Effect of an inclined magnetic field on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip conditions

This paper studies the influence of an inclined magnetic field on peristaltic transport of incompressible Bingham plastic fluid in an inclined symmetric channel with heat transfer and mass transfer. Slip conditions for heat transfer and concentration are employed. The formulation of the problem is presented through, the regular perturbation technique for small Bingham number is used to find the final expression of stream function, the flow rate, heat distribution and concentration distribution. The numerical solution of pressure rise per wave length is obtained through numerical integration because its analytical solution is impossible. Also the trapping phenomenon is analyzed. The effect of the physical parameters of the problem are discussed and illustrated graphically


Introduction
Peristaltic flow in the presence of slip conditions in comparison to non-slip conditions has an important role in many applications especially in the modern material industry (polymer industry where it is given as a microscopic wall slip), medical application (for example polishing artificial hearts),

Adnan and Hadi
Iraqi Journal of Science, 2019, Vol. 60, No. 7, pp: 1551-1574 1552 engineering, and the technological process. Its importance inspired many researchers to study the impact of slip conditions on the peristaltic flow problem for Newtonian and non-Newtonian fluids. Hayat et al. [1] analyzed the effect of an inclined magnetic field on peristaltic flow of Williamson fluid in an inclined channel with convective slip condition. Hayat et al. [2] analyzed the effect of slip conditions on peristaltic flow of Powell-Eyring fluid. Adel and Abdualhhadi [3] studied the peristaltic transport of MHD Powell-Eyring fluid through a porous medium in an asymmetric channel with slip condition. Bhatti et al. [4] investigated the simultaneous effect of slip and MHD on the peristaltic blood flow of a Jeffery fluid model in a non-uniform porous channel. Tanveer et al. [5] discussed the influence of an inclined magnetic field on the peristaltic flow of a hyperbolic tangent nonofluid in an inclined channel which has flexible walls. Adnan and Abdualhhadi [6] investigated the effect of a magnetic field on a peristaltic transport of Bingham plastic fluid in a symmetrical channel. Adnan and Abdualhhadi [7] studied the effect of the magnetic field on a peristaltic flow of Bingham plastic fluid. Ahmed and Abdualhhadi [8] investigated the effect of a magnetic field on peristaltic flow of Jeffery fluid through a porous medium in a tapered asymmetric channel.
In this study the effect of an inclined magnetic field on the peristaltic transport of Bingham plastic fluid through an inclined symmetric channel with the slip conditions of heat and concentration will be investigated. The large wave length and small Reynolds number concept are taken into considersion to simplify the problem. The regular perturbation technique for small Bingham number is used to find the final expression of stream function, heat distribution and concentration distribution. The numerical integration of pressure rise is found by using series solution. Finally the influence of different parameters on velocity axial, pressure gradient, pressure rise, the local shear stress, the temperature distribution, the concentration, and the trapping phenomenon are discussed in details with the use of graphs.

The mathematical model of the problem
Assume the peristaltic flow of incompressible Bingham plastic fluid in a two dimension tapered symmetric channel with thickness ( ) Both the magnetic field and the channel are inclined at angles and The x-axis is taken along the length of the channel and y-axis is the opposite of it [see Figure-1)]. A uniform magnetic field ( ) is applied. The flow is generated by sinusoidal waves propagating along the compliant walls of the channel. The structures of the wall geometry is described as follows Lower wall (2) In which and are the upper and lower wall respectively, is the wave amplitude, is the wave length, is the wave speed and t is the time.

The fundamental equation of the problem
The governing equation for the conservation of mass, momentum, energy and concentration for incompressible Bingham plastic fluid in an inclined symmetric channel can be written as follows: Equation of particles concentration ̅ Where is the Laplace operator. And Where ( )denotes the fluid density, is the electrical conductivity, the specific heat, the thermal diffusion ratio, the coefficient of mass diffusivity, the mean temperature, the gravity effect and the concentration susceptibility respectively. Where T is the temperature of the material. The stress tensor for Bingham plastic fluid is described as follows: And the Cauchy stress tensor denoted by ̅ ̅ ̅ ̅ ̅ where ̅ is the pressure, and ̅ the identity tensor then ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ (8) Writing the system in laboratory frame, the continuity equation can be written as: The ̅ and ̅ components of the equation of motion are respectively given by ( The associated boundary conditions are Applying the mixed condition of heat transfer and mass transfer where is the heat transfer coefficient and is the mass transfer coefficient. is the temperature at the upper wall and is the concentration at the lower wall. The flow is time dependent with respect to the laboratory frame ( ̅ ̅ ̅ ) while in the wave frame with coordinate ( ̅ ̅) moving with the wave speed the flow is considered steady. Where ̅ and ̅ are velocity components and ( ̅ ̅) the pressure in the wave frame. Writing the system equations in a wave frame needs the following transformation between the laboratory frame and wave frame. ] then the resulting system will be reduced as follows: (17) In which is called the geometric parameter (amplitude ratio) Where ( ) ( ) and is called Bingham fluid number.
Finally the dimensionless of the corresponding boundary condition Where , is the heat transfer Biot number and is the mass transfer Biot number. Defining , where is the non-dimensional time mean flow in the fixed frame and is the nondimensional time mean flow in the wave frame. And the expression of non-dimensional pressure difference per wave length is ∫ (24)

Solution to the problem
The resulting system of equations consists of highly nonlinear partial differential equations, because of its hard to find the exact solution. Laminar flow is achieved for a low Reynolds number which produces a free initial term, therefore the solution of this problem is determined by adopting this assumption. The long wave length and a low Reynolds number are widely used in the analysis of peristaltic flow. This approximation of a long wave length is based on that the assumption the wave length of the peristaltic wave is larger than the half width of the channel/ tube. Under this assumption neglecting the wave number of the equations (19), (20), (21), and (22), the system becomes Assuming the dimensionless quantities, the stream function , the flow rate , the temperature distribution and the concentration distribution will be expended by about the small values of Bingham number .
(29) Putting the above quantities (29) into the equations (25), (27) and (28), then collecting the (terms of like) power of , we obtain the following zeroth and first order systems.

Zeroth order system
The coefficients of are defined as follows ( )

First order system
The coefficients of are defined as follows

Along with the corresponding boundary conditions
Solving the above zeroth and first order forms with the corresponding boundary conditions, the final form of the stream function, heat and concentration.

Result and Discussion
In this section, the numerical and computational results are illustrated and plotted for the problem of the peristaltic transport of Bingham plastic fluid in an inclined symmetric channel with convective condition. Analytical results are shown by using the regular perturbation technique for small value of Bingham parameter . The analysis for velocity distribution, pressure gradient, pressure rise, the local shear stress, the temperature distribution, the concentration, and the trapping phenomenon for the peristaltic flow of Bingham plastic fluid in an inclined symmetric channel.

Velocity distribution
The outcomes of the axial velocity in terms of different parameters have been plotted and analyzed in this subsection, and it is clear from these graphs that the velocity profiles attain parabolic in nature except if there are some points of reflection on curves of velocity which versa the situations from the increase or decrease. Figures-(2-6) show the behavior of axial velocity with variation of inclined angle of magnetic field , the Hartman number M, amplitude ratio , the parameter of Bingham fluid , and flow rate.

Pressure gradient
The physical impact of pertinent parameters on the pressure gradient in wave are investigated through Figures-(7-14). The effect of an inclined angle of magnetic field on the pressure gradient is plotted in Figure-7). It has been seen that for large values of , the pressure gradient increases. Figure-8 portrays that ascending values of Hartman number M provide a resistance to the flow rate and the pressure gradient decreases. Figure-9 discusses the influence of increasing the pressure gradient. It shows that the pressure gradient decreases with an increase in the Bingham parameter. Figure-10 shows the effect of amplitude ratio increasing on the pressure gradient, in the vicinity of the channel walls (-1 ) ( ) the magnitude of the pressure gradient decreases but this action reverses in the central part of the channel when the pressure gradient increases. The increase of flow rate on is discussed in Figure-11. It is observed that an increase of the flow rate decreases the magnitude of the pressure gradient. Figures-(12, 13) demonstrate the physical reaction of an inclined angle of the channel and Reynolds number on the magnitude of the pressure gradient, an increase in the pressure gradient is noticed upon the increasing of and . Figure-14 shows that an increase of the Froude number decreases the pressure gradient.

Pressure rise
In this subsection, the linear relationship that relates non dimensional average pressure rise per wave length and dimensionless mean flow rate can be seen in the plots (15)-(21). The influence of various parameters on the pressure rise per wave length against were investigated. The final expression of is obtained by numerical integration of series approximation for by a Mathematica program. The pumping region is divided in to four regions 1.
The co-pumping region where 3.
The augmented region creates for 4.
The free pumping region achieved when . Figure-15 illustrates the effect of on the pumping. It has been seen for large values of that the pumping rate increase in free pumping and the augmented regions. Figure-16 deduced the pumping via variation of it has been noticed that the pumping rate in augmented region and decreases in free pumping and retrograde region. However the opposite results were noticed when the Bingham number increased in Figure-17. It is visualized from Figure-18 that an increase in magnitude of Hartmam number M increases the pumping rate in retrograde region, especially at and decreases the pumping rate in the augmented region and the free pumping region. Figures-(19, 20) show the effect of an increase of the inclined angle of the channel and the Reynolds number on the pressure rise per wave length against respectively. It is noticed that the pumping rate increases in the augmented region, free

Temperature distribution
The effect of different emerging parameters on the temperature distribution ( ) are observed physically at a fixed value of .through Figures-(27-37). It is observed that throughout all the figures, ( ) attains a maximum value near the central part of the channel. Figure-27 shows the effect of on the temperature profile. It is noted that an increase in the angle decreases the temperature profile. Figure-28 shows that the temperature profile is an increasing function with an increase of Bingham number . The variation in temperature with an increase of amplitude ratio is discussed in Figure-29. It is noted that the temperature profile increases. The influence of the flow rate on the temperature profile increase in Figure-30, where it is noted that there is an increase in the central part of the channel. The temperature profile shows an increased function for higher values of Hartman number , Dufour number, Soret number, Schmidt number, Prandtl number, and Brinkman number as illustrated in Figures-(31-36). Figure-37 is devoted to explain the influence of variations of , the heat transfer Biot number. It is observed that ( ) is decreasing with an increase of .

Trapping
An interesting topic in peristaltic transport is the phenomenon of trapping. The formation of an internally circulating bolus of fluid through closed stream lines is called trapping and this trapped bolus is moving ahead through the peristaltic waves with the same speed as the wave. This phenomenon physically