## 2 Problem Formulation  
Consider the flow of a fluid with pressure-dependent viscosity through a porous sediment of depth \(h\) inclined at angle \(\vartheta\) to the horizontal. The flow configuration is illustrated in Figure 1, which shows the orientation of the coordinate system used. It is assumed that the sediment is bounded by impermeable, solid walls, on which the no-slip condition is applied. The flow in the above domain is governed by the equation of continuity (4) and momentum Equation (13), which can be reduced to the following set of equations, wherein \(u=u(y)\) is the tangential velocity component and \(g\) is acting vertically downward:  
\[-\frac{dp}{dx}+\mu\frac{d^{2}u}{dy^{2}}+\frac{d\mu}{dy}\frac{du}{dy}+\rho gsin \vartheta-\frac{\mu}{k}u=0 \tag{14}\]  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]  
with the boundary conditions given by zero-slip on the solid boundaries \(y=0\) and \(y=h\), and a prescribed pressure at \(y=h\) (such as atmospheric pressure, say \(p_{0}\)). The boundary conditions are thus given as:  
\[u(0)=0,\ \ u(h)=0,\ \ \ p(h)=p_{0} \tag{16}\]  
Introduce the dimensionless quantities, which are defined as follows:  
\[y^{*}=\frac{y}{h};\ \ u^{*}=\frac{u}{\sqrt{gh}};\ p^{*}=\frac{p}{\rho gh};\ \mu^{*}=\frac{\mu}{\rho h\sqrt{gh}};\ k^{*}=\frac{k}{h^{2}} \tag{17}\]  
Then, introduce the boundary conditions, (16), and governing Equations (14) and (15) take the following dimensionless forms, respectively, after dropping the asterisks \({}^{a*a*}\):  
\[u(0)=0,\ \ \ u(1)=0,\ \ \ p(1)=p_{0} \tag{18}\]  
\[\mu\frac{d^{2}u}{dy^{2}}+\frac{d\mu}{dy}\frac{du}{dy}+sin\vartheta-\frac{\mu} {k}u=0 \tag{19}\]  
\[\frac{dp}{dy}=-cos\vartheta \tag{20}\]  
Figure 1: Representative sketch. We note that in (17), the velocity scale has been set to \(\sqrt{gh}\), which renders the resulting solution readily applicable as the velocity scale depends on physical parameters of the problem. Furthermore, since \(\sqrt{gh}\) is the velocity scale, \(u^{*}\) is the local Froude number, and \(\mu^{*}=\frac{1}{Ke}\), where \(Re\) is the Reynolds number. The general solution to (20) takes the following form:  
\[p=c-ycos\phi \tag{21}\]  
where \(c\) is an arbitrary constant. Using pressure condition \(p(1)=p_{0}\), we find that \(c=p_{0}+cos\phi\), and (21) takes the form  
\[p=p_{0}+(1-y)cos\phi \tag{22}\]  
In order to solve (19) for \(u(y)\), we assume that the viscosity varies with pressure. We thus let  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]0  
where \(f(p)\) is a function of \(p\) (to be specified). From (23), we obtain  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]1  
From (22), we obtain  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]2  
Through using (25) in (24), we obtain  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]3  
Through using (23) and (26) in (19), we obtain  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]4  
or  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]5  
Equation (28) is a general differential equation that governs the flow of a fluid with pressure-dependent viscosity through a porous domain down an inclined plane. Given \(f(p)\) and \(k\), specific forms of (28) are obtained. A form of interest to this work from a fundamental analysis point of view is the following:  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]6  
where \(A=e^{-\alpha}\), \(\alpha>0,\ \mu_{0}=f(p_{0})>0\) is the viscosity at \(\mathrm{y}=1\), where \(p=p_{0}>0\), and \(n\geq-1\) is an integer. Using (29) in (28), the following governing differential equation is obtained:  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]7  
Let  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]8Then, (17) can be written as follows:  
\[-\frac{dp}{dy}-\rho gcos\vartheta=0 \tag{15}\]9  
where \(B=1+C\) and \(C=\frac{cos\theta}{p_{0}}\). Using (31) and (32), Equation (30) can be written as follows:  
\[u(0)=0,\ \ u(h)=0,\ \ \ p(h)=p_{0} \tag{16}\]0  
Equation (33) is to be solved subject to no-slip conditions in (16). The solution to (33) is obtained numerically using MATLAB (version R2022a).