The viscoplastic circular hydraulic jump

We examine theoretically the spread of a jet impacting on a circular disk and the hydraulic jump of a viscoplastic fluid of the Herschel-Bulkley type. The depth-averaging approach is employed in the supercritical region, and the subcritical flow is assumed to be inertialess of the lubrication type. The jump is treated as a shock, where the balance of mass and momentum is established in the radial direction, including the effect of surface tension across the jump. We show that, in contrast to the Newtonian jet, which requires separate formulations in the developing-boundary layer and fully viscous layers, the supercritical formulation for the fully yielded and pseudo-plug layers is uniformly valid between the impingement zone and the jump. Consequently, a viscoplastic jet does not experience the discontinuity in the film height, pseudo-plug layer velocity gradient, and shear stress, exhibited by a Newtonian film at the transition location. The jump is found to occur closer to impingement, with growing height, as the yield stress increases; the subcritical region becomes invaded by the pseudo-plug layer. The viscosity does not influence sensibly the jump location and height except for small yield stress; only the yielded-layer is found to remain sensitive to the power-law rheology for any yield stress. In particular, shear thickening can cause the fully yielded layer to drop in height despite the jump in the film surface. We also find that the jump would not occur if the disk was smaller than a critical size, but the yield stress tends to enhance the formation of the jump compared to a Newtonian jet. We show that an almost constant local downstream Froude number also exists for a viscoplastic fluid. Finally, our results reduce to the limiting cases of Bingham, power-law, and Newtonian fluids.