The Effect of Wall'S Porous Liner on MHD Couette Flow of Carreau Fluid in an Inclined Channel under the Convective Conditions

Due to its usefulness in numerous fields of industry and technology, non-Newtonian fluids have recently attracted a lot of attention from the scientific community. Blood, honey, shampoo, ketchup, paints, mud, and polymer melts are all examples of non-Newtonian fluids since their viscosity varies with the applied force. Because of their complexity, non-Newtonian fluids can be categorized into two types: those whose shear stress depends solely on the shear rate and those with both elastic and viscous properties. Therefore, the expected behavior of this fluid type is described by many fluids. Many researchers have dealt with various types of non-Newtonian fluids for different flows and using analytical and numerical methods [1-4]. One such model is the Carreau fluid, a generalized Newtonian fluid whose viscosity varies with the shear rate; this model is particularly well-suited to representing fluid movement in the high-shear-rate regime. Carreau pioneered the use of molecular network theory in theological equations [5]. Because of these uses, Carreau fluid flow has been the subject of much research in a large number of different geometries. It is common knowledge that fluid dynamics when surrounded by a magnetic field, play an important role in numerous scientific and technological procedures. (MHD) pumps, accelerators, power generators, aerodynamic heating, and polymer technologies are all examples of these uses. Pump meters, cell separators, heat exchangers, magnetic drugs, and magnetic tracers used for precise homing all make use of the principles of magnetic field engineering (MHD) [6]. A multitude of researchers have looked at MHD. in various fluid flows in light of these potential uses. Akbar et al. [7] a conduit allows for the passage of an electrically conducting viscoelastic fluid as well as the transmission of heat. has been theoretically investigated (MHD). Hayat et al. [8] provide research on the impact of a generated magnetic field on the peristaltic movement of Carreau fluid. Prakash et al. [9] investigated the waveform transport of Carreau fluid in an asymmetric tapered channel using the effects of a uniform magnetic field. The peristaltic movement of a Carreau fluid through a non-uniform conduit is shown to be affected by an angled magnetic field by Ahmed [10], for more references on this side [11-15]. Fluid cooling and heating are common tasks in many manufacturing and chemical processes. Extrusion is a common procedure used in the plastics and food processing industries. The rate of heat transport and the distribution of temperatures must be understood with thermally sensitive goods like food and pharmaceutical products, and the maximum allowable temperature must not be raised. In the polymer business, knowing the rate of heat transmission and temperature distribution profile is crucial since it has a major impact on the uniformity of the finished products [16, 17]. The appropriate pumping of groundwater can help alleviate water scarcity issues, and this is where the research of input through a porous media comes in. Biologists have found the flow of water through porous plant material to be fascinating. Channebasappa et al. [18] studied how the measurement of the porous material's thickness affected the flow in a canal with parallel plates. When a porous coating that is impervious to erosion is installed in the channel walls. Similar to what Kumaraswamy Naidu and others have done [19] was looked at. Jeffrey fluid speed in a parallel plate channel lined with an absorbent material that doesn't erode depends on the depth of the pore structure [20-22]. Couette flow, one of the fundamental flows, has piqued the interest of fluid dynamics experts. As a result of its potential utility in a variety of technological and industrial contexts. What we have here is the movement of two surfaces separated by a viscous fluid, with one of those surfaces moving tangentially with respect to the other. Shear stress is applied to the fluid and produces flow as a result of the motion of the surfaces. Additionally, there may be a pressure gradient that is applied in the direction of flow, albeit this depends on the precise meaning of the term Injection molding [23]. The manufacturing processes under consideration include flow molding, blow molding, and continuous casting., forming plastic, extraction, asthenosphere, and extrusion flows are all examples of one-directional flows utilized in a variety of industries [24]. The governing differential equations for such a flow may also yield accurate solutions, which can be used to complement the outcomes of the flow issue when dealing with complex geometries. You can see further reading in this area [25, 26]. For more references in this field can see the studies of Nadeem et al. [27], Rickert et al. [28], and Niranjana et al. [29]. This work presents the results of a theoretical investigation into the impact or influence of an inclined magnetic field on the Couette steady flow of conducting Carreau fluid in a sloping channel made up of parallel plates. Flow modification due to slip boundary circumstances. The use of convective boundary conditions for heat transport aids in this investigation. By using approximation and numerical techniques, the governing equations of motion and energy can be solved. What follows are the remaining sections of this paper: The mathematical formulation of the problem is presented in section (2). In section (3), we examine the system of motion and energy equations that control the situation. Shear stress constitutive equations for Carreau fluid are presented in section (4). The solutions to the problems are shown in section (5). Sections (6) and (7) analyze the numerical findings of the parameters and the conclusions.

The governing equations for the flow in the fixed frame are:

Continuity equation:

$\frac{\partial \overline{u}}{\partial \overline{x}} =0$           (3)

$\begin{align}  & 0=-\frac{\partial \overline{P}}{\partial \overline{X}}+\frac{\partial }{\partial \overline{Y}}{{\tau }_{\overline{XY}}}-\sigma B_{0}^{2}{{\cos }^{2}}(\beta )\overline{u}+\rho g\sin \alpha   \\ & \rho {{c}_{p}}(\frac{\partial \overline{T}}{\partial \overline{t}}+\overline{u}\frac{\partial \overline{T}}{\partial \overline{X}})={{K}_{1}}(\frac{{{\partial }^{2}}\overline{T}}{\partial {{\overline{Y}}^{2}}})+{{\tau }_{\overline{XY}}}\frac{\partial \overline{u}}{\partial \overline{Y}}+ \\\end{align}$                  (4)

$\sigma B_{0}^{2}{{\cos }^{2}}(\beta ){{(\overline{u} )}^{2}}-\frac{\partial }{\partial \overline{Y}}\overline{qr}$               (5)

in which, $\frac{\partial}{\partial \overline{\bar{Y}}} \overline{q r}=4 \lambda^2\left(T_0-T\right)$.

Considering the following problem-specific border conditions:

$\begin{align}  & \overline{u}={{u}_{0}}              at  \overline{Y}=h     & \frac{\partial \overline{u}}{\partial \overline{Y}}=\frac{{{\alpha }_{1}}}{\sqrt{{{k}_{0}}}}\overline{u}    at  \overline{Y}=-h  \end{align}$                       (6)

$\begin{align}  & -{{k}_{r}}\frac{\partial T}{\partial \overline{Y}}={{h}_{r}}({{T}_{0}}-T)              at  \overline{Y}=-h   & {{k}_{r}}\frac{\partial T}{\partial \overline{Y}}={{h}_{r}}(T-{{T}_{1}})              at  \overline{Y}=h   \end{align}$                  (7)

$\begin{align}  & u=\frac{\overline{u}}{{{u}_{0}}},x=\frac{\overline{X}}{h},y=\frac{\overline{Y}}{h},  p=\frac{h\overline{P}}{{{\mu }_{0}}{{u}_{0}}}, \\ & We=\frac{\Gamma {{u}_{0}}}{h} ,Dr=\frac{h}{\sqrt{{{k}_{0}}}}=\frac{1}{Da},t=\frac{\overline{t}{{u}_{0}}}{h}, \\ & \operatorname{Re}=\frac{\rho h{{u}_{0}}}{{{\mu }_{0}}},M=\sqrt{\frac{\sigma }{{{\mu }_{0}}}}{{B}_{0}}h , \theta =\frac{T-{{T}_{0}}}{{{T}_{1}}-{{T}_{0}}} , \\ & Fr=\frac{u_{0}^{2}}{gh} , {{\rho }_{r}}=\frac{{{\mu }_{0}}c\rho }{{{k}_{1}}},  {{\tau }_{\overline{XY}}}={{\overline{\tau }}_{\overline{YX}}} \\ & =\frac{{{\mu }_{0}}{{u}_{0}}}{h}{{\tau }_{XY}},\overset{\bullet }{\mathop{y}}\,=\frac{h}{{{u}_{0}}}\overline{\overset{\bullet }{\mathop{y}}\,}Ec=\frac{{{u}_{0}}^{2}}{c\rho ({{T}_{1}}-{{T}_{0}})}, \\ & \Gamma =\frac{\operatorname{Re}}{Fr},Bh=\frac{hrh}{kr},Br=\rho rEc=\frac{{{\mu }_{0}}{{u}_{0}}^{2}}{{{k}_{1}}({{T}_{1}}-{{T}_{0}})}, \\ & Nr=\frac{2\lambda h}{\sqrt{{{k}_{1}}}} \\\end{align}$              (8)

In the following table, we are given the physical nomenclature of the used parameters.

5.1 Solution of moment equation

It's obvious that the produced formula for motion (Eq. (22)) is not linear, so it is difficult to find the solution exactly. Thus, we have to solve the equation approximately by using (RPTM), and we obtain a series of solutions for the axial velocity. We expand u for small Weissenberg numbers, the following is true:

$u={{u}_{0}}+{{(we)}^{2}}{{u}_{1}}+O({{(we)}^{2}})$                   (23)

Substituting Eq. (23) into Eq. (22) and boundary conditions (16), the following ranking of systems is obtained.

5.1.1 The zero-based ordering system $w e^{(0)}$

$0=\frac{{{\partial }^{2}}{{u}_{0}}}{\partial {{y}^{2}}}-{{M}^{2}}{{\cos }^{2}}(\beta ){{u}_{0}}+\Gamma \sin \alpha $                   (24)

Together with the related boundary conditions:

$ \left. \begin{align}  & {{u}_{0}}=1 ,  at\left( y=1 \right)  \\ & \frac{\partial {{u}_{0}}}{\partial y}={{\alpha }_{1}}Dr{{u}_{0}} ,   at\left( y=-1 \right) \\\end{align} \right]$                       (25)

5.1.2 System of order two $\left(w e^{(2)}\right)$

$\begin{align}  & 0=\frac{{{\partial }^{2}}{{u}_{1}}}{\partial {{y}^{2}}}+3(\frac{n-1}{2}){{(\frac{\partial {{u}_{0}}}{\partial y})}^{2}}.\frac{{{\partial }^{2}}{{u}_{0}}}{\partial {{y}^{2}}}- \\ & {{M}^{2}}{{\cos }^{2}}(\beta ){{u}_{1}}                                 \\\end{align}$                          (26)

$ \left. \begin{align}  & {{u}_{1}}=1 ,  at\left( y=1 \right)  \\ & \frac{\partial {{u}_{1}}}{\partial y}={{\alpha }_{1}}Dr{{u}_{1}} ,   at\left( y=-1 \right) \\\end{align} \right]$                     (27)

5.1.3 Solution of order zero

The solution of Eq. (24) subject to the corresponding boundary conditions (25) is taken the form:

${{\text{u}}_{0}}={{\text{e}}^{{{\text{N}}_{1}}y}}{{\text{a}}_{1}}+{{\text{e}}^{-{{\text{N}}_{1}}y}}{{\text{a}}_{2}}+\frac{\Gamma \text{Sin}\left[ \alpha  \right]}{{{\text{N}}_{1}}^{2}}; $                        (28)

5.1.4 Solution of order two

The solution of Eq. (26) subject to the corresponding boundary conditions (27) is taken the form:

$\begin{align}  & {{\text{u}}_{1}}=\frac{3}{16}{{\text{a}}_{2}}^{3}{{\text{e}}^{-3{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2}+\frac{3}{8}{{\text{a}}_{1}}{{\text{a}}_{2}}^{2}{{\text{e}}^{-{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2}+\frac{3}{8}{{\text{a}}_{1}}^{2}{{\text{a}}_{2}}{{\text{e}}^{{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2} \\ & +\frac{3}{16}{{\text{a}}_{1}}^{3}{{\text{e}}^{3{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2}-\frac{3}{16}{{\text{a}}_{2}}^{3}{{\text{e}}^{-3{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{2}-\frac{3}{8}{{\text{a}}_{1}}{{\text{a}}_{2}}^{2}{{\text{e}}^{-{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{2} \\ & -\frac{3}{8}{{\text{a}}_{1}}^{2}{{\text{a}}_{2}}{{\text{e}}^{{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{2}-\frac{3}{16}{{\text{a}}_{1}}^{3}{{\text{e}}^{3{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{2}+\frac{3}{4}{{\text{a}}_{1}}{{\text{a}}_{2}}^{2}{{\text{e}}^{-{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{3}y \\ & -\frac{3}{4}{{\text{a}}_{1}}^{2}{{\text{a}}_{2}}{{\text{e}}^{{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{3}y-\frac{3}{4}{{\text{a}}_{1}}{{\text{a}}_{2}}^{2}{{\text{e}}^{-{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{3}y+\frac{3}{4}{{\text{a}}_{1}}^{2}{{\text{a}}_{2}}{{\text{e}}^{{{\text{N}}_{1}}y}}n \\ & {{\text{N}}_{1}}^{3}y+{{\text{e}}^{{{\text{N}}_{1}}y}}{{\text{b}}_{1}}+{{\text{e}}^{-{{\text{N}}_{1}}y}}{{\text{b}}_{2}};    \\\end{align}$                        (29)

where, b_j(j=1,2) an optional constant can be found by using boundary conditions (27), therefore, the solution of fluid's velocity can be obtained by:

$\begin{align}  & u={{a}_{2}}{{\text{e}}^{-{{\text{N}}_{1}}y}}+{{a}_{1}}{{\text{e}}^{N1y}}+w{{e}^{2}}({{b}_{2}}{{\text{e}}^{-{{\text{N}}_{1}}y}}+{{b}_{1}}{{\text{e}}^{N1y}}+\frac{3}{16} \\ & {{a}_{2}}^{3}{{\text{e}}^{-3{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2}+\frac{3}{8}{{a}_{1}}{{a}_{2}}^{2}{{\text{e}}^{-{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2}+\frac{3}{8}{{a}_{1}}^{2}{{a}_{2}}{{\text{e}}^{{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2}+ \\ & \frac{3}{16}{{a}_{1}}^{3}{{\text{e}}^{3{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{2}-\frac{3}{16}{{a}_{2}}^{3}{{\text{e}}^{-3{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{2}-\frac{3}{8}{{a}_{1}}{{a}_{2}}^{2}{{\text{e}}^{-{{\text{N}}_{1}}y}} \\ & n{{\text{N}}_{1}}^{2}-\frac{3}{8}{{a}_{1}}^{2}{{a}_{2}}{{\text{e}}^{{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{2}-\frac{3}{16}{{a}_{1}}^{3}{{\text{e}}^{3{{\text{N}}_{1}}y}}n{{\text{N}}_{1}}^{2}+\frac{3}{4}{{a}_{1}}{{a}_{2}}^{2} \\ & {{\text{e}}^{-{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{3}y-\frac{3}{4}{{a}_{1}}^{2}{{a}_{2}}{{\text{e}}^{{{\text{N}}_{1}}y}}{{\text{N}}_{1}}^{3}y-\frac{3}{4}{{a}_{1}}{{a}_{2}}^{2}{{\text{e}}^{-N1y}}n{{\text{N}}_{1}}^{3}y \\ & +\frac{3}{4}{{a}_{1}}^{2}{{a}_{2}}{{\text{e}}^{N1y}}n{{\text{N}}_{1}}^{3}y)+{{n}_{1}}^{2}\Gamma Sin[\alpha ];    \\\end{align}$                         (30)

5.2 Thermodynamic equation and its solution

The solution of the energy equation (Eq. (20)) can be solved numerically using the method of "Explicit Euler" with size step by (0.1) by using the Mathematica program. The numerical results have mentioned in Tables 1-11 for different types of parameters.

Table 1. Variation of temperature θ with y for different values of M

Table 2. Variation of temperature θ with y for different values of Nr

Table 3. Variation of temperature θ with y for different values of β

Table 4. variation of temperature θ with y for different values of α

Table 5. Variation of temperature θ with y for different values of Da

Table 6. Variation of temperature θ with y for different values of α₁

Table 7. Variation of temperature θ with y for different values of Br

Table 8. Variation of temperature θ with y for different values of Bh

Table 9. Variation of temperature θ with y for different values of Γ

Table 10. Variation of temperature θ with y for different values of we

Table 11. Variation of temperature $\theta$ with y for different values of n

The present study focuses on the investigation of a fundamental flow known as Couette flow. This flow involves an incompressible MHD Carreau fluid that is confined between parallel plates within an inclined channel. Notably, the bottom section of the channel is lined with a porous material. In this study, the slip boundary conditions on the flow are carefully considered. The present study also includes an examination of heat transfer, wherein convective boundary conditions are employed. The primary focus of this investigation revolves around identifying the key issues associated with the topic at hand.

1. Velocity is an increasing function of the variables (β, Γ, α) while the situation is covered by the variables (M, n, α₁, Dr, we).
2. The temperature profile is an increasing function of the variables (M, Γ, α, Dr, Br, α₁, n, Nr) and the trend is reversed of the variables (β, we, Bh).
3. The findings of the Newtonian model of fluids are captured from our analysis by putting (n=1 or we=0).