Multiple solutions for natural convective flows in an internally heated, vertical channel with viscous dissipation and pressure work

Abstract This paper considers the vertical flow of an internally heated Boussinesq fluid in a vertical channel with viscous dissipation and pressure work. In the absence of internal heating and with no applied pressure gradient, two solutions are obtained; the first is the expected solution with no flow. The second adiabatic solution has temperatures less than the wall temperature and a large downward velocity. For moderate values of heat addition two solutions are obtained; an upper branch with hot temperatures and upward velocities and a lower branch with downward velocities and cool temperatures. When the non-dimensional heat addition parameter A = Hh 2 α 2 g 2 v 2 c p 2 κ 2 reaches a critical value just under 1000 the solutions bifurcate and four solutions are obtained. For large values of A the solutions are examined using the method of matched asymptotic expansions. The equation of the inner solution is of the form of the Painleve transcendent. In the limit of very large A an infinite number of solutions are found for the inner problem.