When is natural convection completely passive?

Abstract

Momentum and energy equations for vertical flow with viscous dissipation are derived and shown to require that the cross-section mean density is taken as the reference density for calculation of buoyancy forces under the Boussinesq approximation. Solutions are obtained for flow between parallel plane walls, with and without the pressure work as an explicit term in the energy equation. Both walls are at the same temperature, so there is no thermal forcing, but solutions are obtained for all admissible values of dynamic pressure gradient. The passive convection condition, whereby the flow is driven entirely by buoyancy forces resulting from heat generated by the flow's own viscous dissipation, is found on one branch of the dual solutions. However, while theoretically possible, passive convection is not physically realisable with any real fluid.

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

“…On the other hand, when investigating flow in a porous medium in a vertical channel, the same authors [8] find that pressure work is important, in particular producing a cooling effect in upward flow greater than the heating due to viscous dissipation. A similar effect was found by Kay [1] in forced convection of a Newtonian fluid in a vertical channel when the forcing is relatively small; as in the discussion of Rayleigh-Bénard convection by Barletta and Nield [9], it was noted that pressure work is linear in the vertical velocity whereas viscous dissipation is quadratic, so that pressure work must have greater magnitude in the limit of small velocities.…”

“…A thorough discussion of the governing equations for convection in vertical ducts was given in [1], and the reader is referred there for the derivation of the equations which follow. We consider only steady, fully developed flow of a Newtonian fluid in a channel of width 2L between parallel, vertical walls located at Y = ±L.…”

“…where the reference temperature T m and reference density ρ m are required to be the cross-section means of these quantities [1,12]. Other fluid properties that are also assumed constant are viscosity µ, thermal conductivity k and specific heat c. The momentum equation is a balance between the applied pressure gradient (which must be uniform in fully developed flow), the buoyancy force and the viscous stress.…”

“…In [1] velocity and temperature scales were defined from a balance between heat conduction and viscous dissipation. While we are not expecting viscous dissipation to be a leading-order effect in most cases, a nondimensionalisation based on these scales does have the virtue that it can be used in every scenario to be considered below, whether for free or forced convection, and whether temperatures or heat fluxes are specified at walls.…”

“…In a recent paper [1] the present author solved equations for forced convection in a Newtonian fluid confined between parallel vertical plates held at equal temperatures. In the energy equation, temperature changes in the fluid were accounted for by heat conduction, viscous dissipation and pressure work.…”