (007) Navaneeswara Reddy et al (0) have investigated the peristaltic flow of a Prandtl fluid through a porous medium in a channel Subba Reddy and Nadhamuni Reddy (0) have studied the peristaltic flow of a non-newtonian fluid through a porous medium in a tube with variable viscosity using Adomian decomposition method The study of heat transfer analysis is another important area in connection with peristaltic motion, which has industrial applications like sanitary fluid transport, blood pumps in heart lungs machine and transport of corrosive fluids where the contact of fluid with the machinery parts are prohibited There are only a limited number of research available in literature in which peristaltic phenomenon has discussed in the presence of heat transfer (Mekheimer, Elmaboud, 007; Vajravelu et al, 007; Radhakrishnamacharya and Srinivasulu, 007; Srinivas, Kothandapani, 008; vasudev et al, 00) However, the flow of a Jeffrey fluid through a porous medium in an inclined asymmetric channel with peristalsis has received diminutive attention Hence, an attempt is made to study the peristaltic flow of a Jeffrey fluid through a porous medium in an inclined asymmetric channel under the assumptions of long wavelength The expressions for the velocity and pressure gradient are obtained analytically The effects of various pertinent parameters on the pumping characteristics and temperature field are studied in detail with the aid of graphs MATHEMATICAL FORMULATION We consider the flow of an incompressible Jeffrey fluid through a porous medium in an inclined asymmetric channel induced by sinusoidal wave trains propagating with constant speed c along the channel walls The channel walls are inclined at an angle to the horizontal The lower and upper walls of the channel are maintained at constant temperatures T and T 0, respectively Figure- shows the schematic diagram of the problem 96

2 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved The channel walls are characterized by Y HX, td bcos X ct (upper wall) (a) (lower wall) (b) Y HX, t d bcos X ct where b, b are the amplitudes of the waves, is the wavelength, d is the width of the channel, q is the phase difference which varies in the range 0 q p, q = 0 corresponds to a symmetric channel with waves out of phase and defines the waves with in phase q= p ( XY, ) In fixed frame, the flow is unsteady but if we choose moving frame ( x, y), which travel in the X- direction with the same speed as the peristaltic wave, then the flow can be treated as steady The transformation between two frames are related by x = X - ct, y = Y, u = U - c, v = V and px ( ) = PXt (, ) () ( uv, ) ( UV, ) where and are the velocity components, p and P are the pressures in wave and fixed frames of reference, respectively The pressure p remains a constant across any axial station of the channel, under the assumption that the wavelength is large and the curvature effects are negligible The constitutive equation for stress tensor t in Jeffrey fluid is m t = ( g& + l g& ) + l (3) where is the ratio of relaxation time to retardation time, is the retardation time, is the dynamic viscosity, is the shear rate and dots over the quantities indicate differentiation with respect to time t In the absence of an input electric field, the equations governing the flow field in a wave frame are Figure- The physical model u v 0 x y u u p xx xy u v ucgsin x y x x y k0 v v p yx yy u v vgcos x y y x y k T T k u v u v u v T v x y x y y x () (5) (6) (7) where is the density, k 0 is the permeability of the porous medium, is the specific heat at constant volume, v is kinematic viscosity of the fluid, k is thermal conductivity of the fluid, g is the acceleration due to gravity, is the inclination angle and T is temperature of the fluid The boundary conditions are u c at y H, H T T 0 (8) at y H (9) T T at y H (0) In order to write the governing equations and the boundary conditions in dimensionless form the following non - dimensional quantities are introduced 963

3 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved x y u v d d p ct H x ; y ; u, v,, p, t, h, d c c c d h H, b, b, d d d T T 0,Pr, Ec c T T k ( T T ) 0 0 () where is the wave number and and are amplitude ratios In view of (), the Equations () - (7), after dropping bars, reduce to u v 0 x y () u u p xy Re Re xx u v u sin x y x x y Da Fr 3 v v p yx yy Re Re u v v cos x y y x y Da Fr (3) () Re u v x y Pr x y u Ec u v u v x y x y x where Re dc is the Reynolds number, Fr k0 Da d é l cdæ öù u = d + u + v ( + l ) ê a çè x y øú x is the Froude number, t t xx xy ë (5) c is the Darcy number, gd é l cdæ öæ ù u vö = + u + v + d, ( + l ) ê a ç è x y øè úç y x ë û ø t d é l cdæ öù ê ç v + a x y ú ë û y yy = + u + v ( l ) ê ç è ø and ( d <<) Under the assumptions of long wave length, the Equations (3) - (5) become û, The non-dimensional boundary conditions are u =- y= h, h at (9) 0 at y h x at (0) y h x () Equation (7) implies that, hence p is only function of x Therefore, the Equation (6) can be rewritten as dp sin dx y Da Fr u Re u () The rate of volume flow rate through each section in a wave frame, is calculated as q h h udy p¹ p( y) (3) The flux at any axial station in the laboratory frame is p u Re u x y Da Fr p 0 y u Ec Pr y y 0 sin (6) (7) (8) (T=, () h The average volume flow rate over one period ) of the peristaltic wave is defined as T Q Qdt q T (5) 0 h Q x t u dy qh h / c 96

4 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved The dimensionless pressure rise per one wavelength in the wave frame is defined as dp p dx (6) 0 dx 3 SOLUTION Solving Equation () using boundary conditions (9), we get dp Re u sin c cosh y c sinh y (7) dx Fr where sinh h sinh h, c Da sinh h h cosh h cosh h c sinh h h and Substituting Equation (7) in the Equation (8) and Solving Equation (8) using the boundary conditions (0) and (), we obtain c c cosh y (8) dp Re c3 cye Pr sin cc sinh y 8 dx Fr c c y where E Pr c 5 3 ch dp Re c sin 8N dx Fr, EPrc6 c5 dp Re c sin, h h 8h hn dx Fr c c c coshh cc sinhh c c h, and cosh sinh 5 c 6 c c h cc h c c h The volume flow rate q in the wave frame of reference is given by cosh ( hh) ( )sinh dp Re h h h h q 3 sin h h dx Fr sinh h h (9) From (33), we have dp dx 3 qhhsinh h h coshh h h h sinhh h Re sin Fr (30) The dimensionless pressure rise per one wavelength in the wave frame are defined as dp p dx (3) 0 dx DISCUSSION OF THE RESULTS In order to get the physical imminent of the problem, pumping characteristics and temperature field are computed numerically for different values of various emerging parameters are presented in Figures -5 Figure- depicts the variation of pressure rise p different values of with Re 0, Fr, 05, 07, Da 0,, and It is observed that, the Q decreases with increases in the pumping region, whereas the Q increases in both the free pumping and copumping regions with increasing The variation of pressure rise p different values of with Re 0, Fr, 05, 07, Da 0, 03, and is depicted in Figure-3 It is found that, the Q decreases with increasing phase shift in all the three regions, vz, pumping region p 0, free pumping region p 0 and co-pumping region p 0 Moreover, the Q increases with increasing p 0 appropriately chosen Figure- shows the variation of pressure rise for p different values of Da with 965

5 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved Re 0, Fr, 05, 07,, 03, and It is noted that, in the pumping region p 0, the Q decreases with increasing Da whereas it increases with Da in both free pumping p 0 and co-pumping p 0regions The variation of pressure rise p different values of with Re 0, Fr, Da 0, 07,, 03, and is shown in Figure-5 It is found that, the Q increases with an increase in in both pumping and free pumping regions But in the co-pumping region, the Q decreases with increasing, for an appropriately chosen p 0 Figure-6 illustrates the variation of pressure rise p different values of Re 0, Fr, 05, Da 0, and with, 03, It is noted that, as increases, the Q increases in both pumping and free pumping regions, while in co-pumping region, the Q decreases as increases, for an appropriately chosen p 0 The variation of pressure rise p different values of with Re 0, Fr, 05, 07,, 03, and Da 0 is shown in Figure-7 It is found that, the Q increases in all the pumping, free pumping and copumping regions with an increase in Further, it is found that the Q is more for vertical channel than that of Horizontal channel Figure-8 depicts the variation of pressure rise p different values of Re with Da 0, Fr, 05, 07,, 03, and It is observed that, the Q increases in all the pumping, free pumping and copumping regions with increasing Re The variation of pressure rise p different values of Fr with Da 0,Re 0, 05, 07,, 03, and is shown in Figure-9 It is noted that, the Q decreases in all the pumping, free pumping and co-pumping regions with an increase in Fr Figure-0 shows the temperature profiles for different values of with Fr,, 05, 07, Re 0, Da 0, q, x 0, and Pr E It is found that, the increases with increasing Temperature profiles for different values of phase shift with Da 0, Fr, 05, 07, Re 0, 03, q, x 0, and Pr E is shown in Figure- It is observed that, the oscillates with increasing Figure- illustrates the temperature profiles for different values of Darcy number Da with Fr,, 05, 07, Re 0, 03, q, x 0, and Pr E It is observed that, the decreases with increasing Da Temperature profiles for different values of with Fr,, Da 0, 07, Re 0, 03, q, x 0, and Pr E is shown in Figure-3 It is observed that, the increases with increasing Figure- depicts the temperature profiles for different values of withfr,, 05, Da 0, Re 0, 03, q, x 0, and Pr E the increases with increasing It is observed that, 966

6 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved Temperature profiles for different values of Pr E with Fr,, 05, 07, Re 0, 03, q, x 0, Da 0 and is depicted in Figure-5 It is observed that, the increases with increasing Pr E 5 CONCLUSIONS In this chapter, we investigated the effects of Heat transfer on the peristaltic flow of a Jeffrey fluid in inclined asymmetric channel under the assumptions of long wavelength The expressions for the velocity filed and temperature field are obtained It is found that, in the pumping region the time averaged flux Q increases with increasing,,re and while it decrease with increasing,,fr and Da It is observed that the temperature field increases with increasing,, and Pr E, while it decreases with increasing Da Whereas the temperature field oscillates with increasing Figure-3 The variation of pressure rise p different values of with Re 0, Fr, 05, 07, Da 0, 03, and Figure- The variation of pressure rise p different values of with Re 0, Fr, 05, 07, Da 0,, and Figure- The variation of pressure rise p different values of Da with Re 0, Fr, 05, 07,, 03, and 967

7 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved Figure-5 The variation of pressure rise different values of Da 0, 07, p with Re 0, Fr,, 03, and Figure-7 The variation of pressure rise p different values of with Re 0, Fr, 05, 07,, 03, and Da 0 Figure-6 The variation of pressure rise different values of p with Re 0, Fr, 05, Da 0,, 03, and Figure-8 The variation of pressure rise p different values of Re with Fr, Da 0, 05, 07,, 03, and 968

8 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved Figure-9 The variation of pressure rise p different values of Fr with Re 0, Da 0, 05, 07,, 03, and Figure- Temperature profiles for different values of phase shift with Da 0, Fr, 05, 07, Re 0, 03, q, x 0, and Pr E Figure-0 Temperature profiles for different values of, Re 0 Da 0, q, x 0, with Fr,, 05, 07, and Pr E Figure- Temperature profiles for different values of Darcy number Da with Fr,, 05, 07, Re 0, 03, q, x 0, and Pr E 969

9 VOL, NO 9, MAY 07 ISSN Asian Research Publishing Network (ARPN) All rights reserved Figure-3 Temperature profiles for different values of with Fr,, 07, Da 0, Re 0, 03, q, x 0, and Pr E Figure-5 Temperature profiles for different values of Pr E with Fr,, Re 0, 05, 07, 03, q, x 0, Da 0 and REFERENCES EL Shehawey, EF, Eldabe, NT, Elghazy EM and Ebaid A 006 Peristaltic transport in an asymmetric channel through a porous medium, Appl Math Comput 8: 0-50 Eytan O and Elad D 999 Analysis of Intra - Uterine fluid motion induced by uterine contractions, Bull Math Bio 6: -38 Eytan O, Jaffa AJ and Elad D 00 Peristaltic flow in a tapered channel: application to embryo transport within the uterine cavity, Med Engng Phys 3: 73-8 Figure- Temperature profiles for different values of with Fr,, Re 0, 05, Da 0, 03, q, x 0, and Pr E Hayat T, Ali N, Asghar S and Siddiqui A M 006 Exact peristaltic flow in tubes with an endoscope, Appl Math Comput 8: Mekheimer KhS, Abd elmaboud Y 007 The influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus: application of an endoscope Physics A -9 Mishra M and Ramachandra RAO A 003 Peristaltic transport of a Newtonian fluid in an asymmetric channel, Z Angew Math Phys (ZAMP) 5: Nagendra N, Madhava Reddy N and Subba Reddy M V and Jayaraj B 008 Peristaltic flow of a Jeffrey fluid in a tube, Journal of Pure and Applied Physics 0: 89-0 Navaneeswara Reddy S, Viswanatha Reddy G and Subba Reddy M V 0 Peristaltic flow of a Prandtl fluid through a porous medium in a channel, Int J Mathematical Archive 3():

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