Direct numerical simulations have been conducted to investigate turbulent Rayleigh– Bénard convection (RBC) of liquid metal in a cuboid vessel with aspect ratio $\varGamma =5$ under an imposed horizontal magnetic field. Flows with Prandtl number $Pr=0.033$, Rayleigh numbers ranging up to $Ra\leq 10^{7}$, and Chandrasekhar numbers up to $Q\leq 9 \times 10^6$ are considered. For weak magnetic fields, our findings reveal that a previously undiscovered decreasing region precedes the enhancement of heat transfer and kinetic energy. For moderate magnetic fields, we have reproduced the reversals of the large-scale flow, which are considered a reorganization process of the roll-like structures that were reported experimentally by Yanagisawa et al. (Phys. Rev. E, vol. 83, 2011, 036307). Nevertheless, the proposed approach of skewed-varicose instability has been substantiated as insufficient to elucidate fundamentally the phenomenon of flow reversal, an occurrence bearing a striking resemblance to the large-scale intermittency observed in magnetic channel flows. As we increase the magnetic field strength further, we observe that the energy dissipation of the system comes primarily from the viscous dissipation within the boundary layer. Consequently, the dependence of Reynolds number $Re$ on $Q$ approaches a scaling as $Re\,Pr/Ra^{2/3} \sim Q^{-1/3}$. At the same time, we find the law for the cutoff frequency that separates large quasi-two-dimensional scales from small three-dimensional ones in RBC flow, which scales with the interaction parameter as ${\sim }N^{1/3}$.

In an effort to capture the continuous hydraulic jump and flow structure for a jet impinging on a disk, we recently proposed a composite mean-field thin-film approach consisting of subdividing the flow domain into three distinct connected regions of increasing gravity strength (Wang et al., J. Fluid Mech., vol. 966, 2023, A15). In the present study, we further validate our approach, and examine the characteristics and structure of the circular jump and recirculation. The influence of the disk radius is found to be significant, especially in the subcritical region. Below a disk radius, the jump transits from type Ia to type 0 after the recirculation zone has faded. The supercritical flow and jump location are insensitive to the disk size, but the jump length and height as well as the vortex size are strongly affected, all decreasing with decreasing disk radius, exhibiting a maximum with the flow rate for a small disk. The jump is relatively steep with a strong recirculation zone for a high obstacle at the disk edge. Comparison against the Navier–Stokes solution of Askarizadeh et al. (Phys. Rev. Fluids, vol. 4, 2019, 114002; Intl J. Heat Mass Transfer, vol. 146, 2020, 118823) for the weak and intermediate surface tension suggests that the surface tension effect is unimportant for a high obstacle for a jump of type 0 or type Ia. The film thickness at the disk edge for a freely draining film is found to comprise, in addition to a static component (capillary length), a dynamic component: ${h_\infty }\sim {(Fr/{r_\infty })^{2/3}}$ that we establish by minimizing the Gibbs free energy at the disk edge, and, equivalently, is also the consequence of the flow becoming supercritical near the edge. By assuming negligible film slope and curvature at the leading edge of the jump and maximum height at the trailing edge, we show that the jump length is related to the jump radius as ${L_J}\sim Re{(F{r^2}/{r_J}^5)^{1/3}}$. The vortex length follows the same behaviour. The energy loss and conjugate depth ratio exhibit a maximum with the flow rate, which we show to originate from the descending and ascending branches of the supercritical film thickness. The presence of the jump is not necessarily commensurate with that of a recirculation; the existence of the vortex closely depends on the upstream curvature and steepness of the jump. The surface separating the regions of existence/non-existence of the recirculation is given by the universal relation $R{e^{10/3}}F{r^2} = 9r_\infty ^9/50$. The jump can be washed off the edge of the disk, particularly at low viscosity and small disk size. The flow in the supercritical region remains insensitive to the change in gravity level and disk size but is greatly affected by viscosity.

Plane unsteady potential flows of an ideal incompressible fluid with a free boundary are considered in the absence of external forces and surface tension. Examples of exact solutions in situations where the entire boundary of the domain occupied by the fluid is completely free are constructed. There may be polar singularities of the complex velocity function inside the fluid, which corresponds to the presence of a source or a sink there.

A theoretical and computational study analysing the initiation of yield-stress fluid percolation in porous media is presented. Yield-stress fluid flows through porous media are complicated due to the nonlinear rheological behaviour of this type of fluid, rendering the conventional Darcy type approach invalid. A critical pressure gradient must be exceeded to commence the flow of a yield-stress fluid in a porous medium. As the first step in generalising the Darcy law for yield-stress fluids, a universal scale based on the variational formulation of the energy equation is derived for the critical pressure gradient which reduces to the purely geometrical feature of the porous media. The presented scaling is then validated by both exhaustive numerical simulations (using an adaptive finite element approach based on the augmented Lagrangian method), and also the previously published data. The considered porous media are constructed by randomised obstacles with various topologies; namely square, circular and alternatively polygonal obstacles which are mimicked based on Voronoi tessellation of circular cases. Moreover, computations for the bidispersed obstacle cases are performed which further demonstrate the validity of the proposed universal scaling.

The first stages of the path instability phenomenon affecting the buoyancy-driven motion of gas bubbles rising in weakly or moderately viscous liquids are examined using a recently developed numerical approach designed to assess the global linear stability of incompressible flows involving freely evolving interfaces. Predictions for the critical bubble size and frequency of the most unstable mode are found to agree well with reference data obtained in ultrapure water and in several silicone oils. By varying the bubble size, stability diagrams are built for several specific fluids, revealing three distinct regimes with different bifurcation sequences. The spatial structure of the unstable modes is analysed, together with the variations of the bubble shape, position and orientation. For this purpose, displacements of the bubble surface are split into rigid-body components and volume-preserving deformations, allowing us to determine how the relative magnitude of the latter varies with the fluid properties and bubble size. Predictions obtained with freely deformable bubbles are compared with those found by maintaining the bubble shape determined in the base state frozen during the stability analysis. This comparison reveals that deformations leave the phenomenology of the first bifurcations unchanged in low-viscosity fluids, especially water. Hence, in such fluids, bubbles behave essentially as freely moving rigid bodies submitted to constant-force and zero-torque constraints, at the surface of which the fluid obeys a shear-free condition. In contrast, deformations change the nature of the primary bifurcation in oils slightly more viscous than water, whereas, somewhat surprisingly, they leave the near-threshold phenomenology unchanged in more viscous oils.