The free-surface flow formed by a circular jet impinging on a rotating disk is analysed theoretically. The study explores the effects of rotation and inertia on the thin-film flow. Both boundary-layer height and film thickness are found to diminish with rotation speed. A maximum film thickness develops in the supercritical region, which reflects the competition between the convective and centrifugal effects. Unlike the flow on a stationary disk, an increase in the wall shear stress along the radial direction is predicted, at a rate that strengthens with rotating speed. Our results corroborate well existing measurements. The location and height of the hydraulic jump are determined subject to the value of the thickness at the edge of the disk, which is established first for a stationary disk based on the capillary length, and then for a rotating disk using existing analyses and measurements in spin coating. The case of a stationary is revisited in an effort to predict the location and height of the jump uniquely. The formulated value of the height at the edge of the disk seems to give excellent results for a jet at moderately high flow rate (or low viscosity) where the jump structure is well identifiable in reality.

The free-surface flow formed by a circular jet impinging on a stationary disk is analysed theoretically. We develop a simple and coherent model to predict the location and height of the jump for high-viscosity liquids. The study explores the effect of gravity in the supercritical flow. The formulation reduces to a problem, involving only one parameter: $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$ , where $Re$ and $Fr$ are the Reynolds and Froude numbers based on the flow rate and the jet radius. We show that the jump location coincides with the singularity in the thin-film equation when gravity is included, suggesting that the jump location can be determined without the knowledge of downstream flow conditions such as the jump height, the radius of the disk, which corroborates earlier observations in the case of type I circular hydraulic jumps. Consequently, there is no need for a boundary condition downstream to determine the jump radius. Our results corroborate well existing measurements and numerical simulation. Our predictions also confirm the constancy of the Froude number $Fr_{J}$ based on the jump radius and height as suggested by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, 54002). We establish theoretically the conditions for $Fr_{J}$ to remain independent of the flow rate. The subcritical flow and the height of the hydraulic jump are sought subject to the thickness at the edge of the disk, comprising contributions based on the capillary length and minimum flow energy. The thickness at the edge of the disk appears to be negligibly small for high-viscosity liquids.

When a laminar inclined circular jet impinges on a horizontal surface, it forms a non-circular hydraulic jump resulting from the non-axisymmetric flow. In this study, we develop an integral approach in the boundary-layer (near impingement) and thin-film regions to theoretically analyze the flow field and the hydraulic jumps structure. We particularly explore the interplay among inertia, gravity, viscosity, and the effective inclination angle on the non-axisymmetric flow. The boundary-layer height exhibits an azimuthal dependence at a strong gravity level only; however, the thin film thickness as well as the hydraulic jump profile shows a strong non-axisymmetric behavior at all gravity levels. In contrast to the existing literature, the present study accounts for the presence of the boundary layer near impingement and the azimuthal flow. We demonstrate that the azimuthal flow component cannot be neglected in the presence of gravity. The theory is validated against existing experimental results for the inclined jet of water.

The present work investigates the causes of the circular hydraulic jump for both low- and high-viscosity liquids in an effort to address a recent dispute in the research community. We first validate our numerical model against existing experiments and then study the effects of different parameters involved in the problem. The influences of viscosity, gravity, and surface tension on the formation of the jump are comprehensively explored. We observe a significant difference in the mechanisms behind the hydraulic jump for low- and high-viscosity liquids, which have rarely been reported. Surface tension is found to be responsible for the low-viscosity jump, while gravity dominates the high-viscosity jump, which partially resolves the recent noise regarding the cause of the jump in a consistent manner.

The influence of inertia and elasticity on the onset and stability of three-dimensional thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The onset and the stability of steady convective patterns, namely rolls, hexagons and squares, are studied in the post-critical range of the Rayleigh number by using an amplitude equation approach. The square pattern is found to be unstable. In contrast to Newtonian fluids, the hexagonal pattern can be stable for a certain range of elasticity.