Poiseuille-Rayleigh-Bénard instability of a channel flow with uniform cross-flow and thermal slip

Stability characteristics of a pressure-driven Poiseuille type flow are explored in a horizontal channel with porous walls. The flow is modified with a thermal gradient, and the temperature slips alongside a uniform cross-flow for three different configurations: (i) the flow having heat influx/outflux and temperature jump across the channel walls with internal heat generation in the fluid, (ii) the flow with constant wall temperatures (not necessarily equal) and no internal heat source, and (iii) the flow having a temperature jump/slip at the upper wall and lower wall with a constant temperature. The Reynolds, Péclet, and Rayleigh numbers govern the behavior of the thermal and velocity profiles along with the physical aspects of the flow. The modified Orr-Sommerfeld and energy equations are derived for the perturbed system by the normal mode analysis and solved using the Chebyshev collocation method, while the energy budget analysis is used to further illustrate the stability characteristics of the system. The resulting eigenvalues and eigenfunctions are used to analyze the growth rate characteristics, neutral stability ranges, velocity isolines, and temperature isotherms for each flow configuration. Further, the energy from various sources is calculated by making use of eigenfunctions corresponding to the unstable eigenmodes. An increase in the Rayleigh and Reynolds numbers promotes the flow instability; whereas, a higher cross-flow Reynolds number suppresses the instability by raising the upward cross-flow. The porous walls, characterized by the velocity slip and wall shear, inhibit the Poiseuille-Rayleigh-Bénard instability of the system. The higher temperature difference/lower wall temperature and the stronger thermal slip, respectively, destabilize and stabilize the flow. Conclusively, the flow instability varies depending on the configuration.