On the appearance of Moffatt eddies in viscous cavity flow as the aspect ratio varies

Abstract

Two-dimensional Stokes flow in a rectangular driven cavity is studied. Flow in very tall or shallow cavities has many counter-rotating eddies lying along the cavity centerline. This structure is investigated by constructing asymptotic approximations to the flow based on the assumptions A⪡1 and A⪢1, where A is the cavity’s aspect ratio. We show that the number of eddies increases as A tends to infinity or zero and derive asymptotic formulas for the values of the aspect ratio at which the streamline topology bifurcates and a new eddy appears. We concentrate particularly on flow driven by translation of the top and bottom cavity walls with equal and opposite velocities. For this benchmark problem our asymptotics are able to connect existing computations (performed in the approximate range of 10−1≤A≤101) with Moffatt’s 1964 theory for an infinite channel (i.e., A=0 or ∞). We show that tall cavities have sequences of eddies which match the infinite channel flow, as observed in the previous computations. In contrast, shallow cavities have only half the number of eddies and a more complicated streamline topology. In both cases our asymptotic approximations give analytic formulas for the number of eddies and the shape of the streamlines. Other flows, corresponding to different driving at the boundaries, are also discussed and can be treated by the asymptotic methods we derive.