Weakly nonlinear analysis of viscous dissipation thermal instability in plane Poiseuille and plane Couette flows

The weakly nonlinear stability analysis of plane Poiseuille flow (PPF) and plane Couette flow (PCF) when viscous dissipation is taken into account is considered. The impermeable lower boundary is considered adiabatic, while the impermeable upper boundary is isothermal. The linear stability of this problem has been performed by Barletta and Nield (J. Fluid Mech., vol. 662, 2010, pp. 475–492) for PCF and by Barletta et al. (J. Fluid Mech., vol. 681, 2011, pp. 499–514) for PPF. These authors found that longitudinal rolls are the preferred mode of convection and the onset of instability is described through the governing parameters ).