# The Role of Buoyancy Induced Instability in Transpirational Cooling Applications
## 6 Conclusions  
Linear stability analysis was performed for a simplified system modeling transpirational cooling applications. Analysis was performed utilizing three models of increasing detail: Case 1, a Horton-Rogers-Lapwood (HRL) type configuration, Case 2, a HRL configuration with non-linear (cubic) density dependence on temperature, and Case 3, both a cubic density function and a base through-flow velocity. In all cases, linear stability analysis was performed to yield governing equations for the system. These equations were then solved utilizing multiple numerical methods, comparing results from a combination of analytical solutions, finite difference, power series, and Chebyshev methods. Results demonstrated excellent consistency in predictions across these methods.  
The addition of a cubic density dependence on temperature had several effects as compared to Case 1 (HRL problem). First, accounting for the non-linear density variation during the transpirational heating process shifts the marginal stability curve to a higher Grashof-type number resulting in increased stability. This is due to sharp density gradients near the top wall decreasing the density such that an overlaying fluid is not heavy as compared to the linear density case. In addition, a shift to higher wavenumbers is observed. The cubic density profile also shifted velocity peak toward the top of the porous media. Finally, Case 3 studied the effects of the addition of a base through-flow velocity--the base flow is in the same direction as the gravity vector. A base through-flow velocity sweep revealed that initially increasing the base through-flow destabilizes followed by a re-stabilizing effect. However, it is worth noting that all base flows for Case 3 considered in this work were more unstable as compared to Case 2, i.e., the highest base velocity resulted in a lower critical Grashof-type number as compared to Case 2. In both Cases 2 and 3, accounting for the non-linear variation in density increases the stability of the system.  
The implications of an additional mode of convection (buoyancy-induced convection) is increased mixing, i.e., increased heat transfer inside the porous medium. Accounting for this additional mode of convection in cases where instability ensues is important when it comes to thermal management of the system. Thus, an awareness that sharp density gradients at the top of the domain from non-linear density variation in the system, as well as increased through-flow have a stabilizing effect on the system are pertinent to design and modeling considerations.  
The framing of the problem was carried out by [PERSON]. The governing equations, non-dimensionalization, perturbed equations, and base state solutions were independently generated by both authors. [PERSON] implemented the power series and Chebyshev solution techniques while [PERSON] developed the finite difference scheme and solution. The thermodynamic property analysis using real-fluid equations of state along with the non-linear density implementation was completed by [PERSON]. Both authors contributed to the writing, editing, and figure generation of the manuscript. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.