Buoyant Poiseuille–Couette flow with viscous dissipation in a vertical channel

Abstract

Steady combined forced and free convection is investigated in a vertical channel having a wall at rest and a moving wall subjected to a prescribed shear stress. The moving wall is thermally insulated, while the wall at rest is kept at a uniform temperature. The analysis deals with the fully-developed parallel flow regime. The governing equations yield a boundary value problem, that is solved analytically by employing a power series expansion of the velocity field with respect to the transverse coordinate. It is shown that the nonlinear interplay between buoyancy and viscous dissipation may determine the existence of dual solutions of the boundary value problem corresponding to fixed values of the applied shear stress on the moving wall and of the hydrodynamic pressure gradient. It is shown that a nontrivial fully separated flow may occur such that the hydrodynamic pressure gradient is zero and the shear stress vanishes on both walls. (2000). 76R10, 80A20, 76D05, 34B15. Mathematics Subject Classification

When is natural convection completely passive?

“…We have avoided this question by obtaining solutions both with and without the pressure work term in the energy equation. In particular, we have found passive convection solutions when it is retained as well as reproducing the solution found by Barletta, Lazzari, and Magyari [14] when it is neglected.…”

“…Our procedures might be considered more philosophically satisfying, since the volume flux is the consequence, not the cause, of the pressure gradient. In further work Barletta et al [14] did use the procedure we recommend, with the reference temperature taken to be the cross-section mean and with the pressure gradient introduced through a shear stress boundary condition; indeed, they even obtained the true solution for completely passive convection. Miklavčič and Wang were apparently unaware of this solution, maybe because it is contained within an analysis of a much more general problem.…”

“…2. [Note that a plot of this function can also be found in [14], Fig. 2, where it is the curve for = 0, but upside-down since our λ is the same as −η in [14].]…”

“…If the Boussinesq approximation is valid and the flow is vertical, the dominant contribution to the pressure work term in (14) will be from the hydrostatic pressure gradient, so…”

“…Here we have applied the Boussinesq approximation to replace ρ in (14) with the reference density ρ m , and we have allowed for Barletta's [10] formulation by replacing c p with an unspecified but constant specific heat c. Note that the term W G in (22) has the same mathematical form (W × constant) as the pressure work term in (17). However, the constants in (17) are fixed and the term is present regardless of the boundary conditions, whereas G may only be required to be non-zero when heat fluxes are prescribed on all boundaries, and it is then diagnosed from the solution for the flow in the duct.…”