Gas slug between parallel plates—D (C). Fig. P10.1 shows the cross section through a large bubble OPQ between two parallel vertical plates AA and BB that are separated by a distance 2a. For purposes...

Gas slug between parallel plates—D (C). Fig. P10.1 shows the cross section through a large bubble OPQ between two parallel vertical plates AA and BB that are separated by a distance 2a. For purposes of experimental observation, the bubble is held stationary by a downward flow of an inviscid liquid whose velocity is U far upstream of the bubble. The motion is two-dimensional, so that the flow pattern is the same in all planes parallel to that of the diagram.

The following approximate relation has been proposed for the stream function in the liquid: ψ = Uy − U a π eπx/a sin πy a . Verify that this stream function satisfies:

(a) Laplace’s equation.

(b) The boundary condition far above the bubble.

(c) The boundary conditions along the walls AA and BB.

Prove that for small values of y, the equation for the free surface OPQ is: x = πy2 6a .

What boundary condition must be satisfied along this free surface? Show that if this boundary condition is satisfied for small values of y, then the bubble-rise velocity in an otherwise stagnant liquid is: U = ga 3π .