## 1 Introduction  
The main objective of this work is to study the effects of viscosity variations due to pressure on the flow through a porous structure of an incompressible fluid governed by the generalised Brinkman equation. In particular, viscosity, \(\mu\), is taken as an exponential function of pressure, \(p\), of the form:  
\[\mu(p)=f(p)=\mu_{0}e^{((\frac{p}{p_{0}})^{\alpha}-1)} \tag{1}\]  
where \(\alpha\geq 0\), \(p_{0}>0\), \(n\geq-1\), and \(\mu_{0}=f(p_{0})>0\) is the constant viscosity when \(\alpha=0\) and \(p=p_{0}\). The realisation that changes in flow conditions (such as changes in temperature and pressure) result in corresponding changes in the viscosity of a flowing fluid dates back to the works of [PERSON] [1] and [PERSON] [2]. With increasing temperature, the viscosity of a fluid decreases, while an excessive increase in pressure could result in an increase in fluid viscosity. [PERSON] [2] suggested that viscosity is an exponentially increasing function of pressure of the form:  
\[\mu=\mu_{0}e^{\alpha(p-p_{0})} \tag{2}\]  
For small values of \(\alpha\) or small pressure differences, relationship (2) between viscosity and pressure can be expressed as [3]:  
\[\mu=\mu_{0}[1+\alpha(p-p_{0})] \tag{3}\]  
It has been customary to assume that in the flow of Newtonian liquids in the absence of temperature variations, shear viscosity is constant (cf. [4] and the references therein). This assumption ignores the effects of pressure on viscosity when the fluid pressure is of the order of 1000 atm [3; 5]. While the above relationships, (2) and (3), are suitable for fluids with small molecules, long-chain molecules (the likes of polymers and some oil mixtures) require other forms of viscosity-pressure relations [3; 6; 7]. It has also been said that the effect of pressure on the density of a fluid is small compared to the effect of pressure on viscosity [6; 7; 8]. Accordingly, it suffices in many applications to study incompressible fluid flow with pressure-dependent viscosity as governed by the equation of continuity and the Navier-Stokes equations, written here for steady flow, in the absence of body forces, in the following forms [3; 9; 10], respectively:  
\[\
abla\cdot\overset{\rightarrow}{u}=0 \tag{4}\]  
\[\rho\overset{\rightarrow}{u}\cdot\
abla\overset{\rightarrow}{u}=-\
abla p +\
abla\cdot\overset{\rightarrow}{T}+\rho\overset{\rightarrow}{g} \tag{5}\]  
where  
\[\overset{\rightarrow}{T}=-p\overset{\rightarrow}{I}+2\mu(p)\overset{ \rightarrow}{K} \tag{6}\]  
is the Cauchy stress tensor in which  
\[\overset{\rightarrow}{K}=\frac{1}{2}\Big{(}\
abla\overset{\rightarrow}{u} +\Big{(}\
abla\overset{\rightarrow}{u}\Big{)}^{T}\Big{)} \tag{7}\]  
is the symmetric part of the velocity gradient, \(\overset{\rightarrow}{u}\) is the velocity field, \(\rho\) is the constant fluid density, and \(\rho\overset{\rightarrow}{g}\) is the gravitational force per unit volume. Governing Equations (4) and (5), with \(\overset{\rightarrow}{T}\) and \(\overset{\rightarrow}{K}\) defined by (6) and (7), represent an underdetermined system of four equations in the five unknowns \(\overset{\rightarrow}{u}=(u,v,w)\), \(p\), and \(\mu\). In the absence of additional conservation principles to provide an additional equation, viscosity can be expressed as a function of pressure (such as in (2) and (3)) to provide an additional condition to render the governing system of equations determinate. In addition to the flow conditions' influence on viscosity, there are flow domain conditions that could result in changes in the viscosity of a flowing fluid. For instance, in fluid flow through domains with constrictions and in porous structures, it has been suggested by a large number of researchers that the effective viscosity of a fluid in a porous structure is different than the viscosity of the base fluid (cf. [11] and the references therein). It can be argued that [PERSON]'s equation accounts for viscosity change due to domain constrictions by introducing an effective viscosity term.