The Instability of Flow Through a Slowly Diverging Pipe With Viscous Heating

The nonparallel linear stability analysis of flow through a slowly diverging pipe undergoing viscous heating is considered. The pipe wall is maintained at constant temperatures and Nahme’s law is applied to model the temperature dependence of the fluid viscosity. A one-parameter family of velocity profiles for the basic state is obtained for small angles of divergence. The nonparallel stability equations for the disturbance velocity coupled to a linearized energy equation are derived and solved using a spectral collocation method. Our results indicate that increasing viscous heating, characterized by increasing Nahme number, is destabilizing. The Prandtl number has a negligible effect on the linear stability characteristics. The Grashof number stablizes the flow for $Gr>106$⁠, below which it has a negligible effect.