Gravity-Capillary Two-Dimensional Free Surface Flows in the Presence of Rigid Walls.

Two-dimensional free surface flows in a domain bounded on the left by an infinite wall and on the right by a semi-infinite wall and a free surface is considered. The acceleration of gravity g and surface tension T are included in the free surface condition. The angles between the right (left) wall and the horizontal is denoted by alpha(beta), and the angle between the free surface and the right wall at the separation point by gamma. Numerical solutions are computed via series truncation. The flows are characterized by the parameters H = (Q^2/gw^3) ^{1/3}, alpha and beta when T = 0. Here Q and w are the total flux and the distance between the separation point and the left wall. It is shown that for given values of beta and alpha > pi/3, there is a critical value H _{c} of H such that H = H_{c} for gamma = 2pi/3, H > H_{c} for gamma = pi and H < H_{c} for gamma = pi-alpha. However when alpha <=q pi/3, there is only one configuration with gamma = pi . When alpha = pi-beta and 0 < beta < pi, the motion of a two-dimensional bubble rising at a constant velocity U in a tube of width h is considered. The same configuration describes also a jet emerging from a nozzle and falling down along an inclined wall. The problem is characterized by the Froude number F = U/sqrt {gh} = H^{3/2}, the Weber number omega = rho U^2h/T, beta and the supplement vartheta of gamma. Here rho is the density of the fluid. Numerical solutions are obtained for all values of 0 < beta < pi. When a small amount of T is included, a particular value F* of F is selected. The numerical values of F* and the corresponding free surface profiles for which T = 0 are found to be in good agreement with experimental data for bubbles rising in an inclined tube when 0 < beta<=qpi/2.