Stability analysis of some fully developed mixed convection flows in a vertical channel

Abstract

Stability of fully developed mixed convection flows, with significant viscous dissipation, in a vertical channel bounded by isothermal plane walls having the same temperature and subject to pressure gradient is investigated. It is shown that one of the dual solutions is always unstable and that both are unstable when the total flow rate is big enough. The completely passive natural convection flow is shown to be unstable.

Pressure work and viscous dissipation in the equations of thermal convection in a vertical channel

“…We can now use the momentum equation (2) to eliminate the temperature from either form of the energy equation, yielding…”

“…With any given value of applied pressure gradient, there exist either dual solutions or no solution at all; in particular, with zero applied pressure gradient, one solution branch yields static fluid with no temperature variation, whereas the second solution branch yields "passive convection", in which viscous dissipation provides the heat to maintain (through buoyancy forces) the flow that gives rise to the dissipation. Setting up the flow specified by this second solution branch would be very difficult, and indeed Miklavčič [2] has recently shown that solutions on this branch are unstable. Even on the stable first branch, flow velocities and temperature gradients are only of reasonable magnitude if the applied pressure gradient is of moderate magnitude; furthermore, large applied pressure gradients may violate the Boussinesq approximation (made in all the calculations) which requires any vertical dynamic pressure gradient to be small compared to the hydrostatic pressure gradient.…”

On fully developed mixed convection with viscous dissipation in a vertical channel and its stability

“…Stability analyses of the dual solutions emerging in viscous dissipation buoyant flows are quite rare in the literature. Besides the analysis by Barletta and Rees , relative to a fluid saturated horizontal porous channel, the only study dealing with the stability of dual buoyant flows with viscous dissipation in a vertical channel is that recently published by Miklavčič . This paper presents a nonlinear stability analysis focusing on special modes of perturbation preserving the parallel velocity, or fully‐developed, flow regime.…”

“…The author considers a vertical plane channel where no temperature difference between the boundaries is prescribed, so that the viscous dissipation is the unique source of the buoyancy force and, thus, of the thermal instability. The governing equations analyzed by Miklavčič are based on the Oberbeck‐Boussinesq approximation where the constrained wall temperature is taken as the reference temperature for the linearization of the fluid equation of state. Although being the simplest choice to carry out the analysis, the physical model behind this choice can lead to increasingly large errors when the internal heat input provided by the viscous dissipation is large enough.…”

“…The aim of our contribution is to reconsider the analysis carried out by Miklavčič , in order to estimate the extent to which a different choice of the reference temperature in the formulation of the Oberbeck‐Boussinesq approximation can influence the dual solutions for the basic flow, as well as the results of the stability analysis. In particular, the recommendation expressed and motivated by Barletta and Zanchini , to take the average fluid temperature as the reference temperature, is adopted in our study.…”