Influence of Varying Temperature and Concentration on Magnetohydrodynamics Peristaltic Transport for Jeffrey Fluid with a Nanoparticles Phenomenon through a Rectangular Porous Duct

A mathematical model constructed to study the combined effects of the concentration and the thermodiffusion on the nanoparticles of a Jeffrey fluid with a magnetic field effect the process of containing waves in a three-dimensional rectangular porous medium canal. Using the HPM to solve the nonlinear and coupled partial differential equations. Numerical results were obtained for temperature distribution, nanoparticles concentration, velocity, pressure rise, pressure gradient, friction force and stream function. Through the graphs, it was found that the velocity of fluid rises with the increase of a mean rate of volume flow and a magnetic parameter, while the velocity goes down with the increasing a Darcy number and lateral walls. Also, the velocity behaves strangely under the influence of the Brownian motion parameter and local nanoparticle Grashof number effect.


Introduction:
The Peristaltic flow is a mechanics for pumped fluids into tubes when the wave-out of the contraction zone or expansion spreads along an expandable and shrinking tube containing fluid. The peristalses have very important applications in many industries and physiological systems. They include the transfer of urine and food through the urinary tract and digestive system respectively, blood circulation through blood movement, the menstrual movement for the egg in the fallopian tube. The major industrial application for this phenomenon is the design of rotary pumps used in pumping fluids without contamination due to contact with pumping munitions (1). Furthermore, the peristaltic movement study has acquired many applications, for example, ship movements, mud transport, sensitive or corrosive liquids, healthy fluids, and harmful fluids in the nuclear industry. In (2) Kothandapani and Srinivas analyzed the effect of a magnetic field in the peristaltic transport for a Jeffrey fluid in an asymmetric canal and discussed the problem in wave frame moved at a stable axial velocity under the approximations of low Reynolds number and long wavelength. The influence of wall properties and heat transfer on the peristaltic transport of a Jeffrey fluid through a porous medium in the magnetic field has been investigated by Al-Khafajy and Abdulhadi in (3). The effect of lateral walls on peristaltic flow in a rectangular duct has been investigated by (4). They noted that the uterine cross-section of the uterus may be preferably approximated by a tube than a rectangular section of a two-dimensional canal. Nadeem et al. (5) analyzed a "mathematical model for the peristaltic flow of Jeffrey fluid with nanoparticles phenomenon through a rectangular duct".
Nanotechnology has basic and important applications in the modern industry in nano-size exhibit unmatched physical and chemical properties. Gasoline, oil, ethylene glycol and water are common examples of essential fluids used for liquid nanoparticles. Nano-fluids make a huge contribution to heat transfer such as fuel cells, microelectronics, hybrid engines, refrigerators, nuclear reactor radiators, space technology and many other cases. Suitable to the spacious thermal properties, nanofluid has attracted the consideration of researchers to the fabrication of heat transfer fluids in hotness exchangers, in plants and in auto cooling chillers. In literature, many researchers and industrialists are studied nanofluid and their applications, (6)(7)(8)(9)(10)(11). Nadeem and Maraj (6) described the mathematical analysis for peristaltic flow of a nanofluid in a curved channel with compliant walls. Hayat et al. (7) investigated the flow of MHD from a saturated porous space of Williamson fluid. The effect of thermal radiation on unstable free heat flow (MHD) for the rotation of Jeffrey nanofluid that passes through the porous medium was studied in (8). In (9) Latha and Kumar described the mathematical analysis to study the effects of Joule heating and Hall current by heat radiation on the peristaltic flow of a nanofluid in a channel with flexible walls. In (10) Hayat et al. investigated the magneto peristalsis of Jeffrey nanomaterial in vertical asymmetric compliant channel walls with considering nonlinear thermal radiation. In (11) Ansari et al. analyzed the problem of unstable laminar boundary layer flow, caused by propellant expansion sheet and thermal transfer to Jeffrey nanofluid.
Stimulated by this, assume a mathematical model for analyzing the combined effects of concentration and heat diffusion on nanoparticles with the influence of the magnetic field on the process of containing waves in a rectangular porous medium channel in 3D. This paper consists of five sections, the first section includes formulating the governing equations with the boundary conditions in addition to displaying the dimensionless transformations for facilitation the governing equations with assuming a very small Reynolds number or a very large wavelength to solve the problem. In the second section, the dimensionless equations are analytically solved by the HPM, the expressions are obtained for velocity profile, temperature distribution, pressure rise, pressure gradient, nanoparticles concentration and friction force. The third section includes the effects of various emerging parameters that are discussed through graphs in detail. The fourth section discusses the trapping phenomenon and the parameters that affect the increase and decrease, appear or disappear of the trapping bolus. The last section briefly reviews the most important parameters (Schmidt number, Grashof number, Prandtl number, Darcy number, magnetic parameter) that affect the movement of the fluid.

Mathematical formulation:
Considered the peristaltic flow of a non-Newtonian (Jeffrey) incompressible fluid with the concentration of nanoparticles in a cross-section of a normal rectangular three-dimensional canal (4). The flow is generated by the propagation of sinusoidal waves along the axial direction of the canal with c (constant velocity), Fig.1.

Figure 1. Schematic diagram for peristaltic flow in a rectangular duct
The peristaltic waves on the walls as represented (4): whereas a and b are wave amplitudes, t is time and X is the wave propagation direction. The walls are still parallel to XZ-plane that is unobstructed and not subject to any wave movement. Assuming that the side speed is zero as there is no change in the lateral direction of the transverse channel, that is The governing equations in three-dimensional for flow velocity of the nanofluid problem has the following form: where is the component of the velocity in Xdirection, is the component of the velocity in Zdirection, T is the temperature, C is the concentration of the fluid, is the dynamic viscosity, is the permeability, 0 is the magnetic field, is the electrical conductivity, is the thermal conductivity, is the specific heat capacity at constant pressure, is the coefficient of mass diffusivity, is the mean fluid temperature and = ( ) ( ) is the ratio of the active thermal capacity of the nanoparticle to the thermal capacity of the base fluid. Also S represents the structural relations for Jeffrey fluid (12): where 1 the ratio of relaxation to retardation times, ̇ is the shear rate, 2 is the retardation time and is the viscosity of the fluid.
To Compensate equations (10) into equations (1)-(9), and using the assumption of long-wavelength ≪1 and low Reynolds number, lead to simplifying the equations to the following form:  where represents the flow function.

Problem solving by HPM:
The solution of the nonlinear partial differential equations (11)-(15) have been found by the HPM. The deformity equations for the problem are defined as (Ji-Huan, 2010).
2 )} = 0. … (22) Here, r is the parameter included that has the range 0 ≤ ≤ 1, provided that for = 0, obtained the primary solution and for = 1, reached the final solution. Here, is the linear operator that is taken Replacing the equations (25) into equations (20)-(22) and then the similar forces are compared with , the following problems are produced with corresponding boundary conditions, i.e. For 0 : For 1 : The corresponding solutions for the above equation systems determines after three iterations and uses equations (25) as = lim →1 = 0 + 1 + 2 + ⋯ = lim →1 Θ = Θ 0 + Θ 1 + Θ 2 + ⋯ Ω = lim →1 Φ = Φ 0 + Φ 1 + Φ 2 + ⋯ Obtained: The volumetric flow rate is The average volume flow rate over one period ( = ) of the peristaltic wave is From solving equation (30) after compensating the equation (29), the pressure gradient is obtained; Numerical integration of the pressure gradient along one wave, gives us a pressure rise ∆ formula, i.e.
The dimensionless friction force F at the wall per wavelength is given by: The corresponding stream function can be obtained by integrating equation (26)  and on the velocity. It is found that the velocity profile u, achieve its maximum height at z =0, the speed of the fluid begins to increase and tends to be fixed in the walls ∓ℎ( ) as particularly at the boundary conditions.      directly proportional to the difference of but inversely related to , view (b). From the figures below, observed the movement from -h(x) to 0, the curves are declined, but as proceed, those begin to rise and get stable in h(x).   and on the friction force vs. ∅ and , respectively. Figure 12 illustrates the effect of the parameters , 1 , , , , , and on the friction force vs. ∅. Observed that the distribution of friction force gives an inverse behavior compared to the distribution of pressure rise versus the amplitude ratio factor. Also, observed that the distribution of friction force gives an inverse behavior compared to the distribution of pressure rise versus the average flow rate in Fig.  13.  .

Trapping phenomena:
The effects of ∅, 1 , , , , , , , and on trapping bolus can be seen through Figs. 14 -23. Figure 14 shows that the size of the trapped bolus grows and increases with the increasing of ∅, the effect of 1 on trapping bolus is similar to the effect of ∅ on trapping bolus which can be seen in Fig. 15. The effect of lateral walls on trapping bolus is analyzed in Fig. 16. It can be deduced that the size of the trapped bolus in the channel is contracted and decreases when increases, also at = 1.535 the upper bolus disappears while at =1.5545 the lower bolus is disappeared. Figures 17 and 18 show that the size of the trapped bolus shrinks and decreases with the increases of and , respectively. The effect of thermophoresis parameter on trapping bolus is analyzed in Fig. 19. It can be deduced that the size of the trapped bolus in the channel shrinks and decreases when increases. While in Fig motion parameter on trapping is inversion of effect of on the trapped bolus. The effect of on the trapping is analyzed in Fig. 21, note that the dose size increases and expands with an increased . The effect of on trapping analogous to effect on trapping, Observe that in Fig. 22. And the effect of on trapping bolus is analyzed in Fig. 23. It can be deduced that the size of the trapped bolus in the channel is contracted and decreases when increases, also at = 1.191 the upper bolus disappears while at = 1.241 the lower bolus disappears.

Concluding remarks:
The peristaltic flow of a nanofluid for Jeffrey fluid is deemed in a cross-section of rectangular porous medium duct to portray the mathematical results under convection is the phenomenon of heat transfer and the concentration of nanoparticles with the magnetic field. Current analysis can serve as a model that may help to understand the mechanism of physiological flows in a loop for fluids acting like nanofluids. From the mechanic's point of view, it is interesting to note how the peristaltic movement of the applied pressure gradient is affected. The exact expressions for axial velocity of the fluid, axial pressure gradient, pressure rise and stream function are obtained analytically. All governing equations are designed under the long wavelength approximation and the number of Reynolds negligible. The flow is measured in a reference frame moving at constant speed c along the axial direction of the canal. Analytical results were obtained using the HPM and all physical parameters affecting the phenomenon were discussed. The main findings can be summarized as follows: 1-The velocity is an increasing function vs.
∅, , 1 and , respectively, but decreasing 294 function vs. and both for two and three dimensional analysis. 2-The velocity is a decreasing function vs. , and , respectively, when -0.5 < z < 0.5, while the function is an increasing function when z(-1,-0.5) ∪ (0.5,1). 3-The velocity is an increasing function vs.
when -0.5 < z < 0.5, while the function is a decreasing when z ∈ (-1,-0.5) ∪ (0.5,1). 4-The temperature distribution is changing inversely vs. ∅ and , respectively. And the discussion previously mentioned that temperature curves decrease with increases in and when -1 < z < 0 while increases when 0 < z < 1, respectively. 5-The nanoparticles concentration rising up with the increase of , while it reveals opposite relation with ∅, and . 6-The pressure gradient profile displays direct relation with and , while reverses variation with 1 , , , and . Also, the pressure gradient profile directs with and reverses with in middle part of the canal, whilst in the both sides the fact is reversed. Furthermore the pressure gradient is positive in middle part of the canal, whilst negative on both sides of the canal. 7-The peristaltic pumping rate increases vs. ∅ with the increase in , 1 , , and , while decreases with the increase in , and , respectively. Moreover, observed that the relationship between the pressure rise function and the amplitude ratio parameter is a parabola. 8-The peristaltic pumping rate decreases vs. the flux , with the increase in , , and , respectively, while increases with the increases of in . Moreover, observed that the relationship between the pressure rise function and the flux is a linear. Also, it is concluded that peristaltic retrograde pumping (∆ <0) occurs when 0< <0.2, free pumping (∆ =0) occurs near =0.2 and peristaltic pumping region (∆ >0) occurs when >0.2. 9-The size of the trapped bolus is growing and increasing with the increasing of ∅, , 1 , and , respectively, while the trapped bolus is contracting and decreasing with increasing in , , , and . In general, the size of trapped bolus in upper half is greater than of lower half.