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.
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