Peristaltic Slip Flow of a Viscoelastic Fluid with Heat and Mass Transfer in a Tube

The paper discusses the combined effect of slip velocity and heat and mass transfer on peristaltic flow of a viscoelastic fluid in a uniform tube. This study has numerous applications. It serves as a model for the chyme movement in the small intestine, by considering the chyme as a viscoelastic fluid. The problem is formulated and analysed using perturbation expansion in terms of the wave number as a parameter. Analytic solutions for the axial velocity component, pressure gradient, temperature distribution, and fluid concentration are derived. Also, the effects of the emerging parameters on pressure gradient, temperature distribution, concentration profiles, and trapping phenomenon are illustrated graphically and discussed in detail.


Introduction
Peristalsis is an important mechanism for mixing and transporting fluid which is generated by a progressive wave of contraction or expansion moving on the wall of the tube.It occurs widely in many biological and biomedical systems.In physiology, it plays an indispensable role in various situations.For examples, the transport of urine from kidney to the bladder, the movement of chyme in the gastrointestinal tract, transport of spermatozoa in the ducts efferentes of the male reproductive tracts, movement of ovum in the female fallopian tube, transport of lymph in the lymphatic vessels, vasomotion of small blood vessels such as arterioles, venules, and capillaries, and so on.
The peristaltic flow of non-Newtonian fluids has gained considerable interest during the recent years because of its applications in industry and biology.In biology, it is well known that most physiological fluids behave like non-Newtonian fluids.Hence, the study Mathematical Problems in Engineering of peristaltic transport of non-Newtonian fluids may help to get a better understanding for some biological systems.Now, several theoretical and numerical investigations have been carried out to understand the peristaltic mechanism in different situations.Some of the recent studies on peristaltic flow of non-Newtonian Fluids can be seen through references 1-10 .Recently, investigations of heat and mass transfer in peristalsis have been considered by some researchers due to its applications in the biomedical sciences.Srinivas and Kothandapani 11 investigated the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls.Eldabe et al. 12 studied the mixed convective heat and mass transfer in a non-Newtonian fluid at a peristaltic surface with temperature-dependent viscosity.The influence of radially varying MHD on the peristaltic flow in an annulus with heat and mass transfer has been studied by Nadeem  Problems that involve slip boundary conditions may be useful models for flows through pipes in which chemical reactions occur at the walls, flows with laminar film condensation, and certain two phase flows.Motivated by this, several studies were made to investigate the effect of slip velocity on peristaltic transport.Some of these studies have been done by Sobh 17  It is noticed from the available literature that no analysis has been made yet for the peristaltic flow of a viscoelastic fluid with heat and mass transfer in a tube in the presence of slip conditions on the tube wall.For this purpose, the peristaltic slip flow of an Oldroyd fluid, as a viscoelastic fluid, in a uniform tube is considered here in the presence of heat and mass transfer.This analysis can model movement of the chyme in the small intestine by considering chyme as an Oldroyd fluid.The flow analysis is developed in a wave frame of reference moving with the same velocity of the wave travelling down the tube wall.The perturbation technique is used to obtain an analytic solution for the governing equations in terms of the wave, Reynolds, and Weissenberg numbers.The derived solutions for pressure gradient, temperature field, and concentration profiles are plotted and analyzed in detail.The trapping phenomenon is also discussed.

Mathematical Modeling
The continuity and momentum equations for an incompressible fluid, in the absence of body forces, are given by div V 0, where ρ is the density of the fluid, V is the velocity vector, p is the pressure, τ is the extra stress tensor, and d/dt is the material time derivative.The constitutive equation for Oldroyd fluid is given by 22 in which τ ij , i, j, k 1, 2, 3 are the components of the extra stress tensor, g ii and g jj are respectively the diagonal components of covariant and contravariant metric tensor, v i are the velocity components, μ is the fluid viscosity, Γ is relaxation time, and γij are the components of strain-rate tensor.

Formulation of the Problem
Consider the peristaltic flow of an incompressible Oldroyd fluid in an axisymmetric tube of a sinusoidal wave travelling down its wall.The wall of the tube is maintained at temperature T 0 and concentration C 0 .In the fixed cylindrical coordinate system R, Z , the geometry of the problem, as can be seen in Figure 1, is where Z is the axis lies along the centreline of the tube, R is the distance measured radially, a is the radius of the tube, b is the wave amplitude, λ is the wavelength, and c is the propagation velocity.
Let us introduce a wave frame r, z moving with velocity c away from the fixed frame R, Z by the transformation where U, W , u, w are the velocity components in the fixed and wave frames, respectively.For the case of axisymmetric tube, the constitutive equations 2. ∂C ∂z where C is the concentration of the fluid, T is the temperature, T m is the temperature of the medium, D m is the coefficient of mass diffusivity, K T is the thermal diffusion ratio, μ is the viscosity, c p is the specific heat at constant volume, k is the thermal conductivity, δ is the dimensionless wave number assumed to be small, Re is the Reynolds number, Wi is the Weissenberg number, Pr is the Prandtl number, E is the Eckert number, Sr is the Soret number, Sc is the Schmidt number, and Br E Pr is the Brinkman number.
The dimensionless boundary conditions are u 0, ∂w ∂r 0, ∂T ∂r 0, ∂C ∂r 0, at r 0, where K n K n /a, is the dimensionless slip parameter.

Perturbation Solution
We begin the construction of the solution by expanding the following quantities as power series in the small parameter δ as follows: where f 2 h 0 rwdr is the dimensionless mean flow rate in the wave frame which is related with the mean flow rate in the fixed frame θ by the relation 4 Substituting the expansions 4.1 into 3.6 and 3.7 and collecting terms of like powers of δ we obtain the following systems of coupled differential equations.

4.4
The solution of 4.3 , subject to the boundary conditions 4.4 , is where a 1 , a 2 , a 3 , and a 4 are stated in the appendix.

First Order System
Substituting the zero order solution into 4.9 , we obtain τ 1 13 in the form Substituting 4.14 together with the zero order solution into 4.8 and integrating subject to the boundary condition 4.12 , taking into account that ∂p 1 /∂r 0, we obtain a differential equation for w 1 r, z in the form

4.15
The solution of 4.15 , subject to the boundary condition 4.13 is given by The results of our analysis can be expressed to first order by defining then substituting into zero and first order solutions and neglecting all terms of higher than O δ , we find w r, z 1 4

Discussion of Results
It is clear that our results allow calculation of the velocity, the pressure gradient, the temperature, and the concentration field without any restrictions on the Reynolds and Weissenberg numbers but we have used a small wave number.Moreover, we note that the approximation we have used small wave number, δ < 1 holds for our application as the values of various parameters for transporting the chyme in the small intestine are 23 .

5.1
This agrees with the small wave number approximation.In order to have an estimate of the quantitative effects of the various parameters involved in the results of the present analysis, Figures 2-23 are prepared using the MATHEMATICA package.

Pumping Characteristics
The effect of the slip parameter K n on the pressure gradient for both Newtonian and Oldroyd fluids is shown in Figures 2 and 3, respectively.It is evident that the pressure gradient decreases by increasing the slip parameter K n .Furthermore, from the two figures it can be noticed that in the wider part of the tube z ∈ 0, 0.3 and 0.6, 1 , the pressure gradient is small.This means that the flow can easily pass without imposition of a large pressure gradient.On the other hand, in the narrow part of the tube z ∈ 0.3, 0.6 , a large pressure gradient is required to maintain the flow to pass it.δ 0, 0.02, 0.04 and θ 0.1, ϕ 0.6, K n 0.05, Wi 0.04, δ 0.01, Re 0, 15, 30 , respectively.The figures reveal that the pressure gradient increases by increasing both wave number and Reynolds number.
Figure 6 depicts the effect of the Weissenberg number Wi on the pressure gradient of the viscoelastic fluid at θ 0.1, ϕ 0.6, K n 0.05, Re 10, and δ 0.02.We can conclude that an increase in the Weissenberg number decreases the pressure gradient.In Figure 9, we consider the variation of the temperature with r for z 0.2, Br 1, Pr 1, θ 0.5, ϕ 0.4, K n 0.02, δ 0.156, Re 10, and Wi 0, 0.04, 0.08 .The figure shows that an increase of the Weissenberg number lowers the temperature.

Temperature Distribution
The effects of the wave and Reynolds numbers on temperature distribution are shown in Figures 10 and 11, respectively.One can observe that the temperature profiles increase with increasing both wave and Reynolds numbers.In Figure 12, the temperature distribution is graphed versus r for z 0.2, Re 10, Pr 1, θ 0.5, ϕ 0.4, K n 0.02, Wi 0.04, and Br 0.5, 0.7, 1 .We notice that the temperature profile increases with increasing Brinkman number Br.  z 0.2, Br 1, Pr 1, θ 0.5, ϕ 0.4, Re 10, δ 0.02, Wi 0.03, Sr 0.3, Sc 0.3, and K n 0, 0.02, 0.04 .It is observed that the concentration profiles are increasing with increasing slip parameter K n .This means that the concentration for slip flow is greater than for no-slip flow.Figures 14,15,and 16 show the effects of the Weissenberg, the wave, and the Reynolds numbers on the concentration profiles.It is seen that the concentration profiles increase as the Weissenberg number increases while it decreases by increasing the wave and the Reynolds numbers.

Streamlines and Trapping Phenomenon
The phenomenon of trapping is another interesting topic in peristaltic transport.The formulation of an internally circulating bolus of the fluid by closed streamline is called  trapping.This trapped bolus is pulled ahead along with the peristaltic wave.The effect of slip parameter on trapping can be seen in Figure 20.It is observed that the trapping is symmetric about the centre line and the volume of the trapped bolus decreases with increasing K n .

Mathematical Problems in Engineering
Figures 21 and 22 illustrate the effects of the wave and Reynolds numbers on the streamline at fixed values of other parameters.It is evident that the volume of the trapped bolus increases by increasing δ and Re.Moreover, we notice from the two figures that when δ andRe increase, another trapped bolus arises.Finally, Figure 23 shows the graph of streamlines for θ 0.3, ϕ 0.3, δ 0.05, Re 1, K n 0.05, and Wi 0, 0.04, 0.08 .As shown, there is no effect of the Weissenberg number on the behaviour of streamlines.

Conclusion
The study examines the combined effect of slip velocity and heat and mass transfer on peristaltic transport of a viscoelastic fluid Oldroyd fluid in uniform tube.The problem can be considered as an application to the movement of chyme in small intestine.Using perturbation technique, analytical solutions for velocity, pressure gradient, temperature, and concentration fields have been derived without any restrictions on Reynolds number and Weissenberg number.The main results are summarized as follows.
1 The pressure gradient decreases by increasing the slip parameter K n .
2 The pressure gradient increases with increasing the wave and Reynolds numbers while it decreases with increasing the Weissenberg number.
3 The temperature profiles decrease as the slip parameter K n increases.4 The temperature profiles decrease by increasing the Weissenberg number.
6 The concentration profiles are increasing when the slip parameter and Weissenberg number increase while it is decreasing when the wave, the Reynolds, the Brinkman, the Eckert, and the Soret numbers are increasing.
7 The trapped bolus decreases with increasing slip parameter and increases with increasing wave number and Reynolds number.A.1

Appendix
and Akbar 13 .Moreover, Akbar et al. 14 investigated the effect of heat and mass transfer on the peristaltic flow of hyperbolic tangent fluid in an annulus.Moreover, Hayat et al. 15 investigated the peristaltic flow of pseudoplastic fluid under the effects of an induced magnetic field and heat and mass transfer in a channel.Furthermore, Hayat et al. 16 studied the heat and mass transfer effects on the peristaltic flow of Johnson-Segalman fluid in a curved channel with compliant walls.

Figure 1 :
Figure 1: Geometry of the problem.

Figures 7 -
are devoted to explain the effect of emerging parameters on the temperature distribution.The effect of slip parameter K n on the temperature distribution T at z 0.2, Br 1, Pr 1, θ 0.5, ϕ 0.2 is shown in Figures7 and 8for both Newtonian and Oldroyd fluid, respectively.As shown, the temperature decreases as the slip parameter increases.The two figures also reveal that the behaviour of the temperature profiles is the same for both Newtonian and Oldroyd fluids.

Figures 13 -
illustrate the behaviour of the fluid concentration for different values of the physical parameters.Figure3depicts the concentration field with the variation of r for

Figure 13 :Figure 14 :
illustrate the behaviour of the fluid concentration for different values of the physical parameters.Figure3depicts the concentration field with the variation of r for
, and 19 for different values of other physical parameters.The figures reveal that the concentration field decreases with increasing Br, Sr, and Sc.
16ing zero order solution together with the first order solution of w 1 r, z , 4.16, into 4.10 , 4.11 and applying the boundary conditions, we get − Br Re a 12 r 8 − h 8 a 13 r 6 − h 6 a 14 r 4 − h 4 a 15 r 2 − h 2 − Br Wi a 16 r 6 − h 6 a 17 r 5 − h 5 a 18 r 4 − h 4 a 19 r 3 − h 3 , Br Sr ScRe a 20 r 8 − h 8 a 21 r 6 − h 6 a 22 r 4 − h 4 a 23 r 2 − h 2 Br Sr Sc Wi a 16 r 6 − h 6 a 17 r 5 − h 5 a 18 r 4 − h 4 a 19 r 3 − h 3 , Re a 5 r 6 a 6 r 4 a 7 r 2 a 8 Wi a 9 r 4 a 10 r 2 a 11 , 4.16 where a 5 , . ..a 11 are stated in the appendix.The dimensionless mean flow rate in the wave frame f 1 is given by where a 12 , . . .a 19 are defined in the appendix.