## 1 Introduction
However, this does not explicitly take into account spatial variations of viscosity or viscosity variations due to changes in pressure, which is a function of position (cf. [12] and the references therein). When the flow is taken through a porous medium, Navier-Stokes equations are microscopically valid in the pore space. However, they are hard to track due to the complexity of the pore space. Furthermore, changes in flow quantities at the microscopic scale are at times insignificant at the macroscopic scale. The equations governing the flow of a fluid with pressure-dependent viscosity through a porous structure are the continuity equation, of the form of (4), and the momentum equation proposed by [PERSON] et al. [10], referred to as the generalised Brinkman's equation. For the purposes of this work, the generalised Brinkman's equation is conveniently written in the following form:  
\[\rho\Big{(}\overset{\rightarrow}{u}\cdot\
abla\Big{)}\overset{ \rightarrow}{u}=-\
abla p+\
abla\cdot\mu\Big{(}\
abla\overset{ \rightarrow}{u}+(\
abla u)^{T}\Big{)}-\Gamma(p)\overset{\rightarrow}{u}+ \rho\overset{\rightarrow}{g} \tag{8}\]  
where \(\overset{\rightarrow}{u}\) in (4) and (8) now stands for the Brinkman velocity vector, \(\mu=\mu(p)\), and \(\Gamma(p)\) is the pressure-dependent drag coefficient. Both \(\mu(p)\) and \(\Gamma(p)\) control variations in viscosity due to pressure and variations in pressure due to variations in porous parameters. In particular, [PERSON] and [PERSON] [13] identified \(\mu(p)\) by measuring the resistance between fluid layers, while \(\Gamma(p)\) is a measure of the friction between the solid and fluid at the pore. Various forms and combinations of \(\mu(p)\) and \(\Gamma(p)\) have been suggested and used by various researchers in the literature, popular among which are the following forms [14]:  
\[\mu(p)=Ae^{\mu p}\ \ \ \ \ \Gamma(p)=Be^{bp} \tag{9}\]  
\[\mu(p)=Ae^{\mu p}\ \ \ \ \Gamma(p)=B\bigg{(}\frac{p}{p_{0}}\bigg{)}^{m} \tag{10}\]  
\[\mu=\mu_{0}e^{\alpha(p-p_{0})} \tag{2}\]0  
\[\mu=\mu_{0}e^{\alpha(p-p_{0})} \tag{2}\]1  
wherein \(A,B,a,\ b,\ m,\ n\) are constants. Considering that flow through a porous structure is influenced in part by permeability to the flow, [PERSON] et al. [11] and [PERSON] et al. [12] suggested that \(\Gamma(p)=\frac{\mu(p)}{k}\), where \(k\) is the permeability, and provided the following form of momentum Equation (8):  
\[\mu=\mu_{0}e^{\alpha(p-p_{0})} \tag{2}\]2  
It is conceivable that the use of (8) or (13) is predicated by the type of practical application being considered and by the porous microstructure. In general, flow through porous media has well-established applications that include the study of groundwater movement, oil and gas recovery, flow through membranes and kidney dialysis, air conditioning and heat transfer, and the drying of solids (cf. [15] and the references therein). Flow with high pressures arise in a number of industrial applications that involve chemical and process technologies, such as medical tablet production, crude and fuel oil pumping, food processing, fluid film lubrication theory, and in microfluidics [3; 4; 5; 16; 17; 18; 19; 20]. The form of governing equations used must accommodate and describe the different types of fluid flow, flow conditions, and the anticipated porous structure [12; 21; 22]. For the current work, the objective was to study the effects of porous medium permeability, and the form of dependence of viscosity on pressure, on the characteristics of the flow. In particular, relationship (1) between viscosity and pressure is employed in this work in order to study the effects of \(n\). Cases where \(n\) is a positive integer are investigated, together with a case wherein \(n=-1\), which results in a decrease in the exponent with increasing pressure distribution. For comparison, the case of \(n=0\) is considered. This case corresponds to a flow with constant fluid viscosity.