© 1955 by Oxford University Press
THE EFFECT OF VISCOSITY UPON THE CRITICAL FLOW OF A LIQUID THROUGH A CONSTRICTION
( Trinity College Cambridge )
The motions of a viscous liquid under gravity over a broad-crested weir, and with swirl under pressure through a convergent-divergent nozzle, are analysed on the supposition that the whole of the discharge passes through a laminar boundary layer. The constrictions are assumed to be gradual; and the pressure gradients in the direction of streaming being favourable, the velocity distributions over the cross-sections are taken to be of polynomial forms. When the momentum integral equations are satisfied, first-order differential equations are obtained which show that, as with inviscid flow, the motion is ‘critical’ in the sense that arbitrary inlet conditions may not be specified. The flow must be determined from the conditions that prevail at the critical cross-sections, which are displaced downstream from the throats of the constrictions. The equations are difficult to solve because at the critical cross-sections, where their step-by-step integration has to be begun, they take the indeterminate form 0/0, but a numerical example of each type of motion is given. It is proved that a long wave of small amplitude can maintain itself stationary on the moving liquid at the critical cross-sections. A comparison is made with inviscid flow through both kinds of constriction.