Parameter values excluded by existence conditions for buoyant dissipative motions in vertical channels

The nonexistence of steady, fully-developed solutions for frictionally-heated buoyant flow in vertical channels is established analytically. Explicit bounds on the values of parameters beyond which solutions to this nonlinear problem cannot exist are derived by integral estimation and by a priori comparison methods. The a priori bound shows that no solution exists if a linear combination of the pressure gradient, constant heat source and wall temperature difference is larger than a specific number proportional to the lowest eigenvalue of a linear comparison equation. The integral estimation procedure leads to the delineation of excluded regions in a parameter space; for parameter values in these regions, no solution exists. For boundary conditions permitting a symmetric solution, the integral estimates exclude further regions in the parameter space, while the power series solution leads to an excluded line of values.