Temperature and Concentration Effects on Oscillatory Flow for Variable Viscosity Carreau Fluid through an Inclined Porous Channel

The aim of this paper is to study the combined effects of the concentration and the thermo-diffusion on the unsteady oscillation flow of an incompressible Carreau fluid through an inclined porous channel. The temperature is assumed to affect exponentially the fluid's viscosity. We studied fluid flow in an inclined channel under the non-slip condition at the wall. We used the perturbation series method to solve the nonlinear partial differential equations. Numerical results were obtained for velocity distribution, and through the graphs, it was found that the velocity of fluid has a direct relation with Soret number, Peclet number, and Grashof number, while it has a reverse variation with chemical reaction, Schmidt number, frequency of oscillation, and Froude number.


Introduction
The oscillatory flow is very important to study blood dynamics in the arteries and veins to prevent cardiovascular diseases, as a flow of Carreau fluid in a diagonal channel indicates the nature of blood flow in the human body. Mathematical modelling of blood dynamics helps to understand this phenomenon in vivo. Theoretical and practical investigations can reveal important relationships between the oscillatory flow of the Carreau fluid in the closed duct and the movement of blood in the arteries and other blood vessels. Many researchers presented their scientific results related to ISSN: 0067-2904
The viscosity of fluids is of great importance in the fields of industrial food, medicine, and other sciences. There are many studies in the scientific literature on the movement of viscous fluids in the canal. Person [7] discussed the effect of high temperature on the flow of viscous fluids through channels. Al-Khafajy [8] developed a model that treats the oscillating flow of a viscous Jeffrey fluid through a porous channel and found that the velocity of a variable viscosity fluid is lower than that of a fluid with a fixed viscosity, under the same conditions for both cases. Peristalsis of Sisko fluids with variable viscosities was investigated by Tanveer et al. [9]. The influence of heat transfer on MHD oscillatory flow for Williamson fluid with variable viscosity through a porous medium was investigated by Khudair and Al-khafajy [4]. Increased interest has led to study the effects of temperature with the focus on fluid movement in recent years. Most research confirmed that the increase in temperature increases the velocity of the fluid, while the fluid velocity changes in an unclear manner with the difference in concentration and according to the location of the fluid in the canal [10][11][12][13]. In nature, most of the flow channels, especially in the human body, are inclined channels. For this reason, there was a great interest in the recent period on fluid flow through inclined channels, as many researchers presented mathematical models in different fluid flow values in inclined channels [13][14][15].
The present analysis aims to discuss the combined effects of the concentration and the thermodiffusion on the unsteady oscillatory flow for Carreau fluid with variable viscosity through an inclined porous channel. Such attempt has not yet been explored, to the best of our knowledge. This paper consists of five sections, the first of which is the current one "Introduction". The second section includes formulating the governing equations with the boundary conditions, in addition to the basic equation for Carreau fluid with variable viscosity. The third section includes solving the problem. We discuss the effects of parameters affecting the velocity of the fluid through graphs in the fourth section. The last section briefly reviews the most important parameters on the problem.

MATHEMATICAL FORMULATION
Consider the unsteady oscillation flow of an incompressible Carreau fluid with variable viscosity between two parallel porous plates which are inclined in an upward extending in the x-axial direction. The width (2l) is much smaller than the length of the channel so that the flow is approximately in one direction. As shown in Figure-1, the Cartesian coordinate system may be chosen for the channel, whereas x is taken as the coordinate axis parallel, whereas y is perpendicular, to the channel plates. Also, we set the no-slip boundary conditions for the movement of fluid through the channel while the two walls of the channel are held at different temperatures and (with ) and different concentrations and (with ), respectively. Moreover, there is an inclined channel making an angle with the horizontal (x-axis) and an angle with the vertical (y-axis).
The governing equations of the problem are given by: The continuity equation: (1) The momentum equations: The concentration equation: The temperature equation: ) is a radiation heat flux, is a radiation absorption, ( ), ( ), is a density, is a gravity field, is a specific heat at constant pressure, is a thermal conductivity, is a heat generation, is a coefficient of mass diffusivity and is a thermal diffusion ratio. The velocity, concentration, and temperature at the inclined walls of the channel are given as: The basic equation for Carreau fluid with variable viscosity is given as: where ̅ is extra stress tensor, ̅ is pressure, is unit tensor, is time constant, ∞ is infinite shear rate viscosity and ( ) is variable viscosity. In the case of and ∞ , we can write: The stress components are given by: We rewrite the system of a non-linear partial differential (1) -(4) as follows:

Method of Solution
The governing equations for non-dimensional conditions are: where is mean flow velocity, is Darcy number, is Reynolds number, is Peclet number, is radiation parameter.
is Schmidt number, is Soret number, is heat generation, is mean temperature, is thermal Grashof number with tilt up, is solutal Grashof number with tilt up, is chemical reaction, is Weissenberg number, and is Froude number. By substituting (13) into (8)-(12), we have the following system of the non-dimensional partial differential equations with independent variables and and dependent variables , , and .  (5) and (6)), we have:

Solution of the heat and the concentration equations
Using the separating variables method to solve the heat equation (16) with boundary conditions (19) and the concentration equation (15) with boundary conditions (18), assuming that ( ) ( ) and ( ) ( ), respectively, where is the frequency of oscillation, we obtain the solution of the heat equation and the concentration equation, which are: . where , ( ).

Solution of motion equations
To solve the momentum equation for oscillatory flow, we defined Reynold's variation for the viscosity model with temperatures as ( ) , see [8]; Let ( ) ( ) , (20) By using the Maclaurin series, we get: where is the frequency of oscillation and is a real constant. The viscosity is fixed at . By substituting equations (20) and (21) into the momentum equation (14), we get: It is difficult to solve the nonlinear differential equation (22) and thus we propose a perturbation technique method [16] with boundary conditions ( ) ( ) .

-First-order system
with boundary conditions ( ) ( ) . We note that it is difficult to solve equations (24) and (25), and therefore we use the perturbations method again with respect to the parameter by considering that it is small. So that let: Hence, the fluid velocity is given as:

Results and Discussion
We discussed the effects of temperature and concentration on the unsteady oscillation flow of an incompressible Carreau fluid with variable viscosity through an inclined porous channel. Analytical solutions are acquired for the problem by the perturbation technique up to the second-order using MATHEMATICA program. We discussed graphically all solutions obtained under variations of different relevant parameters. The results obtained are graphically presented through Figures (2-16). Figures-(2-10) illustrate the effects of the parameters , , , , , , , , , , , , , , , , and , respectively, on the velocity. It is found that the velocity profile u achieves its maximum height in the center of the channel (at y=0). The speed of the fluid begins to increase and tends to be fixed in the walls , particularly at the boundary conditions. Figure-2 illustrates the influence of the parameters and on the velocity distribution function u vs. y. It is found that the velocity profile u decreases with increasing and , respectively. Figure-3 shows that the velocity profile u rises with increasing while u decreases with increasing when and u increases with increasing when . We noted that the velocity profile u decreases with increasing while u rises with increasing , as shown in Figure-4. Figure-5 illustrates the influences of the parameters and on the velocity distribution function u vs. y. It is found that the velocity profile u rises with increasing , while u decreases with increasing when and u increases with increasing when . Figure-6 contains the velocity profile behaviour under different and . It is indicated that the velocity profile rises with increasing and when and u decreases with increasing and when . Figure-7 demonstrates that the velocity profile u decreases by increasing , while u increases with increasing when and u decreases with increasing when . We noted that the velocity profile u increases with increasing and u decreases with increasing in Figure-8. Figures-9 and 10 show that the velocity profile u rises with increasing the parameters , , and , respectively. Figures- (11)(12) illustrate the effects of the parameters , , and , respectively, on the temperature function . It is found that the temperature of fluid rises with increasing , while it decreases with increasing in Figure-11. Figure-12 demonstrates that the temperature of fluid is reduced by increasing and rises with increasing . Figures-(13-16) illustrate the effects of the parameters , , , , , , and , respectively, on the concentration function . Figures-13 and 14 indicate that the fluid concentration decreases by increasing the parameters , , and , respectively. In Figure-15, we notice that the concentration increases with the increase in and decreases with the increase in . The last Figure-16 shows that the concentration increases with the increase in and decreases with the increase in .

Concluding Remarks
We refer in this section to the most important parameters that affected the unsteady oscillation flow of an incompressible Carreau fluid through an inclined porous channel. We found the velocity function by using the perturbation technique and MATHEMATICA-12 program. We discussed graphically all solutions obtained under variations of different relevant parameters. The main findings can be summarized as follows:  The velocity flow rises with the increase of , , , , , , and , while decreases with the increase of , , , and .  The velocity is decreasing function vs. , and , respectively, when -1 < y < 0, while it is an increasing function when 0 < y < 1.  The velocity flow rises with the increase of when -1 < y < 0.15, while it decreases when 0.15 < y < 1.  The temperature of fluid rises with increasing and , while it decreases with increasing and .  The concentration of fluid rises with increasing and , while it decreases with increasing , , , , and .