# The Role of Buoyancy Induced Instability in Transpirational Cooling Applications
## 2 Mathematical Model  
### Governing Equations  
A simplified model for transpirational cooling is presented in Figure 1. Here, a fluid-saturated isotropic porous medium of infinite horizontal extent is confined between two permeable boundaries a distance \(d\) apart. The hot gas flowing over bottom layer is modeled as a constant temperature boundary condition as in Ref. [6]. Coolant continuously flows over the top permeable boundary (\(z=d\) in Figure 1), and creates an isothermal condition. Wall suction provides a constant coolant injection aligned with the gravitation vector [3]. This following problem configuration is similar to the Wooding problem [19], where a constant through-flow is heated which gives rise to an additional thermally-induced natural convection of the Benard type. These assumptions, along with the Boussinesq approximation, lead to Dirichlet conditions for velocity and temperature at the top and bottom boundary. Analysis of this problem requires conservation of the mass, momentum, and energy equations subject to the aforementioned boundary conditions. In regards to the momentum equation, several experimental [6,7] and numerical studies [8,9] have determined that transpirational cooling applications are accurately approximated by the Darcy-Forchheimer equation. The curl of the Darcy-Forchheimer equation is presented here to drop the pressure term. The energy equations for the porous material and the fluid phase can be simplified to one equation if local thermal equilibrium exists between the solid and fluid phases, and there is no net conduction between the phases [14]. The resulting single equation uses an overall heat capacity and thermal conductivity per unit volume of the porous material and fluid denoted by the subscript'm'. For example, \((\rho c_{p})_{m}=(1-\psi)(\rho c_{p})_{s}+\psi(\rho c_{p})_{f}\), where \(\psi\) is the porosity and the subscripts's' and 'f' refer to the solid and fluid phase, respectively. [PERSON] and [PERSON] [6] investigated these assumptions along with negligible work due to pressure, negligible viscous heating, and constant density in the energy equation to study transpirational cooling in scramjet engines and found good agreement with experimental results. Thus, the resulting energy equation is implemented in Equation (1). Figure 1: Simplified model of the transpirational cooling process. The governing equations and boundary conditions are shown in Equation (1). \[ \begin{cases}\
abla\cdot\mathbf{u}^{*}=0\\ \frac{\mu}{K}\Big{(}\
abla\times\mathbf{u}^{*}+\frac{c_{F}\sqrt{K}}{\
u}\
abla \times(\mathbf{u}^{*}|\mathbf{u}^{*}|)\Big{)}=\
abla\times\rho g\mathbf{e_{z}} \\ (\rho c_{p})_{m}\frac{\partial T^{*}}{\partial t^{*}}+(\rho c_{p})_{f} \mathbf{u}^{*}\cdot\
abla T^{*}=k_{m}\
abla^{2}T^{*}\\ z^{*}=0:\quad\mathbf{u}^{*}=<0,0,-\lambda\frac{Kg\beta\Delta T}{\
u^{2}}>, \quad T^{*}=T_{w}\\ z^{*}=d:\quad\mathbf{u}^{*}=<0,0,-\lambda\frac{Kg\beta\Delta T}{\
u^{2}}>, \quad T^{*}=T_{0}\end{cases}  \tag{1}\]  
The filtration velocity vector is represented by the three orthogonal velocity components, \(\mathbf{u}^{*}=<u^{*},v^{*},w^{*}>\), corresponding to the Cartesian position vector, \(\mathbf{x}^{*}=<x^{*},y^{*},z^{*}>\). The asterisk here refers to the dimensional quantities. The terms \(\
u\), \(K\), \(c_{F}\), \(\rho\), \(|\mathbf{u}^{*}|\), and '\(g\)' in the Darcy-Forchheimer equation are the kinematic fluid viscosity, permeability of the porous material, dimensionless Darcy-Forchheimer form drag coefficient, fluid density, velocity magnitude, and gravity, respectively. The term \(\mathbf{e_{z}}\) is the unit vector in the \(z\) direction. T\({}^{*}\), \(c_{p}\), \(k\), \(\beta\), and \(\lambda\) are the respective temperature, specific heat, thermal conductivity, coefficient of thermal expansion, and a velocity scaling constant. The subscript '0' and 'w' refer to the constant quantity at the top and bottom boundary, respectively. Here, all the properties are assumed constant except the density associated with buoyant forces.