Prandtl–Batchelor flows revisited

In this paper laminar flows are considered with closed streamlines. For such flows Prandtl [1905. Űber Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III. Internationalen Mathematiker Kongresses, Heidelberg, 1904, pp. 484-491, Teubner, Leizig. See Gesammelte Abhandlungen II, pp. 575–584] and Batchelor [1956. On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190] proved that the vorticity is constant in an inner region separated from the walls by a thin boundary layer. Moreover, Batchelor [1956. On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190] was able to derive the value of the constant vorticity if this wall has a circular shape. The present contribution is concerned with noncircular shapes. In this case no exact result similar to that found by Batchelor [1956. On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190] can be obtained. A useful approximation is derived here based on the condition that the torque on a closed streamline must be constant throughout the boundary layer. The contribution to the torque by the pressure is shown to be small with respect to that by the viscous stresses for a nearly circular shape. From the latter, the vorticity in the inner region can be obtained. For the square cavity, the torque calculation provides an exact value of the vorticity.