Effects of Heat and Mass Transfer on MHD Peristaltic Flow of a Non-Newtonian Fluid through a Porous Medium between Two Coaxial Cylinders

We investigated the influence of heat and mass transfer on the peristaltic flow of magnetohydrodynamic Eyring-Powell fluid under lowReynolds number and long-wavelength approximation.Thefluid flows between two infinite cylinders; the inner tube is uniform, rigid, and rest, while the outer flexible tube has a sinusoidal wave traveling down its wall. The governing equations are solved numerically using finite-difference technique. The velocity, temperature, and concentration distribution are obtained. The features of flow characteristics are analyzed by plotting graphs and discussed in detail.


Introduction
Expansion and contraction of an extensible tube in a fluid generate progressive waves which propagate along the length of the tube, mixing and transporting the fluid in the direction of wave propagation.This phenomenon is known as peristalsis.It is an inherent property of many tubular organs of the human body.In some biomedical instruments, such as heart-lung machines, peristaltic motion is used to pump blood and other biological fluids [1].In addition, peristaltic pumping occurs in many practical applications involving biomedical systems.It has now been accepted that most of the physiological fluids behave like a non-Newtonian fluids.Raju and Devanathan [2] were probably the first to consider this aspect.The analysis of the mechanisms responsible for peristaltic transport for non-Newtonian fluid has been studied by many authors.
Mernone et al. [3] considered the peristaltic flow of rheologically complex physiological fluids when modelled by a non-Newtonian Casson fluid in a two-dimensional channel.Haroun [4] studied the effect of a third-order fluid on the peristaltic transport in an asymmetric channel.In his study, the wavelength of the peristaltic waves is assumed to be large compared to the varying channel width, whereas the wave amplitudes need not be small compared to the varying channel width.Eldabe et al. [5] analyzed the incompressible flow of electrically conducting biviscosity fluid through an axisymmetric nonuniform tube with a sinusoidal wave under the considerations of long wavelength and low Reynolds number.The effect of rotation, porous medium, and magnetic field on peristaltic transport of a Jeffrey fluid in tube is studied by Mahmoud [6].Another work dealing with non-Newtonian peristaltic flow is mentioned in [7][8][9][10][11].
In all the abovementioned studies, no porous medium has been taken into account.But it is well known that flow through a porous medium has practical applications especially in geophysical fluid dynamics.Examples of natural porous media are beach sand, sandstone, limestone, rye bread, wood, the human lung, bile duct, gall bladder with stones and in small blood vessels.In the arterial systems of humans or animals, it is quite common to find localized narrowings, commonly caused by in travascular plaques [12].Afifi [13] was the first who studied the peristaltic flow through a porous medium.The peristaltic motion of a generalized Newtonian fluid through a porous medium is studied by Elshehawey et al. [14].Mekheimer [15] studied the nonlinear peristaltic transport through a porous medium in an inclined planar channel taking the gravity effect on pumping characteristics.Such analysis is of great value in medical research.Hayat et al. [16] investigated the peristaltic transport of electrically conducting Maxwell fluid through a porous medium in a planar channel.
Magnetohydrodynamics (MHD) is the science which deals with the motion of a highly conducting fluids in the presence of a magnetic field.The motion of the conducting fluid across the magnetic field generates electric currents which change the magnetic field, and the action of the magnetic field on these currents gives rise to mechanical forces which modify the flow of the fluid [17].The study of peristaltic flow with heat and mass transfer for an electrically conducting fluid past a porous plate under the influence of a magnetic field has attracted the interest of many investigators in view of its applications in many engineering problems such as MHD generator, plasma studies, nuclear reactors, oil exploration, geothermal energy extractions, and the boundary layer control in the field of aerodynamics [18].Vajravelu et al. [19] examined the flow through vertical porous tube with peristalsis and heat transfer.Srinivas and Kothandapani [20] have studied the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls.The peristaltic transport of Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium is studied by Srinivas and Gayathri [21].For the benefit of the readers in this direction you can take these additional papers ( [22][23][24][25]) into account.
With the above discussion in mind, we propose to study the fluid mechanics effects of MHD peristaltic transport through a porous media in a gap between two coaxial tubes, filled with an incompressible non-Newtonian (Eyring-Powell) fluid.The effects of body temperature and concentration are taken into consideration.The inner tube is rigid and the outer one has a sinusoidal wave traveling down its wall.The flow analysis is developed in a wave frame of reference moving with the velocity of the wave.Our problem can be considered as a mathematical model for the blood flow with catheterization.The problem is solved numerically for the velocity, temperature, and concentration.The effects of various emerging parameters on the flow, temperature, and concentration distributions are shown and discussed with the help of graphs.

Formulation of the Problem
Let us consider the MHD flow with heat and mass transfer of an incompressible electrically conducting non-Newtonian obeying Eyring-Powell model between two coaxial tubes through a porous medium.The inner tube is rigid and uniform, while the outer tube has a sinusoidal wave traveling down its wall.We use cylindrical coordinate system (, , ) with  in the radial direction,  lies along the center line of the inner and outer tubes, the geometry of the wall surfaces is described in (Figure 1), and the equations for the radii are in which  signifies the radius of the outer tube,  indicates the wave amplitude,  is the wavelength,  is the propagation velocity along  direction, and  is the time.The fluid is subjected to a constant magnetic field  0 that acts in  direction.Induced magnetic field and external electric field are neglected.The equations that govern the flow are the balance of mass: the equation of momentum: Maxwell's equations: the equation of energy: and the equation of concentration: The stress is defined by where  is the density,  is the fluid velocity and  are the pressure,  is the stress tensor,  is the plastic viscosity of the fluid,   is the permeability of the porous medium,  is total magnetic field vector,  is current density vector,  is the fluid temperature,  is the fluid concentration,   is the specific heat at constant pressure,   is the thermal conductivity of the fluid,   is the coefficient of mass diffusivity,   is the thermal diffusion ratio,   and   are the mean fluid temperature and concentration,   is the concentration susceptibility,   is the radiative heat flux,  is the electric field vector,  is the electrical conductivity of the fluid,   is the magnetic permeability,  is the reaction rate constant,  is the reaction order, and  1 and  2 are characteristics of Eyring-Powell model.
Since the flow parameters are independent of the azimuthal coordinate ; hence, the velocity is given by  = (, 0, ).A uniform magnetic field  = (0,  0 , 0) is applied in a direction perpendicular to the fluid flow.The governing equations ( 2 The boundary conditions are given by where   and   are the uniform temperature and concentration at the outer tube.
The appropriate nondimensional variables for the flow are defined as In terms of these variables and dropping the star mark for simplicity, (8) becomes Here, Re = / is Reynolds number, Da = / 2 is Darcy number,  =  2 0  2 / is the magnetic field parameter, Thus, the boundary conditions (9) in their dimensionless form read After applying the long-wavelength approximation to our analysis as described in [5], then the field (11) now gives Equation ( 14) indicates that  is not function of .Hence  is function of  and  only.

Method of Solution
We shall solve the system of non-linear ordinary differential equations numerically using the finite difference technique which is discussed by Soundalgekar and Ganesan [26], and so the system of ( 15)-( 17) yields The boundary conditions (12) transform into

Results and Discussion
To  2, the effect of Pr on the temperature distribution  is indicated, and this figure shows that the temperature increases with the increasing of Pr.Also the relation between  and  is approximately linear for small values of Pr (<0.7), but for large values of Pr (>0.7),  increases with increasing  till a finite value  (maximum value), after which  decreases with increasing .The effect of  on the temperature distribution  is shown in Figure 3; it is clear from this figure that the temperature decreases with increasing .It is also noted that the temperature increases with increasing  and reaches a maximum value (at a finite value of :  = 0.85) after which it decreases.Figure 4 gives the effect of Ec on the temperature distribution ; from this figure, we observe that  increases with increasing Ec, and although there is a small difference between the values of Ec, there is a remarkable difference between the four curves of , which were taken at Ec = 0.2, 0.3, 0.4, and 0.5 when the other parameters are constant, that is, Ec affects  strongly.The effect of Da on the temperature distribution is described in Figure 5, it is clear that the temperature increases with increasing Da but the effect of Da on  disappears at the range 0.96 <  < 1.Also, it is clear that the relation between  and  is approximately linear for small values of Da but for large values of Da there is a normal curves.Figure 6 shows the effect of Df on the temperature distribution, and it is seen from this figure that the temperature increases with increasing Df, and, in all curves, the maximum value of  occurs at 0.85 <  < 0.9, and it increases as  increases.In Figure 7, the relation between  and  for various values of  is obtained.From this figure, it is clear that the temperature increases with increasing , and, from this figure, we can see that although there is a big difference between the values of , there is a very small difference between the four curves of , which were taken at  = 0.1, 100, 200, and 300 when the other parameters are constant.Figure 8 shows the effect of Sr on the temperature distribution; it is noted that the temperature increases with increasing Sr.It is also noted that the effect of Sr on  disappears near the boundaries of the tubes and all curves have a maximum value of  in a finite range of  namely 0.88 <  < 0.9.Figures 9 and 10 show that the effect of both Sc and  on the temperature distribution  is similar to the effect of Sr on  given in Figure 8.The effect of / on the temperature distribution is indicated in Figure 11.It is seen from this figure that the temperature decreases with increasing /, and all carves have a maximum value as in Figure 8.
Figures 12-21 show the effect of the various parameters on the concentration distribution.Figure 12 gives the effect of Pr on the concentration distribution.This figure shows that the concentration decreases with the increase of Pr.Also Pr affects the relation between  and , and this relation is approximately linear at the values 0.7 and 1 of Pr, but, for large values of Pr (≥3),  decreases with increasing  till a finite value of  (minimum values), after which  increases with increasing . Figure 13 shows the effect of  on the concentration distribution, it is observed that the concentration increases with increasing  and all curves have a minimum value at the range 0.98 <  < 1.The effect of Ec on the concentration distribution is illustrated in Figure 14, and we can see from this figure that although there is a small difference between the values of Ec, there is a remarkable difference between the four curves of .  for small values of Da, but for large values of Da, there are normal curves.Figure 16 shows the effect of Df on the concentration distribution.It is seen from this figure that the concentration decreases with increasing Df.Also, the concentration  decreases with increasing  in the range from 0 to 0.7 but after which it increases with increasing . that the effect of  on  is similar to the effect of Df on , with the only difference that the obtained curve is closer to those obtained in Figure 16.The effect of Sr on the concentration distribution is shown in Figure 18; from this figure, we can see that  decreases with increasing Sr, and, from this figure, we can see that the effect of Sr on  disappears in the range 0.98 <  < 1.Also, the relation between  and  is approximately linear at the value namely, 1.5 of Sr.All the curves of  were taken at Sr = 1.5, 1.8, 2, and 2.5 when the other parameters are constant.Figure 19 gives the effect of Sc on the concentration distribution, and it is clear that  decreases with increasing Sc; also Sc affects the relation between  and .This relation is approximately linear for small values of Sc, but, for large values of Sc,  decreases with increasing  till a finite value of  (minimum value), after which it increases.Figure 20 describes the relation between  and  for various values of .It is clear from this figure that the concentration decreases with increasing .From all curves in this figure, we obtained that the effect of  on  disappears at the range 0.5 <  < 0.55 and 0.96 <  < 1.All curves were taken at  = 0.1, 0.3, 0.5, and 0.9 when the other parameters are constant.Figure 21  shows the effect of / on the concentration distribution.The calculations were obtained at / = −35, −30, −25, and −20 when the other parameters are constant. increases with increasing /.It is also noted from all curves in this figure that the concentration decreases by increasing  at the range 0.5 <  < 0.65 but after that it increases with increasing .
Figures 22-25 show the effect of the various parameters on the velocity distribution.Figure 22 gives the effect of  on the velocity distribution.This figure shows that the velocity decreases by increasing  at the range 0.5 <  < 0.84 and after that the velocity increases by increasing .Also, it is clear that, in all curves in this figure, the maximum value of  occurs at 0.65 <  < 0.7. Figure 23 illustrates the effect of Da on the velocity distribution.It is clear that  increases by increasing Da in the range 0.5 <  < 0.84 and  decreases by increasing Da in the range 0.84 <  < 1.Also, from this figure, we can see that although there is a big difference between the values of Da, there is a small difference between the four curves of  which were taken at Da = 0.005, 0.01, 0.09, and 0.5, when the other parameters are constant.Figure 24 describes the relation between  and  for various values of .From this figure, it is clear that the velocity decreases with increasing , but the effect of  on  disappears in the range 0.96 <  < 1.All curves in this figure were taken at  = 0.1, 0.3, 0.5, and 0.9 when the other parameters are constant.Figure 25 shows the effect of / on the velocity distribution.The calculation was obtained at / = −35, −30, −25, and −20, and the other parameters are constant.The velocity decreases with increasing / and in all curves the maximum value of  occurs in the range 0.65 <  < 0.7.

Conclusion
In this paper, we studied the effects of the physical parameters of the considered problem on peristaltic transport in the gap between two coaxial tubes, filled with an incompressible non-Newtonian (Eyring-Powell) fluid, and considered the effects of body temperature and concentration.The inner tube is rigid and the outer one has a sinusoidal wave traveling down its wall.The present analysis can serve in understanding the mechanism of many physiological flows.Physically, our model corresponds to the transport of the gastric juice in the small intestine when an endoscope is inserted through it.The system is solved numerically by finite difference technique.The effects of various emerging parameters on the flow, the temperature, and the concentration distributions are shown and discussed with the help of graphs.The main findings can be summarized as follows.
(1) The velocity  rise decreases with the increase of each of ,  and /, whereas it increases as Da increases.But, all the other parameters such as, Pr, Ec, Df, , Sr, and Sc do not affect on the velocity (the flow of fluid).
(2) The temperature  rise increases with the increase of each of Pr, Ec, Da, Df, , Sr, Sc, and , whereas it decreases as  and / increase.
(3) The concentration  rise increases with the increase each of  and /, whereas it decreases as Pr, Ec, Da, Df, , Sr, Sc, and  increase.
(4) The temperature has the opposite behavior compared to the concentration rise behavior at all different parameters in this problem.

Applications
(1) Gastrointestinal motility is the function of gastrointestinal smooth muscle.It is controlled by both the intrinsic and extrinsic nerves of the gastrointestinal tract and, to a lesser degree, the gastrointestinal hormones.Therefore, any abnormality of the above factors, theoretically, can cause gastrointestinal dysmotility.In a clinical situation, commonly seen is gastrointestinal dysmotility caused by either smooth muscle or intrinsic and extrinsic nerves dysfunction.Diseases that cause smooth muscle dysfunction include familial visceral myopathies, nonfamilial visceral myopathies, collagen disease, muscular dystrophies, amyloidosis, and thyroid disease.Diseases that cause enteric nerve dysfunction include familial visceral neuropathies, nonfamilial visceral neuropathies, diabetes mellitus, Chagas' disease, ganglioneuromatosis of the intestine, visceral neuropathy of carcinomatosis, Parkinson's disease.
(2) Gastroparesis syndrome is a recognized complication of longstanding diabetes mellitus and is attributed to reduced gastric contractility due to "autovagotomy." However, motor abnormalities associated with this syndrome may not be limited to the stomach.Abnormal intestinal manometric patterns were observed in twelve out of the fourteen patients.
Our findings show that the small intestine is frequently affected in patients with diabetic gastroparesis, and that the motility disorder both in the stomach and the small bowel is not invariably of a "paretic" type.The occurrence of incoordinated intestinal long bursts and continuous antral activity suggests that disturbed sympathetic innervations participate in the aetiopathogenesis of their upper-gut dysfunction.
(19)uss the effects of various parameters involved in the problem such as Prandtl number Pr, magnetic field parameter , Eckert number Ec , Darcy number Da, Dufour number Df, reaction rate parameter , Soret number Sr, Schmidt number Sc, Eyring Powell parameter , and the pressure gradient /, on the solution of the considered problem, numerical results are calculated by solving(18)with boundary conditions(19)using finite difference technique.The distributions of axial velocity , the temperature , and the concentration  are illustrated in Figures2-25.The values of  are plotted versus the radial coordinate in Figures2-11.In Figure