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

The nonlinear stability of the channel flow of fluid with temperature-depen dent viscosity is considered for the case of vanishing Peclet number for tw o viscosity models, mu(T), which vary monotonically with temperature, T. in each case the basic state is found to lose stability from the linear criti cal point in a subcritical Hopf bifurcation. We find two-dimensional nonlin ear time-periodic flows that arise from these bifurcations. The disturbance to the basic flow has wavy streamlines meandering between a sequence of tr iangular-shaped vortices, with this pattern skewing towards the channel wal l 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 m inimum Reynolds number at which a secondary flow may exist. In contrast to the results for the linear critical Reynolds number, the precise form of mu (T) is not found to be qualitatively important in determining the stability of the thermal flow relative to the isothermal flow. For the viscosity mod els considered here, we find that the secondary flow is destabilized relati ve to the corresponding isothermal flow when mu(T) decreases and vice versa . However, if we remove the bulk effect of the non-uniform change in viscos ity by introducing a Reynolds number based on average viscosity, it is foun d that the form of mu(T) is important in determining whether the thermal se condary flow is stabilized or destabilized relative to the corresponding is othermal flow. We also consider the linear stability of the secondary flows and find that the most unstable modes are either superharmonic or subharmo nic. All secondary disturbance modes are ultimately damped as the Floquet p arameter in the spanwise direction increases, and the last mode to be dampe d is always a phase-locked subharmonic mode. None of the secondary flows is found to be stable to all secondary disturbance modes. Possible bifurcatio n points for tertiary flows are also identified.