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 and , where , and are respectively the Gebhart number, the Péclet number and the Prandtl number. The current study focuses on the near-threshold behaviour of longitudinal rolls by using a weakly nonlinear analysis. We determine numerically up to third order the coefficients of the Landau amplitude equation and investigate in detail the influences on bifurcation characteristics of the different nonlinearities present in the system. The results indicate that for both PPF and PCF configurations (i) the inertial terms have no influence on the nonlinear evolution of the disturbance amplitude (ii) the nonlinear thermal advection terms act in favour of pitchfork supercritical bifurcations and (iii) the nonlinearities associated with viscous dissipation promote subcritical bifurcations. The global impact of the different nonlinear contributions indicate that, independently of the Gebhart number, the bifurcation is subcritical if Pr < 0:25 (Pr < 0:77) for PPF (PCF). Otherwise, for higher Prandtl number, there exists a particular value of Gebhart number, such that the bifurcation is supercritical (subcritical) if ). Finally, for both PPF and PCF, the amplitude analysis indicates that, in the supercritical bifurcation regime, the equilibrium amplitude decreases on increasing Pr and a substantial enhancement (reduction) in heat transfer rate is found for small Pr (moderate or large Pr)..