Effects of Heat Transfer on the Peristaltic Flow of Jeffrey Fluid through a Porous Medium in a Vertical Annulus

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1 J. Basic. Appl. Sci. Res., (7)75-758,, TextRoad Publication ISSN 9-44X Journal of Basic and Applied Scientific Research Effects of Heat Transfer on the Peristaltic Flow of Jeffrey Fluid through a Porous Medium in a Vertical Annulus C.Vasudev, Prof.U.Rajeswara Rao, M.V.Subba Reddy 3, G.Prabhakara Rao 4 Assistant Professor, Department of Mathematics, K.S.Institute of Technology, Bangalore, India Head, Chairman & BOS, Department of Mathematics, Sri Krishna Devaraya University, Anantapur, India. 3 Professor, Department of Information Technology, Sri Venkatesa Perumal College of Engineering, Puttur. 4 Research Scholar, Department of Mathematics, Sri Krishna Devaraya University, Anantapur, India. ABSTRACT In this paper, we studied the peristaltic flow of a Jeffrey fluid through a porous vertical annular region between two concentric vertical tubes under the assumptions of low Reynolds number and long wave length. The expressions for the temperature field, the velocity field, and the pressure gradient and heat transfer coefficient are obtained analytically. Interaction of various pertinent parameters with peristaltic transport is discussed with the help of graphs. KEY WORDS: Heat Transfer, Peristaltic Transport, Jeffrey Fluid, Pressure Gradient, Reynolds Number.. INTRODUCTION Peristaltic transport is an important mechanism for mixing and transporting fluids, which is generated by a progressive wave of contraction and expansion moving on the wall of the tube. This mechanism is found in urine transport from kidney to bladder, the motion of spermatozoa in the cervical canal, the movement of chyme in the gastrointestinal tract, vasomotion of the small blood vessels and in many other glandular ducts. In addition such mechanism has several applications in engineering and in biomedical systems including roller and finger pumps. Shapiro et al. (969) have investigated the inertia free peristaltic flow with long wavelength analysis. The early developments on the mathematical modeling and experimental fluid mechanisms of peristaltic flow were given in a comprehensive review by Jaffrin and Shapiro (97). Most of the theoretical investigations have been carried out by assuming blood and other physiological fluids to behave like a Newtonian fluid. Although, this approach provides satisfactory understanding of the peristaltic mechanism in the ureter, it fails to give a better understanding when the peristaltic mechanism involved in small blood vessels, intestine and in transport of spermatozoa in the cervical canal. It has been accepted that majority of the physiological fluids behave like a non-newtonian fluids. Peristaltic flow of blood in small vessels was investigated using the viscoelastic, power-law, Casson, micropolar fluid models by (Bhome and Friedrich, 983; Radhakridhnamacharya, 98; Srivastava and Srivastava, 984; Srinivasacharya et al., 3). The power-law model was used to study the intra uterine fluid in a sagittal cross section of uterus due to myometrial contraction by Subba Reddy (7). Hayat et al. (6) have investigated the effect of endoscope on the peristaltic flow of a Jeffrey fluid assuming chyme as a non-newtonian fluid. Hayat et al. (8) have analyzed the influence of an endoscope on the peristaltic flow of a Jeffrey fluid under the effective of magnetic filed in a tube. Peristaltic motion of a Jeffrey fluid under the effect of a magnetic field in a tube was discussed by Hayat and Ali (8). Still the actual mechanism for the transport of water from the ground to upper branches of all tall trees is not well understood, it is speculated that peristalsis and free convection contribute to this motion. The diameters of the trunks of the trees are found to vary with time. In view of these, some researchers (Aikmn and Anderson, 97; Canny and Phillips, 963) have studied peristalsis with reference to water transport in trees. The translocation of water involves its motion through the porous matrix of the tree. Radhakrishnamurthy et al. (995) have studied the interaction peristaltic flow with heat transfer for the flow of a viscous fluid through a vertical porous tube. Recently, Vajravelu et al. (7) have investigated the effect of heat transfer on the peristaltic flow of a Newtonian fluid through a vertical porous annulus. Mekheimer and Abd Elmaboud (8) have studied the influence of heat transfer and magnetic field on the peristaltic transport of Newtonian fluid in a vertical annulus. It also found that in trees there is a core region through which water does not flow and water flows only through the outer region. *Corresponding Author: C.Vasudev, Research Scholar, Department of Mathematics, Sri Krishna Devaraya University, Anantapur, India. -raadha@gmail.com 75

2 Vasudev et al., However, the problem of peristaltic flow of a Jeffrey fluid through a vertical porous annuls has received little attention. Hence, an attempt is made to investigate the peristaltic flow of a Jeffrey fluid through a vertical porous annulus. The analysis has been carried out in the wave frame of reference with long wavelength and zero Reynolds number assumptions. The expressions for the temperature field, the velocity field and the heat transfer coefficient are obtained analytically. The effects of various emerging parameters on the pumping characteristics, the temperature field and the heat transfer coefficient are discussed in detail with the help of graphs.. MATHEMATICAL FORMULATION We consider the peristaltic flow of an incompressible Jeffrey fluid through a porous annular region between two coaxial vertical tubes. The flow is generated by sinusoidal wave trains propagating with constant speed c along the wall of the outer rube. The axisymmetric cylindrical polar coordinate system Z, R is chosen such that the Z - coordinate is along the center line of the inner and outer tubes and R - coordinate along the radial coordinate. The inner tube is rigid maintained at a temperature T and the outer tube maintained at a temperature T. Fig. depicts the physical model of the problem. The geometry of the inner and outer walls are defined by R R a (.) R R Z, t a bsin Z ct (.) where a, a are the radii of the inner and outer tubes, b is the amplitude of the wave, is the wavelength and t is the time. The flow is unsteady in the fixed frame( Z, R ). However, in a co-ordinate system moving with the propagation velocity c (wave frame( z, r )), the boundary shape is stationary. The transformation from fixed frame to wave frame is given by z Z ct, r R, w W c, u U (.3) where ( w, u ) and ( W, U ) are the velocity components in the wave and fixed frames respectively. The constitute equation of S for Jeffrey fluid is S (.4) where is the dynamic viscosity, is the ratio of relaxation to retardation times, is the retardation time, is the shear rate and dots over the quantities denote differentiation with time. The equations governing the flow in the wave frame of reference are Fig.. Geometry of the problem 75

3 J. Basic. Appl. Sci. Res., (7)75-758, u u w r r z (.5) u u p Srz u w rsrr u r z r r r z k (.6) w w p Szz u w ( rsrz ) w c g T T r z z r r z k (.7) T T T T T cp u w k Q r z r r r z (.8) k is the permeability of the porous medium, T is the temperature, Q is the constant heat c is the specific heat at constant pressure, k is the where p is the pressure, addition/absorption, is the coefficient of linear thermal expansion of the fluid, thermal conductivity and is the density of the fluid. Introducing the non-dimensional variables defined by r z w u r, z, w, u, a c c pa p as, S c c, T T a,, T T r a r, r b, r sin z,, a a a a 3 g a T T c Gr, Pr p a Q, k k T T p k, ac Da Re, a where Da is the Darcy number, Re is the Reynolds number, is the wave number, is the amplitude ratio, Pr is the Prandtl number, Gr is the Grashof number and is the non-dimensional heat source/sink parameter, into the Equations (.5) (.8), we obtain (after dropping the bars) u u w r r z (.) 3 u u p S Re rz u w rsrr u r z r r r z Da (.) w w p Szz Re u w ( rsrz ) w Gr r z z r r z Da (.) T T T Re Pr u w r z r r r z (.3) where and c u Srr u w a r z r, c w u Srz u w, a r z r z c w Szz u w a r z. z ) and low Reynolds number ( Re p r Using the long wavelength approximation ( (.9) ), assumption the Equations (.) and (.3) become (.4) 753

4 p w r w Gr z ( ) r r r Da r r r From Eq. (.4) and (.5), we have Vasudev et al., dp w ( ) r N w ( 3 ) Gr dz r r r ( ) where N. Da The corresponding non-dimensional boundary conditions are (.5) (.6) (.7) w at r r, r (.8) at r r (.9) at r r (.) The dimensionless volume flow rate in the wave frame is given by h q wrdr (.) The dimensionless instantaneous volume flow rate in the fixed frame of reference is given by h h Q( x, t) WRdr ( w ) rdr q h The dimensionless time mean flow over a period T / c 3. SOLUTION T (.) Q Q( x, t) dt q h dx q T of the peristaltic wave, is defined as (.3) Solving Eq. (.6) using the boundary conditions (.9) and (.), we obtain where (3.) log 4 r c r c 4 r r c r 4log r and c r log r 4 r log r r 4log r Substituting Eq. (3.) in to the Eq. (.7) and solving Eq.(.7) together with the boundary conditions (.8), we get where dp w f 5I Nr f6k Nr N dz f3 f4 Gr I Nr K Nr c log r r 4Da N f f 4 4 f c c log r r 4 N, 4 f c c log r r 4 N,. (3.) 754

5 J. Basic. Appl. Sci. Res., (7)75-758,, f4 fi Nr fi Nr K Nr K Nr I Nr I Nr f f K Nr f K Nr 3 f5, f6 f f I Nr K Nr I Nr K Nr. The volume flow rate q is given by, and dp q f 7 r r Grf 8 (3.3) N dz N f 5 f6 4 r r f7 r I Nr r I Nr r K Nr r K Nr N N where and f3 f4 f 8 r I Nr r I Nr r K Nr r K Nr Mf Nf 4 4 r r r r c 4 4 N From Eq.(3.3), we have The pressure rise r r q r r Grf N dp N dz f7 p per one wave length is given by 8 f r r r r c log r log r 4 dp p dz (3.5) dz The heat transfer coefficient at the outer wall is defined by r r c r z yh r cos 4. DISCUSSION OF THE RESULTS z In order to see the effects of various pertinent parameters on the pumping characteristics and the temperature we have plotted Figs. 9. Fig. shows the variation of pressure rise with. (3.4) (3.6) p with time averaged flux Q for different values of Jeffrey fluid parameter.4,., Da., Gr 3and 5. It is found that, the time - averaged flux Q decreases with increasing the pumping region with increasing. p and free pumping region p, while in the co-pumping region p in, the Q increases. Also, it is found that the pumping is more for Newtonian fluid than that of Jeffrey fluid The variation of pressure rise p with time averaged flux Q for different values of with.4,., Da., Gr 3 and.3 is presented in Fig. 3. It is found that, the time - averaged flux Q increases with an increase in heat source/sink parameter in all the three (pumping, free pumping and co-pumping) regions. 755

6 Fig. 4 depicts the variation of pressure rise.4,.,.3, Gr 3and 5 Vasudev et al., p with time averaged flux Q for different values of Darcy number Da with. It is observed that, in the pumping region the time-averaged flux Q decreases with increasing Da, while Q increases with increasing Da in the free pumping and co-pumping region. The variation of pressure rise p with time averaged flux Q for different values of Grashof number Gr with.4,., Da.,.3 and 5 is shown in Fig. 5. It is noted that, an increase in the Gr increases the time - averaged flux Q in all the three (pumping, free-pumping and co-pumping) regions. Fig. 6 presents The variation of pressure rise p with time averaged flux Q for different values of with.4,.3, Da., Gr 3and 5. It is observed that, the time - averaged flux Q increases with increasing in the pumping region and free pumping region, while in the co-pumping region, the Q decreases with increasing. 5 The variation of pressure rise p with time averaged flux Q for different values of with.3,., Da., Gr 3and 5 is depicted in Fig. 7. It is found that, the time - averaged flux Q increases with an increase in in both pumping and free pumping regions, while in co-pumping region, the Q decreases as increases for an appropriately chosen p. Fig. 8 have been plotted to illustrate the variation of on the temperature distribution. It is observed that, the temperature increases with increasing. In order to see the effect of amplitude ratio on the temperature distribution, we have plotted Fig. 9. It is found that, the temperature decreases with an increase in. Fig. shows the variation of on the temperature distribution. It is found that, the temperature increases with increasing. Table depicts the variation of heat transfer coefficient with for.4 and.. It is observed that, the heat transfer coefficient increases with increasing with. The variation of heat transfer coefficient with for 5 and. is shown in Table. It is found that, the heat transfer coefficient increases with an increase in. Table 3 shows the variation of heat transfer coefficient with for.4 and. coefficient decreases with an increase in.. It is noted that, the heat transfer 756

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8 J. Basic. Appl. Sci. Res., (7)75-758, 5. CONCLUSIONS In this chapter, we modeled the peristaltic flow of a Jeffrey fluid through a porous vertical annular region between two concentric vertical tubes under the assumptions of low Reynolds number and long wavelength. The expressions for the temperature field, the velocity field, the pressure gradient and heat transfer coefficient are obtained analytically. Interaction of various pertinent parameters with peristaltic transport is discussed with the help of graphs. It is found that, the pumping decreases with increasing and Da, whereas the pumping increases with increasing, Gr, and. The temperature distribution increases with increasing and, whereas the temperature distribution decreases with increasing. The heat transfer coefficient increases with increasing and, whereas the heat transfer coefficient decreases with an increase in. ACKNOWLEDGEMENTS We thank the management of Sri Kalki Supreme Constructions Pvt Ltd Hyderabad; Suma Engineering works Bangalore, Sri Sanjana Food Products Chittoor, Swami & sons constructions Bangalore, Sri Sai Educational Institutions Anantapur, India, for their support, valuable guidelines, consistent encouragement and providing me necessary facilities in pursuing the research-work. Aikman, D. P. and Anderson, W. P., Ann. Botany, 35(97), 76. REFERENCES Bohme, G., and Friedrich, R. Peristaltic flow of viscoelastic liquids, J.Fluid Mech., 8 (983), 9-. Canny, M. J. and Phillips, O. M. Ann. Botany, 7(963), 379. Hayat, T., Ali, N., Asghar, S. and Siddiqui, A. M. Exact peristaltic flow in tubes with an endoscope, Appl. Math. Comput. 8 (6) Hayat, T. and Ali, N. Peristaltic motion of a Jeffrey fluid under the effect of a magnetic field in a tube, Communications in Nonlinear Science and Numerical Simulation, 3(8), Hayat, T., Ahamad, N. and Ali, N. Effects of an endoscope and magnetic field on the peristalsis involving Jeffrey fluid, Communications in Nonlinear Science and Numerical Simulation, 3(8), Jaffrin, M.Y. and Shapiro, A. H. Peristaltic Pumping, Ann. Rev. Fluid Mech., 3(97), Mekheimer, Kh. S. and Abd Elmaboud, Y. The influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus: Application of an endoscope, Physics letters A, 37(8), Radhakrishnamacharya, G. Long wavelength approximation to peristaltic motion of a power law fluid, Rheol. Acta., (98), Radhakrishnamurthy, V. Radhakrishnamacharya, G. and Chandra, P. Advances in Physiological Fluid Dynamics, Narosa publishing House, India, 995. Shapiro, A.H., Jaffrin, M.Y and Weinberg, S.L. Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech. 37(969), Srinivasacharya, D., Mishra, M. and Ramachandra Rao, A. Peristaltic pumping of a micro polar fluid in a tube, Acta Mechanica, 6(3), Srivastava, L.M. and Srivastava, V.P, Peristaltic transport of blood: Casson model II, J. Biomech., 7(984), Subba Reddy, M.V., Ramachandra Rao, A. and Sreenadh, S. Peristaltic motion of a power law fluid in an asymmetric channel, Int. J. Non-Linear Mech., 4 (7), Vajravelu, K., Radhakrishnamacharya, G. and Radhakrishnamurthy, V. Peristaltic flow and heat transfer in a vertical porous annulus, with long wave approximation, Int. J. Non-Linear Mech., 4(7),