We examine the structure of the continuous circular hydraulic jump and recirculation for a jet impinging on a disk. We use a composite mean-field thin-film approach consisting of subdividing the flow domain into three regions of increasing gravity strength: a developing boundary layer near impact, an intermediate supercritical viscous layer and a region comprising the jump and subcritical flow. Unlike existing models, the approach does not require any empirically or numerically adjusted boundary conditions. We demonstrate that the stress or corner singularity for a film draining at the edge is equivalent to an infinite slope of the film surface, which we impose as the downstream boundary condition. The model is validated against existing experiment and numerical simulation of the boundary-layer and Navier–Stokes equations. We find that the flow in the supercritical region remains insensitive to the change in gravity level but is greatly affected by viscosity. The existence of the jump is not necessarily commensurate with the presence of recirculation, which is strongly dependent on the upstream curvature and steepness of the jump.

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The transient flow of a circular liquid jet impinging on a horizontal disk, and the hydraulic jump formation, are examined theoretically and numerically. The interplay between inertia and gravity is particularly emphasised. The flow is governed by the thin-film equations, which are solved along with a force balance across the jump. The unsteadiness of the flow is caused by a linearly accelerating jet from an initial to a final steady state. To validate the predicted boundary-layer flow evolution, an analytical development is conducted for small distance from impingement, and for small time. In addition, the predictions of the film profile and jump location are compared against numerical simulation for the transient flow, and are further validated against experiment for steady flow. The evolutions of the film thickness and the wall shear stress in the developing boundary-layer region are found to be similar to those reported for a fluid lying on a stretching surface. The flow responds to the jet acceleration quasi-steadily near impingement but exhibits a long-term transient behaviour near the jump. Analysis of the jump evolution is considered in the range 5 < Fr < 40 for the Froude number (based on the jet radius and velocity). For Fr < 10, the jump reaches the final state instantly when the jet acceleration ceases. At higher Froude number, the jump settles at a later time, exhibiting an overshoot in the thickness due to the dominance of inertia.

An analysis of laminar natural convection in inclined slots subjected to patterned heating has been performed. The imposed heating takes a simple form characterized by a single Fourier mode combined with uniform heating. It is shown that periodic heating applied at the lower plate produces no net flow when the slot is either horizontal or vertical, but a net upward flow is generated when the slot is tilted. Periodic heating applied at the upper plate produces net downward flow in the inclined situation. The addition of uniform heating promotes the upward flow while cooling has the opposite effect. There is a critical inclination angle at which the maximum net flow rate is greatest. Dynamic and thermal boundary layers are present when the wavenumber of the imposed heating is large. The use of heating at both plates, with the same wavenumber, leads to a flow dominated by the plate exposed to a more intense heating; when the two plates are heated equally no net flow is observed irrespective of the inclination angle. Changes of the relative positions of the two patterns can change the net flow rate by up to 50 %. The intensity of the flow increases with reduction of the Prandtl number. If the heating applied to the plates is of different wavelength, but of the same intensity, a wide range of behaviours of the flow system is possible. The details of this response are sensitive to the ratio of the two wavenumbers.