Hydraulic jump in yield stress fluids associated with an abrupt channel deviation

A free surface flow meeting a change of the geometry of the channel undergoes a perturbation of its characteristics; the entity of the perturbation varies greatly, depending on the upstream flow conditions, on the type of geometric change and on the rheologic properties of the fluid. In this work, we study the behaviour of a laminar current of non-Newtonian fluid, facing an abrupt deviation of axis of the wide, rectangular channel in which it is flowing. We consider a Herschel-Bulkley fluid, with a rheology described by a three-parameter constitutive equation and in particular its transition from supercritical to subcritical flow. A system of four transcendental equations is obtained by coupling the mass balance equation and the momentum conservation equations (both parallel and perpendicularly to the wave front) to the fluid constitutive equation. This system allows obtaining the downstream flow conditions and the value of the Mach angle of deviation of the wave front as a function of the upstream Froude number. Starting from the general case of a Herschel-Bulkley fluid, the Bingham, Ostwald-DeWaele (power-law) and Newtonian special cases are studied for the same geometry. We then propose an approximate closed-form solution to evaluate the critical depth, obtained by expanding in Taylor’s series the transcendental equations of the model. Results indicate that upstream flow conditions, described by the Froude number, greatly affect the downstream flow and the Mach angle. Furthermore, an increase of the angle of deviation causes a deceleration of the fluid and an increase of the dowstream depth. An application to data taken from the literature is then illustrated. A kaolin suspension of assigned characteristics flowing in a rectangular flume is subject to an abrupt deviation of 45°; the model indicates a angle of deviation of the wave front of 68°; furthermore, the fluid decelerates significantly and greatly increases its depth. Results obtained are of practical interest in the design of channels conveying mud flows or mine tailings of yield stress behaviour; as these materials may be polluting, an appropriate channel design taking into account hydraulic jumps due to deviations is important in order to avoid overspilling.