Nonlinear equilibrium solutions for the channel flow of fluid with temperature-dependent viscosity

Abstract

The nonlinear stability of the channel flow of fluid with temperature-dependent viscosity is considered for the case of vanishing Péclet number for two viscosity models, μ(T), which vary monotonically with temperature, T. In each case the basic state is found to lose stability from the linear critical point in a subcritical Hopf bifurcation. We find two-dimensional nonlinear time-periodic flows that arise from these bifurcations. The disturbance to the basic flow has wavy streamlines meandering between a sequence of triangular-shaped vortices, with this pattern skewing towards the channel wall which the basic flow skews towards. For each of these secondary flows we identify a nonlinear critical Reynolds number (based on half-channel width and viscosity at one of the fixed wall temperatures) which represents the minimum Reynolds number at which a secondary flow may exist. In contrast to the results for the linear critical Reynolds number, the precise form of μ(T) is not found to be qualitatively important in determining the stability of the thermal flow relative to the isothermal flow. For the viscosity models considered here, we find that the secondary flow is destabilized relative to the corresponding isothermal flow when μ(T) decreases and vice versa. However, if we remove the bulk effect of the non-uniform change in viscosity by introducing a Reynolds number based on average viscosity, it is found that the form of μ(T) is important in determining whether the thermal secondary flow is stabilized or destabilized relative to the corresponding isothermal flow. We also consider the linear stability of the secondary flows and find that the most unstable modes are either superharmonic or subharmonic. All secondary disturbance modes are ultimately damped as the Floquet parameter in the spanwise direction increases, and the last mode to be damped is always a phase-locked subharmonic mode. None of the secondary flows is found to be stable to all secondary disturbance modes. Possible bifurcation points for tertiary flows are also identified.