The effect of viscous heating on the linear and nonlinear stability analysis of a flow through a porous duct

The stability of a stationary fully developed vertical flow across a porous pipe is investigated. The heating due to viscous dissipation is assumed to be non–negligible and also to be the only effect triggering the onset of thermal convection. An innovative scaling is employed to study the case of small Gebhart number. The viscous heating term present inside the energy balance equation yields a basic stationary flow characterised by dual branches of solutions: for a given vertical pressure gradient, two possible velocity profiles are obtained. The linear and nonlinear stability of the stationary dual solutions is performed. The linear stability is investigated in a usual fashion by employing the normal modes method and then solving the eigenvalue problem obtained by using the shooting method. The nonlinear stability is investigated by simulating numerically the evolution in time of the perturbed system. The linear stability analysis allows one to conclude that the critical wavenumber for the onset of instability is zero and the critical dimensionless velocity at the pipe axis is equal to $3.43631$. The results of the nonlinear analysis display subcritical instabilities. Indeed, the onset of instability is obtained for values of the governing parameters which are lower than the critical values obtained by the linear analysis. This feature occurs when the amplitude of the initial disturbance applied to the nonlinear problem is sufficiently high, namely when the dimensionless amplitude is larger than ${10}^{-2}$.