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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Laminar natural convection is investigated in an infinite vertical slot which has one wall with a corrugated profile, and which is subject to either a uniform or periodic heating profile. This configuration has the attractive feature that it enables a study of the effects that may be produced via the interaction of heating and topography patterns. It is found that the addition of the grooves to an isothermal plate leads to a reduction in the vertical fluid flow and an increase of the transverse heat flow. In contrast, imposing sinusoidal heating on a flat surface generates convection that appears as counter-rotating rolls but there is no net vertical flow. The combination of the two effects of corrugation together with periodic heating leads to a plethora of flow patterns involving a combination of rolls and stream tubes that carry the fluid along the slot. The details of this vertical flow are governed by a pattern interaction effect dictated by the relative positions of the heating and corrugation patterns; when hot spots of the imposed heating overlap the peaks in the grooves the net flow is upward; in contrast, when they lie over the troughs the resultant flow is downward. The interplay between the thermal and geometrical effects weakens as the wavelength of the structure is reduced. The inclusion of a sufficiently strong uniform heating also seems to wash away the pattern interaction effect.

In the study a simple model of coagulation for nanoparticles is developed to study the effect of diffusion on the particle coagulation in the one-dimensional domain using the Taylor-series expansion method of moments. The distributions of number concentration, mass concentration, and particle average volume induced by coagulation and diffusion are obtained