A bifurcation study of convective heat transfer in a hele‐shaw cell

Steady‐state multiplicity characteristics of convective heat transfer within a Hele‐Shaw cell are investigated. The Navier‐Stokes equations and the energy equation are averaged across the narrow gap, d, of the cell. The resulting two‐dimensional, stationary equations depend on the following parameters: (i) the length to height aspect ratio γ, (ii) the tilt anle ϕ (iii) the Prandtl number Pr, (iv) an inertia parameter ξ = d²/ 12a², and (v) the Grashof number. Gr = Q_gβga⁵/kv². Here a is the height of the cell and Q, is the heat generation rate per unit volume. The complete structure of symmetric and asymmetric stationary solutions are traced using recent algorithms from bifurcation theory. In the double limit of ξ → 0 and Gr → ∞ such that Ra = 4GrPrξ remains finite (where Ra is the Rayleigh number for the Darcy model) the Hele‐Shaw model reduces to that of the Darcy model.