# Thermal Stability and Entropy Generation Analysis for
## 2 Mathematical Formulation  
Consider the steady flow of an incompressible, variable-viscosity, reactive third-grade fluid through an inclined channel filled with a homogeneous and isotropic porous medium, as shown in Figure 1. The walls of the channel are assumed to be porous and to experience fluid slippage. The flow is assumed to be induced by an applied axial constant-pressure gradient and buoyancy force. Neglecting the reacting viscous fluid consumption, the model equations emanating from the momentum and heat balance can be written as [15]  
\[\rho V_{0}\frac{d\bar{u}}{d\bar{y}}=-\frac{d\bar{P}}{d\bar{x}}+\frac{d}{d\bar{ y}}\bigg{(}\bar{\mu}(T)\frac{d\bar{u}}{d\bar{y}}\bigg{)}+6\beta_{3}\frac{d^{2} \bar{u}}{d\bar{y}^{2}}\bigg{(}\frac{d\bar{u}}{d\bar{y}}\bigg{)}^{2}-\frac{\bar {\mu}(T)\bar{u}}{K}+\rho\beta(T-T_{0})sin(\omega), \tag{1}\]  
\[\rho C_{p}V_{0}\frac{d\bar{T}}{d\bar{y}}=k\frac{d^{2}T}{d\bar{y}^{2}}+\bigg{(} \frac{d\bar{u}}{d\bar{y}}\bigg{)}^{2}\bigg{[}\bar{\mu}(T)+2\beta_{3}\bigg{(} \frac{d\bar{u}}{d\bar{y}}\bigg{)}^{2}\bigg{]}+\frac{\bar{\mu}(T)\bar{u}^{2}}{K }+QC_{0}A\bigg{(}\frac{hT}{vl}\bigg{)}^{m}e^{-\frac{E}{RT}}, \tag{2}\]  
with appropriate boundary conditions  
\[ \begin{array}{ll}\bar{y}=-a:&\lambda_{1}\bar{u}=\mu(T)\frac{d\bar{u}}{d\bar{ y}}+2\beta_{3}\bigg{(}\frac{d\bar{u}}{d\bar{y}}\bigg{)}^{3},&T=T_{0}\\ \bar{y}=a:&\lambda_{2}\bar{u}=-\mu(T)\frac{d\bar{u}}{d\bar{y}}+2\beta_{3}\bigg{(} \frac{d\bar{u}}{d\bar{y}}\bigg{)}^{3},&T=T_{0}\end{array}  \tag{3}\]  
Figure 1: Schematic diagram of the problem. \(\bar{P}\) is the modified fluid pressure, \(\bar{x}\) and \(\bar{y}\) are the axial and normal coordinates to the inclined channel, \(\bar{u}\) is the fluid velocity, \(T_{0}\) is the fluid initial temperature, \(T\) is the fluid temperature, \(g\) is the acceleration due to gravity, \(\omega\) is the angle of inclination, \(\beta_{3}\) is the third-grade material coefficient, \(k\) is the thermal conductivity, \(K\) is the porous-medium permeability, \(\bar{\mu}\) is the fluid dynamic viscosity, \(\rho\) is the density, \(Q\) is the heat generated internally, \(C_{0}\) is the initial concentration of the reactant species, \(A\) is the reaction rate constant, \(\lambda_{1}\) and \(\lambda_{2}\) are the slip coefficients at the lower and upper channel walls, respectively, \(E\) is the activation energy, \(h\) is the [PERSON]'s constant, \(l\) is the [PERSON]'s number, \(R\) is the universal gas constant, \(\
u\) is the vibration frequency, \(\beta\) is the volumetric thermal expansion coefficient, and \(m\) is the numerical exponent such that its three values represent numerical exponents for sensitized, Arrhenius, and bimolecular kinetics, respectively, as \(m\in\{-2,0,0.5\}\)[16; 17]. The temperature-dependent viscosity \(\bar{\mu}(T)\) can be expressed as  
\[\bar{\mu}(T)=\mu_{0}e^{-b(T-T_{0})} \tag{4}\]  
where \(\mu_{0}\) is the initial fluid viscosity at temperature \(T_{0}\) and \(b\) is the viscosity variation parameter. Introducing the following non-dimensional variables into Equations (1)-(4):  
\[\left. \begin{array}{l} y=\frac{\bar{y}}{a},\ \ x=\frac{\bar{x}}{a},\ \ f= \frac{\bar{u}\rho a}{\mu_{0}},\ \ \alpha=\frac{bRT_{0}^{2}}{E},\ \ G=-\frac{dP}{dx},\ \ \Omega=\frac{\beta_{3}\mu_{0}}{\rho^{2}a^{4}},\\ Da=\frac{K}{a^{2}},P=\frac{a^{2}\rho\bar{P}}{\mu_{0}^{2}}, \epsilon=\frac{RT_{0}}{RT_{0}},\ \theta=\frac{E(T-T_{0})}{RT_{0}^{2}},\\ Gr=\frac{\rho^{2}g\beta a^{3}RT_{0}^{2}}{m_{0}^{2}E},\ \ \gamma=\frac{\mu_{0}^{3}}{QA\rho^{2}a^{4}C_{0}} \bigg{(}\frac{\
u l}{kT_{0}}\bigg{)}^{m}e^{\frac{E}{kT_{0}}},\delta^{2}=\frac {1}{Da},\\ \beta_{1}=\frac{\mu_{0}}{\lambda_{1}a^{4}},\ \ \beta_{2}=\frac{\mu_{0}}{ \lambda_{2}a^{4}},\ \ Bi=\frac{ah_{1}}{k},\ \ \lambda=\frac{QEAa^{2}C_{0}}{kRT_{0}^{2}}\bigg{(}\frac{hT_{0}}{\
u l} \bigg{)}^{m}e^{-\frac{E}{kT_{0}}},\end{array}