UPSTREAM INFLUENCE AND THE FORM OF STANDING HYDRAULIC JUMPS IN LIQUID-LAYER FLOWS ON FAVORABLE SLOPES

Steady planar flow of a liquid layer over an obstacle is studied for f avourable slopes. First, half-plane Poiseuille flow is found to be a n on-unique solution on a uniformly sloping surface since eigensolutions exist which are initially exponentially small far upstream. These hav e their origin in a viscous-inviscid interaction between the retarding action of viscosity and the hydrostatic pressure from the free surfac e. The cross-stream pressure gradient caused by the curvature of the s treamlines also comes into play as the slope increases. As the interac tion becomes nonlinear, separation of the liquid layer can occur, of a breakaway type if the slope is sufficiently large. The breakaway repr esents a hydraulic jump in the sense of a localized relatively short-s caled increase in layer thickness, e.g. far upstream of a large obstac le. The solution properties give predictions for the shape and structu re of hydraulic jumps on various slopes. Secondly, the possibility of standing waves downstream of the jump is addressed for various slope m agnitudes. A limiting case of small gradient, governed by lubrication theory, allows the downstream boundary condition to be included explic itly. Numerical solutions showing the free-surface flow over an obstac le confirm the analytical conclusions. In addition the predictions are compared with the experimental and computational results of Pritchard et al. (1992), yielding good qualitative and quantitative agreement. The effects of surface tension on the jump are also discussed and in p articular the free interaction on small slopes is examined for large B ond numbers.