International Journal of Applied Mathematics and Phsics, 3(), Jul-December 0, pp. 55-67 Global Research Publications, India Effects of Chemical Reaction with Heat and Mass Transfer on Peristaltic Flow of Jeffre Fluid through Porous Medium in an Inclined Asmmetric Channel with Hall Currents Nabil T. M. El-dabe *, Sallam N. S. *, Mona A. A. *, Mohamed Y. A. * & Assmaa A. H. * ABSTRACT In this stud,the Hall Effect, chemical reaction as well as the thermal diffusion and diffusion thermo effects are taken in our consideration through studing the peristaltic flow of non-newtonian which obes Jeffer model with heat and mass transfer in an inclined channel. This sstem is stressed b an external magnetic field. The analtical solutions have been obtained in the case of long wave length approximation which transformed our nonlinear sstem of equations to linear sstem of ordinar differential equations. The effects of the phsical parameters of the problem were discussed numericall and illustrated graphicall through a set of figures.. INTRODUCTION Peristalsis is derived from a Greek word meaning clasping and compressing. It s used to describe a progressive wave of contraction along a channel or tube whose cross-sectional area consequentl varies. Peristalsis is regarded as having considerable relevance in biomechanics and especiall as a major mechanism for fluid transport in man biological sstems. Peristaltic transport of an incompressible viscous ûuid in an inclined asmmetric channel through a porous medium is studied b Kothandapani et al., []. Some investigators [-4] have studied a peristaltic flow in asmmetric channel dealing different kinds of non-newtonian fluid models. Peristaltic flow of a Newtonian fluid with heat transfer in a vertical asmmetric channel through porous medium is studied under long-wavelength and low-renolds number assumptions b Srinivas et al., [5]. The effect of variable viscosit on the peristaltic flow of a Newtonian fluid in an asmmetric channel has been discussed b Haat et al., [6] Haat et al., [7] studied the influence of Hall current on peristaltic flow of a Maxwell fluid in a porous medium. The effect of Hall currents on interaction of pulsatile and peristaltic transport induced flows of a particle-fluid suspension was investigated b Gad [8]. Nadeem et al., [9]. studied the peristaltic flow of a Jeffre fluid in a rectangular duct. The influence of heat transfer and temperature dependent viscosit on peristaltic flow of a Jeffre-six constant fluid in a non-uniform vertical tube was investigated b Nadeem et al., [0]. Peristaltic flow of a Jeffre fluid through the gap between concentric uniform tubes has been investigated with particular reference to an endoscope effects b Haat et al., []. Haat [] studied the MHD peristaltic motion of a Jeffre fluid in a tube with sinusoidal wave travelling down its wall. Nadeem [3] investigated the peristaltic flow of a Williamson model in an asmmetric channel. M. Kothandapani [4] investigated the effects of heat and mass transfer on MHD peristaltic transport in a porous space with compliant walls. Nadeem [5] investigated the peristaltic motion of Johnson Segalman fluid of a non-uniform tube. Slip and heat transfer effects on the peristaltic flow in an asmmetric channel studied b Haat et al., [6]. [7] Nadeem discussed the effects of partial slip on the peristaltic flow of a * Math Department. Facult of Education, Ain shams, Universit Heliopolis, Cairo, Egpt.
56 International Journal of Applied Mathematics and Phsics (IJAMP) MHD Newtonian fluid in an asmmetric channel. The influence of an endoscope on the peristaltic flow of a Jeffre fluid through tubes has been studied b Haat. et al., [8]. The present article studies the influence of heat and mass transfer on a peristaltic flow of Jeffre fluid in an inclined asmmetric channel with Hall currents through porous medium. The governing equations of Jeffre fluid in Cartesian coordinates have been modeled. The equations are simplified using long wavelength and low Renolds number approximations. The sstem of differential equations of the fluid flow is solved analticall subject to relevant boundar conditions. At the end of the article graphical results are displaed to show the phsical behavior of different parameters of interest.. FORMULATION OF THE PROBLEM Consider the peristaltic transport of an incompressible Jeffer fluid in a two dimensional channel of width (d + d ). The flow is generated b sinusoidal wave trains propagating with constant speed c along the channel walls. The geometr of the wall surface is defined as seen in Figure (). The equation of upper wall is: π Y H d + a cos() X ct λ. The equation of is: π Y H d a cos () X ct + ϕ λ where a and b are the amplitudes of the waves, λ is the wave length, d + d is the width of the channel, c is the velocit of propagation, t is the time and x is the direction of wave propagation. The phase difference ϕ varies in the range in which ϕ 0 corresponds to smmetric channel with waves out of phase and ϕ π the waves are in phase, further a, b, d, d and ϕ satisfies the condition a + b + a b cos ϕ (d + d ) []. Figure : Geometr of the Problem
Effects of Chemical Reaction with Heat and Mass Transfer on Peristaltic Flow of Jeffre Fluid... 57 3. BASIC EQUATIONS The equations governing the flow of an incompressible non-newtonian fluid are given b: The continuit equation: V 0. () The momentum equation: Heat Equation: V µ ρ + () V V P + V S + J B0 V t k. () Mass Equation: T cpρ + () V T k T + Dm C t. (3) C + () V C Dm C + Dm kt T kc t. (4) Where V is the velocit, ρ is the densit, µ is the dnamic viscosit, k is the permeabilit, A uniform magnetic field with magnetic flux densit vector B (0, 0, B 0 ) is applied. The Hall effect is taken into account, the expression for the current densitj including the Hall effect is given b: J σ[vb 0 m(jb 0 )]. B0 Where m σ is t he Hall parameter, e is the electric charge and n e is the number densit of electrons. ene Also T and C are the temperature and concentration of the fluid. k is the thermal conductivit, D m is the coefficient of mass diffusivit, c p is the specific heat capacit at constant pressure and k is constants of chemical reaction. The constitutive equations for an incompressible Jeffre fluid are µ S () γ + λγ + λ (5) where S is the extra stress tensor, P is the pressure, I is the identit tensor, λ is the ratio of relaxation to retardation times, λ is the retardation time, µ is the dnamic viscosit, γ is the shear rate and dots over the quantities indicate differentiation with respect to time. For unstead two-dimensional flows, the velocit, temperature and concentration can be written as a function of x, and t V (u (x,, t), v (x,, t) 0) (6) T T (x,, t), C C (x,, t). (7) Let u and v be the longitudinal and transverse velocit components of the fluid, respectivel. Now the sstem of equations (-4) take the following form: u v + 0 x (8)
58 International Journal of Applied Mathematics and Phsics (IJAMP) u u u p S S xx x µ σb u v 0 ρ + + u () mv usin g t x + + + + ρ α x x k + m (9) v v v p Sx S µ σb u v 0 ρ + + v () mv ucos g t x + + + ρ α x k + m (0) T T T k T T k T C C + u + v + + + t x cpρ x cp cs x () C C C C C Dm k T T T + u + v Dm + k + + C t x x T x () The corresponding boundar conditions are q ψ at() hcos() x + a πx q ψ at() h h x ψ at() h xcos() d + b π x + φ C C h 0 at at 0 at at T T h C C h C C h (3) Let us introduce the following dimensionless quantities as follow: * πx * * πct x,, t, λ d λ * u * v * πd p u, v, p, c δ c µ cλ T T C C µ θ, ϕ, ν, T T C C ρ d a b d a b,,, d a d H H * d S * k h, h, S, k d d µ c d (4) B using equation (4), the sstem of equations (8-) can be written in dimensionless form after dropping star mark as follows:
Effects of Chemical Reaction with Heat and Mass Transfer on Peristaltic Flow of Jeffre Fluid... 59 u u u p Sxx Sx λ + u + δ v + + u + t x R δ x R x R δ k R δ πcρ e e e e 0 λg () m v u sin σb + δ + α + m πc v v v p S S λg + u + v + + v cos α t x R δ R x R δ k R δ c δ xx x e δ e e e (5) (6) θ θ θ θ θ ϕ ϕ δ + u + v D m t x δ + + δ + pr x x (7) ϕ ϕ ϕ ϕ ϕ θ θ δ + u + v S r t x δ + + δ + γϕ Sc x x (8) Where D γ (the Schmidt number), µ c K (prandtl number), R e cd γ (the Renolds number), DKT () T T0 T () c c m 0 kλ() c c0 γ () πc T T πd And δ λ 0 (Soret number), (chemical reaction parameter), (the non-dimensional parameter). Using the long wavelength approximation and neglecting the wave number along with low-renolds number, equations from (6-9) can be written as: [] S x u Ha p u 0 k + m x (9) p 0 (0) θ ϕ D f + 0 Pr ()
60 International Journal of Applied Mathematics and Phsics (IJAMP) ϕ ϕ S r + γϕ Sc () δ Where H a d B0 (Hartman number) µ ψ ψ We shall define the stream function ψ such that ( u, v x ) δ, equations(9) and (0) after eliminating the pressure and using the stream function ψ can be written as: ψ ψ v 0 + λ where and k ν + H a + m S x δ cλδ ψ ψ ψ ψ + + d t x x δ + λ x (4) The solutions of equations (-3) satisfing the corresponding boundar conditions (4) are: a a ψ c + c + c 3 e + c 4 e (5) D f θ () c B B 5 e c 6 e + + c + c (6) P r B B ϕ () c7e + c8e (7) Where c [ q c 3 A c 4 B] h 3 q ah ah c ch c3e c4e c 3 c 4 ah ()( h3 + q ah3e ) B ah ah ah ah 3 3 [() ] + a h e e aabh ah ah ah ah ah 3 3 3 ()(() ) + ah e B a h e e aabh ah ah 3 3 3 (()( + ))() + h 3 (h h ) h q ah e B ah e A
Effects of Chemical Reaction with Heat and Mass Transfer on Peristaltic Flow of Jeffre Fluid... 6 A (e ah e ah ) B (e ah e ah ) c 5 e bh bh e 3bh + bh c 6 e bh bh + bh e e bh b b b b b f P S D S r γ P r c r D f bh bh bh bh b 7 [()()] c 5 e e c 6 e e + P r 4. RESULTS AND DISCUSSION D f bh bh b 8 [ c e c e + ] c h P r 5 6 7 A numerical discussion is used to show the effect of various parameters involved in the problem such as the magnetic field parameter h a, the permeabilit parameter k, the material parameter λ, the phase difference φ, the Prandtl number P r, the Schmidt number, the Soret number and chemical reaction γ. Numerical results are calculated from formula (7) for velocit. The stream function ψ is shown in Figures (-5) π α.5, x, and γ. In Figure () the stream function (ψ) distributions is plotted versus for various values of permeabilit parameter k. It is clear that the stream function (ψ) decreases in a narrow part of the tube in the region.8 < < 0.9, when the permeabilit parameter k increases when the region <, in the wide part of the tube as the permeabilit parameter k increases. In Figure (3), it is observed that the stream function (ψ) distribution increases as the material parameter λ increases for.8 < < 0.7 while it decreases as the material parameter λ increases in the region < 0.7. From Figure (4), it is found that the stream function ( ψ) increases as the hall effect parameter (Hartman number) h a decreases in the region.8 < < 0.7, while it decreases in the region <, as h a decreases. In Figure (5), the stream function (ψ) distributions increases as φ increases. Figures (6-9) illustrate the effects of the parameters and D f on the temperature distribution. From Figure (6), we see that the temperature distribution increases with the increasing of Prandtl number. In Figure (7), it is found that the increase of Schmidt number leads to a decrease in the temperature of fluid. Figure (8) illustrates the effect of Soret number on the temperature distribution. It is obvious that the increase of leads to a decrease in the temperature of fluid. From the Figure (9), it is obvious that with increasing of D f the temperature of fluid decreases. Figures (0-4) illustrate the effects of phsical parameters entering in the problem on the concentration π distribution of fluid C. The expression (30) is evaluated b taking α.5, x, and γ and the values
6 International Journal of Applied Mathematics and Phsics (IJAMP) k 0.5 k 0.5 k 0.35 Figure : The Variation of the Stream Function () Distribution with Different Values of K 0. 4 Figure 3: The Variation of the Stream Function () Distribution with Different Values of h a 0.5 h a.5 h a.5 Figure 4: The Variation of the Stream Function () Distribution with Different Values of h a
Effects of Chemical Reaction with Heat and Mass Transfer on Peristaltic Flow of Jeffre Fluid... 63 4 3 3 5 Figure 5: The Variation of the Stream Function () Distribution with Different Values of 0.55 0.535 0.545 Figure 6: The Variation of Temperature with Different Values of Prandtl Number 0.5 0.6 0.7 Figure 7: The Variation of Temperature with Different Values of Schmidt Number p c
64 International Journal of Applied Mathematics and Phsics (IJAMP) 0.7 0.8 0.9 Figure 8: The Variation of Temperature with Different Values of Soret Number D f D f. D f. Figure 9: The Variation of Temperature with Different Values of D f 0.5 0.55 0.95 Figure 0: The Variation of Concentration with Different Values of Prandtl Number
Effects of Chemical Reaction with Heat and Mass Transfer on Peristaltic Flow of Jeffre Fluid... 65 5 7 9 Figure : The Variation of Concentration with Different Values of Chemical Reaction 0.5 5.5 0.5 Figure : The Variation of Concentration with Different Values of Soret Number 0. 0.3 0.5 Figure (3): The Variation of Concentration with Different Values of Schmidt Number
66 International Journal of Applied Mathematics and Phsics (IJAMP) D f 0. D f D f Figure (4): The Variation of Concentration with Different Values of D f C are plotted versus. in Figure (0), it is observed that the concentration distribution of fluid increases with the increasing of Prandtl number. In Figure (), we find that the increasing of the chemical reaction γ leads to an increasing in the concentration of the fluid. Figure () illustrates the effects of the thermaldiffusion parameter (Soret number) on the concentration dstribution of the fluid. It is clear that the concentration distribution of fluid increases with the increasing of. In Figure (3), it is found that the concentration distribution C increases with the increasing of Schmidt number. In Figure (4), it is observed that the concentration distribution C increases as D f increases. 5. CONCLUSION The stud of the non-newtonian fluid through porous medium has become the basis for man scientific and engineering applications. The results of the problem are of great interest in geophsics. In this work, we have studied the motion of a Jeffre fluid under the effect of Hall current with heat and mass transfer through porous medium in asmmetric channel with dnamic boundar condition. The governing sstem of partial differential equations of this problem, subject to the appropriate boundar conditions is solved analticall b using a perturbation technique under the consideration of long wave length approximation and low Renolds number. The analsis of the results has shown that the the stream function decreases with increasing Hartman number, h a while it increases with increasing the permeabilit parameter k. the temperature increases as both increasing of Prandtl number, Schmidt number while it decreases when the Soret number and D f increases. The concentration increases as the Schmidt number, Soret number and D f increases. REFERENCE [] S. Srinivas, and M. Kothandapani, (008), Non-Linear Peristaltic Transport of a Newtonian Fluid in an Inclined Asmmetric Channel Through a Porous Medium, Phsics Letters A, 37(8): 65-76. [] S. Srinivas, and V. Pushparaj, (008), Non-Linear Peristaltic Transport in an Inclined Asmmetric Channel, Communications in Nonlinear Science and Numerical Simulation, 37(3): 78-795. [3] Nasir Ali, and Tasawar Haat, (008), Peristaltic Flow of a Micropolar Fluid in an Asmmetric Channel, Computers and Mathematics with Applications, 55(8): 589-608. [4] S. Nadeem, and Safia Akram, (00), Peristaltic Flow of a Williamson Fluid in an Asmmetric Channel, Commun Nonlinear Sci. Numer. Simulat., 5: 705-76. [5] S. Srinivas *, and R. Gaathri, (009), Peristaltic Transport of a Newtonian Fluid in a Vertical Asmmetric Channel with Heat Transfer and Porous Medium, Applied Mathematics and Computation, 5: 85-96.
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