Multiplicity of solutions for laminar, fully-developed natural convection in inclined, parallel-plate channels

Natural convection in inclined channels is a rather common flow configuration: it occurs in solar energy systems, ventilated roofs as well as in many industrial applications and chemical processes. Analytical solutions for laminar, fully-developed natural convection in inclined parallel-plate channels are presented in this paper. The Boussinesq approximation is applied and viscous energy dissipation is neglected. One specific thermal configuration is addressed, where one wall is perfectly insulated and a constant, uniform heat flux is released to the fluid from the other wall. The resulting set of governing equations is non-linear, as the mean velocity is not assigned a priori but determined as part of the solution. Depending on the channel inclination angle and on the imposed heat flux conditions, either no solution, one solution, multiple or infinite solutions exist. Under restrictive assumptions velocity profiles are self-similar with respect to the channel inclination, while the temperature profile is independent of the inclination. The two-dimensional, hydraulically- and thermally-developing natural convection channel flow is simulated numerically for some combinations of channel inclination angle and heating intensity to identify the most physical between the many solutions.