Hele Shaw Convection with Imposed Shear Flows: Boundary‐Layer Formulation

The nonlinear convection forced by the boundaries of a Hele Shaw cell to align perpendicular to an imposed shear flow was analytically investigated by the boundary‐layer method. The imposed shear flow may be a Couette flow that extends throughout the convecting layer or flow confined to a boundary, depending on the geometry of the Hele Shaw cell. This study examined the case in which the imposed shear flow has a boundary‐layer structure and its interaction with the convecting interior. Analytical solutions for both the boundary layer and interior were obtained. The study revealed the following.For large aspect ratio A , the interaction of the imposed shear flow and convection is confined to the boundary layer. The boundary layer is a viscous rather than a thermal layer. The results showed that the range of validity of the Hele Shaw equations used in the literature is of order 1/ A 2 . For an asymptotically large aspect ratio A up to order 1/ A 2 , the velocity in the y ‐direction must be zero. The velocity in the x ‐direction and the z ‐direction has a parabolic dependence on y , but the temperature perturbation does not depend on y . These results may have implication for convection in porous media.