Effects of wall properties and heat transfer on the peristaltic transport of a jeffrey fluid in a channel

Transcription

1 Available online at Advances in Applied Science Research,, 4(6):59-7 ISSN: CODEN (USA): AASRFC Effects of wall properties and heat transfer on the peristaltic transport of a jeffre fluid in a channel M. Arun Kumar *, S. Sreenadh * and A. N. S. Srinivas ** *Department of Mathematics, Sri Venkateswara Universit, Tirupati(A.P), India **School of Advanced Sciences, VIT Universit, Vellore(T.N), India ABSTRACT A mathematical model is constructed to stud the effect of heat transfer and elasticit of flexible walls in swallowing of food bolus through the oesophagus. The food bolus is supposed to be Jeffre fluid and the geometr of wall surface of oesophagus is considered as peristaltic wave. The expressions for temperature field, axial velocit, transverse velocit and stream function are obtained under the assumptions of low Renolds number and long wavelength. The effects of thermal conductivit, Grashof number, rigidit, stiffness of the wall and viscous damping force parameters on velocit, temperature and stream function have been studied. It is noticed that increase in λ results in increase of velocit thermal conductivit β, Grashof number Gr and the Jeffre parameter distribution. It is found that that the size of the trapped bolus increases with increaseλ. Kewords: Peristaltic transport, Jeffre fluid, Oesophagus, food bolus, channel. INTRODUCTION Peristaltic transport is a mechanism of pumping fluids in tubes when progressive wave of area contraction or expansion propagates along the length on the boundar of a distensible tube containing fluid. Peristalsis has quite important applications in man phsiological sstems and industr. It occurs in swallowing food through the oesophagus, chme motion in the gastrointestinal tract, in the vasomotion of small blood vessels such as venules, capillaries and arterioles, urine transport from kidne to bladder. In view of these biological and industrial applications, the peristaltic flow has been studied with great interest. Man of the phsiological fluids are observed to be non-newtonian. Peristaltic flow of a single fluid through an infinite tube or channel in the form of sinusoidal wave motion of the tube wall is investigated b Burns and Parkes [], Hanin [],Shapiro et al.[] etc,. In the literature some important analtical studies on peristaltic transport of non Newtonian fluids are available Devi and Devanathan [4], Shukla and Gupta [5], Srivastava and Srivastava [6], Usha and Rao [7], Vajravelu et al. [8,9], Haat et al. [,,]. Further an interesting fact is that in oesophagus, the movement of food is due to peristalsis. The food moves from mouth to stomach even when upside down. Oesophagus is a long muscular tube commences at the neck opposite the long border of cricoids cartilage and extends from the lower end of the pharnx to the cardiac orifice of the stomach. The swallowing of the food bolus takes place due to the periodic contraction of the esophageal wall. Pressure due to 59

2 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7 reflexive contraction is exerted on the posterior part of the bolus and the anterior portion experiences relaxation so that the bolus moves ahead. The contraction is practicall not smmetric, et it contracts to zero lumen and squeezes it marvelousl without letting an part of the food bolus slip back in the opposite direction. This shows the importance of peristalsis in human beings. Mitra and Prasad [] studied the influence of wall properties on the Poiseuille flow under peristalsis. Mathematical model for the esophageal swallowing of a food bolus is analzed b Mishra and Pande [4]. Kavitha et al., [5] analsed the peristaltic flow of a micropolar fluid in a vertical channel with longwave length approximation. Redd et al., [6] studied the effect of thickness of the porous material on the peristaltic pumping when the tube wall is provided with non-erodible porous lining. Lakshminaraana et al., [7] studied the peristaltic pumping of a conducting fluid in a channel with a porous peripheral laer. Radhakrishnamachara and Srinivasulu [8] studied the influence of wall properties on peristaltic transport with heat transfer. Rathod et al., [9] studied the influence of wall properties on MHD peristaltic transport of dust fluid. A new model for stud the effect of wall properties on peristaltic transport of a viscous fluid has been investigated b Mokhtar and Haroun [], Srinivas et al., [] studied the effect of slip, wall properties and heat transfer on MHD peristaltic transport. Sreenadh et al., [] studied the effects of wall properties and heat transfer on the peristaltic transport of food bolus through oesophagus. Afsar Khan et al., [] analzed the peristaltic transport of a Jeffre fluid with variable viscosit through a porous medium in an asmmetric channel. In view of the importance of non-newtonian phsiological fluid motion b peristalsis we consider a mathematical model to stud the effects of wall properties and heat transfer in swallowing the food bolus through the oesophagus. The simplest non-newtonian phsiological fluid is taken as Jeffre fluid. The results are analzed for different values of phsical parameters. Mathematical Formulation Consider the peristaltic flow of an incompressible Jeffre fluid in a flexible channel with flexible induced b sinusoidal wave trains propagating with constant speed c along the channel walls. The wall deformation is given b π = ( ) () λ H ( x, t ) a φ Cos x ct where h, x, t, a, φ, λ and c represent transverse vibration of the wall, axial coordinate, time, half width of the channel, amplitude of the wave, wavelength and wave velocit respectivel. Y H(x) is the wall c a O λ/ λ X Figure. Phsical Model 6

3 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7 The governing equations of motion of incompressible Jeffre fluid are given as p µ u u ρ + u + v u = ρgα( T T ) t x x + λ x ρ + u + v v = + + t x + λ x p µ v v u v + = x T T ρcp + u + v T = K + +Φ t x x () () (4) (5) where ρ is the fluid densit, u axial velocit, v Transverse velocit, transverse coordinate, p pressure, µ fluid viscosit, g acceleration due to gravit, α coefficient of linear thermal expansion of fluid, T temperature, c p specific heat at constant pressure, K thermal conductivit and Φ constant heat addition/absorption. The velocit and temperatures at the central line and the wall of the peristaltic channel are given as T = T at = T = T at = h where T is the temperature at centre is line and T is the temperature on the wall of peristaltic channel. The governing equation of motion of the flexible wall ma be expressed as * L = p p (6) * where L is an operator, which is used to represent the motion of stretched membrane with viscosit damping forces such that * L = τ + m + c x t t Continuit of stress at = h and using momentum equation, ield x x + x t x * p µ u u L ( h ) = = + g ( T T + ρ α ) ρ + u + v u λ (7) Here τ is the elastic tension in the membrane, m is the mass per unit area, C is the coefficient of viscous damping forces. Introducing the following non-dimensional quantities, 6

4 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7 x u v a a p ct h ψ Q x =, =, u =, v =, δ =, p =, t =, h =, ψ =, Q = λ a c cδ λ µ cλ λ a ac ac φ ρ caδ gpa α ( T T ) T T a Φ µ c φ =, Re =, Gr =, θ =, β =, pr = a µ cµ T T k ( T T ) k p (8) where δ is the length of the channel, ψ is the Stream function, Q is the Volume flow rate, Reis Renolds number, Gr is the Grashof number, θ dimensionless temperature, β is the dimensionless heat source/sink parameter and Pr is Prandtl number, we obtain the dimensionless governing equations and boundar conditions as follows h ( x, t ) = φ C o s π ( x t ) (9) p u u Re + u + v u = + δ + Grθ + t x x + λ x () p 4 v v Re δ + u + v v = + δ + δ t x + λ x () u x v + = () ( Re) ( P r) θ θ + u + v ( θ ( T T ) + T ) = δ + + β ( T T ) t x x () δ u u + + Gr θ R e u v u E + + = + E + E ( h) + λ x + λ t x x x t x t (4) u = at = u = at = h v = at = θ = a t =, θ = at = h (5) Solution of the problem Under the assumptions of long wavelength δ and low Renolds number, equations (9)-(5) reduce to h( x, t) = φcos π( x t) (6) + p = p u = + Gr x λ θ (7) (8) 6

5 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7 u v + = x θ + β = (9) () θ = at =, θ = at = h () u h h h + Gr θ = E + E + E + λ x x t x t () The following boundar conditions are imposed on the governing equations to model the problem under consideration: u = at = () u = at = h (4) v = at = (5) Equation (8) shows that P is not a function of. Now on differentiating equation (7) with respect to, the compatibilit equation as follows u θ + Gr = λ + (6) u h h h + Gr θ = E + E + E + λ x x t x t (7) The closed form solution for equations (7) and () with the boundar conditions (), () and (4) is given b β θ = + ( h ) (8) h { ( ) ( )} π φ( h ) E Sin π( x t) Cos π( x t) ( E + E ) 4 πcos π( x t) Sinπ ( x t) u = ( + λ ) Gr β ( h h+ ) ( h ) 6 4 h (9) Integrating the continuit equation with respect to, using the above equation and the boundar condition (5), we obtain transverse velocit as 6

6 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7 π φ h π E Sin π x t E + E π Cos π x t h v = ( + λ ) π φh { ECos4 π ( x t) + ( E + E) ( π Sin π ( x t) )} x 4 Gr β 4 h 4h h h x { 4 ( 4 ( ) ( )4 ( ))} () Stream function can be obtained b integrating equation and using the condition ψ = at =. It is given b { ( ( ) ( )) ( ) ( 4 ( ) ( ))} π φ h E Sin π x t Cos π x t E + E πcosπ x t Sinπ x t ψ = ( + λ ) Gr β 4 h h h h 4 () RESULTS AND DISCUSSION In order to observe the quantitative effects of various parameters involved in the analsis, the velocit, temperature and stream functions are calculated for various values of these phsical parameters. The numerical evaluations of the analtical results and some significant results are displaed graphicall from Figures () - (4). From Figures (), () and (4), it is observed that increase in thermal conductivit β, Grashof number Grand the Jeffre parameter λ results in increase of velocit distribution. Figure (5) displas the effect of rigidit parameter in the presence of ( E ) and viscous damping force ( E ) stiffness. It is noticed that the velocit increases with increase in rigidit parameter. A similar observation is made for different values of E in the presence of other parameters i.e., rigidit and viscous damping force which is shown in Figure (6). From figure (7), we can see the influence of viscous damping force on velocit distribution in the presence of rigidit and stiffness. One can observe that the velocit decreases with the increase in E. The variation in temperature for various values of thermal conductivit is shown in Figure (8). The temperature increases with the increase in β. An interesting phenomenon of peristalsis is trapping in which streamlines split to trap a bolus in the wave frame. The effect of thermal conductivit on trapping is analzed in Figure (9). It can be concluded that the size of the trapped bolus in the left side of the channel decreases when β increases where as it has opposite behavior in the right hand side of the channel. The influence of Grashof number on trapping is analzed in Figure (). It shows that the size of the left trapped bolus decreases with increase in Gr where as the size of the right trapped bolus increases with increase in Gr. The effect of λ on trapping can be seen in Figure (). We notice that the size of the bolus increases with increase λ. The effect of E on trapping can be seen in figure (). We notice that the size of the bolus increases with increase in E. Figure () shows the influence of E on trapping. We observe that the size of the trapped bolus decreases with increase in E. The effect of E on trapping is shown in figure (4). It is shown that the size of the left bolus decreases where as the right bolus increases with increase in E. 64

7 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6): Gr Gr. Gr. u 4 9. u Gr Fig. Velocit distribution for different values of β with E =.7, E =.5, E =., t =.5, β =, =.5, λ = Fig. Velocit distribution for different values of Gr with E =.7, E =.5, E =., t =.5, β =, =.5, λ =.. u u 5 4 E.5 E E.5 E Fig 4. Velocit distribution for different values ofλ with E =.7, E =.5, E =., t =.5, β =, =.5, Gr =. u E. E.5 E E.5 Fig 6. Velocit distribution for different values of E with E =.7, E =., λ =., t =.5, β =, =.5, Gr = Fig 5. Velocit distribution for different values of E with E 4 =.5, E =., λ =., t =.5, β =, =.5, Gr =. E. E.5 u E E Fig 7. Velocit distribution for different values of E with E =.7, E =.5, λ =., t =.5, β =, =.5, Gr =. 65

8 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6): ?.4? Fig 8: The temperature distribution for different values of θ with?.4? E =.7, E =.5, λ =., t =.5, =.5, Gr =. (a)?.4? (b) (c) Fig 9: Effect of β on Trapping (a) β = (b) β = 4 (c) β = 8 for E =.7, E =.5, E =, λ =., t =., Gr =. 66

9 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7?.4? ?.4? (a)?.4? (b) (c) Fig : Effect of Gr on Trapping (a) Gr = (b) Gr = (c) Gr = 4 for E =.7, E =.5, E =, =., t =., =. λ β 67

10 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7? ?.4? (a)?.4? (b) Fig : Effect of (c) λ on Trapping (a) λ = ( b) λ =. (c) λ =.4 E =.7, E =.5, E =, Gr =, t =., β =. for 68

11 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7? ?.4? (a)?.4? (b) Fig : Effect of (c) E on Trapping (a) E = (b) E =.5 (c) E = for E =.5, E =, λ =., Gr =, t =., β =. 69

12 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7? ? (a)?.4? (b) Fig : Effect of (c) E on Trapping (a) E =. (b) E =.5 (c) E =.9 E =.7, E =, λ =., Gr =, t =., β =. for 7

13 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7? ?.4? (a) (b)?.4? Fig 4: Effect of (c) E on Trapping (a) E = (b) E =.5 (c) E = E =.7, E =., λ =., Gr =, t =., β =. CONCLUSION The present stud deals with the combined effect of wall properties and heat transfer on the peristaltic transport of a Jeffre fluid in a two dimensional channel. We obtained the analtical solution of the problem under long wavelength and low Renolds number assumptions. Some of the interesting findings are. The velocit increases with increase in thermal conductivit β, Grashof number Gr and the Jeffre parameter λ.. It is found that that the size of the trapped bolus increases with increase λ.. The coefficient of temperature increases with increasing values of thermal conductivit. for 7

14 M. Arun Kumar et al Adv. Appl. Sci. Res.,, 4(6):59-7 REFERENCES [] Burns JC and Parkes T, Journal of Fluid Mechanics 967, 9, [] Hanin M., Israel Journal of Technolog, 968, 6, [] Shapiro AH, Jaffrin MY and Weinberg SL, Journal of Fluid Mechanics, 969, 7, [4] Devi G and Devanathan R, Proceedings of Indian Academ Science, 975, 8A, [5] Shukla JB and Gupta SP, Journal of Biomechanical Engineering. 98, 4, [6] Srivastava LM and Srivastava VP, Journal of Biomechanics, 984, 7, [7] Usha S and Rao AR, Journal of Biomechanics, 995, 8, [8] Vajravelu K, Sreenadh S and Ramesh Babu V, International Journal of Nonlinear Mechanics, 5a, 4, 8-9. [9] Vajravelu K, Sreenadh S and Ramesh Babu V, Applied Mathematics and Computation, 5b, 69, [] Haat T and Ali N, Communication in Nonlinear Science and Numerical Simulation, 8,, 4-5. [] Haat W, Sallem N and Ali N, Communication in Nonlinear Science and Numerical Simulation, a, 5, [] Haat T, Sajjad R and Asghar S, Communication in Nonlinear science and Numerical simulation, b, 5, [] Mitra T K and Prasad S N, Journal of Biomechanics, 97, 6, [4] Misra JC and Pande SK, Mathematical and computer modelling,,, [5] Kavitha A, Hemadri Redd R, Sreenadh S, Saravana R and Srinivas ANS, Advances in applied Science Research,, (), [6] Hemadri Redd R, Kavitha A, Sreenadh S, and Hariprabhakaran P, Advances in applied Science Research,,(), [7] Lakshminaraana P, Sreenadh S and Sucharitha G, Advances in Applied Science Research,, (5), [8] Radhakrishnamachara G and Srinivasulu Ch, Computer Rendus Mecanique, 7, 5, [9] Rathod VP and Pallavi Kulkarni, Advances in applied Science Research,, (), [] Mokhtar A Abd Elnab and Haroun MH, Communication in Nonlinear Science and Numerical simulation, 8,, [] Srinivas S, Gaathri R and Kothandapani M, Computer Phsics Communications, 9, 8, 6-. [] Sreenadh S, Uma Shankar C and Raga Pallavi A, Int.J.of Appl.Math and Mech.,, 8(7), 9-8. [] Afsar khan A, Ellahi R, and Vafai.K., Advances in Mathematical Phsics,, -5. 7