'Note on 'Flows induced by a plate moving normal to stagnation-point flow' by P. D. Weidman and M. A. Sprague (Acta Mech. 219, 219-229, 2011)'

摘要

In a recent paper of Weidman and Sprague (Acta Mech., 2011), the unsteady flows generated by an impermeable infinite flat plate advancing with constant velocity V toward, or receding from an orthogonal (plane or axisymmetric) stagnation-point flow, have been investigated by an exact similarity reduction of the Navier-Stokes equations. It has been shown that in the co-moving reference frame of the plate, the induced flow appears as a steady flow, with an additional term R f″ in the governing equation of the similar stream function f (η). The Reynolds number R involved in this additional term is proportional to the plate velocity V. The present paper shows, however, that with the aid of a simple transformation, the additional term R f″ can be removed from the governing equation, its effect being transferred in the boundary condition for f (η). As a consequence, the unsteady flow problems of Weidman and Sprague reduce to the classical steady stagnation-point flow problems for permeable surfaces with a uniform lateral suction or injection of the fluid, so that the transpiration parameter f (0) coincides with R for the plane and with R/2 for the axisymmetric flow, respectively. The main benefit of this approach is that all the results of the latter well-investigated problems can simply be transcribed for the problems formulated by Weidman and Sprague (Acta Mech, 2011).