The classical problem of steady rarefied gas flow past an infinitely thin circular disk is revisited, with particular emphasis on the gas behaviour near the disk edge. The uniform flow is assumed to be perpendicular to the disk surface. An integral equation for the velocity distribution function, derived from the linearised Bhatnagar–Gross–Krook model of the Boltzmann equation and subject to diffuse reflection boundary conditions, is solved numerically. The numerical method fully accounts for the discontinuity in the velocity distribution function that arises due to the presence of the edge. It is found that a kinetic boundary layer forms near the disk edge, extending over several mean free paths, and that its magnitude scales as $\textit{Kn}^{1/2}$ as the Knudsen number $\textit{Kn}$ (defined with respect to the disk radius) tends to zero. A thermal polarisation effect, previously studied for spherical geometries, is also observed in the disk case, with a more pronounced manifestation near the edge that exhibits the same $\textit{Kn}^{1/2}$ scaling. The drag force acting on the disk is computed over a wide range of Knudsen numbers and shows good agreement with existing results for a hard-sphere gas and in the near-free-molecular regime.

In Navier–Stokes (NS) turbulence, large-scale turbulent flows inevitably determine small-scale flows. Previous studies using data assimilation with the three-dimensional (3-D) NS equations indicate that employing observational data resolved down to a specific length scale, $\ell ^{\rm 3\text{-}D}_{\ast }$, enables the successful reconstruction of small-scale flows. Such a length scale of ‘essential resolution of observation’ for reconstruction $\ell ^{\rm 3\text{-}D}_{\ast }$ is close to the dissipation scale in three-dimensional NS turbulence. Here, we study the equivalent length scale in two-dimensional (2-D) NS turbulence, $\ell ^{\rm 2\text{-}D}_{\ast }$, and compare with the three-dimensional case. Our numerical studies using data assimilation and conditional Lyapunov exponents reveal that, for Kolmogorov flows with Ekman drag, the length scale $\ell ^{\rm 2\text{-}D}_{\ast }$ is actually close to the forcing scale, substantially larger than the dissipation scale. Furthermore, we discuss the origin of the significant relative difference between the length scales, $\ell ^{\rm 2\text{-}D}_{\ast }$ and $\ell ^{\rm 3\text{-}D}_{\ast }$, based on inter-scale interactions, ‘cascades’ and orbital instabilities in turbulence dynamics.

We introduce a description of passive scalar transport based on a (deterministic and hyperbolic) Liouville master equation. Defining a noise term based on time-independent random coefficients, instead of time-dependent stochastic processes, we circumvent the use of stochastic calculus to capture the one-point space–time statistics of solute particles in Lagrangian form deterministically. To find the proper noise term, we solve a closure problem for the first two moments locally in a streamline coordinate system, such that averaging the Liouville equation over the coefficients leads to the Fokker–Planck equation of solute particle locations. This description can be used to trace solute plumes of arbitrary shape, for any Péclet number, and in arbitrarily defined grids, thanks to the time reversibility of hyperbolic systems. In addition to grid flexibility, this approach offers some computational advantages as compared with particle tracking algorithms and grid-based partial differential equation solvers, including reduced computational cost, no Monte-Carlo-type sampling and unconditional stability. We reproduce known analytical results for the case of simple shear flow and extend the description of mixing in a vortex model to consider diffusion radially and nonlinearities in the flow, which govern the long time decay of the maximum concentration. Finally, we validate our formulation by comparing it with Monte Carlo particle tracking simulations in a heterogeneous flow field at the Darcy (continuum) scale.

Dense granular flows exhibit both surface deformation and secondary flows due to the presence of normal stress differences. Yet, a complete mathematical modelling of these two features is still lacking. This paper focuses on a steady shallow dense flow down an inclined channel of arbitrary cross-section, for which asymptotic solutions are derived by using an expansion based on the flow’s spanwise shallowness combined with a second-order granular rheology. The leading-order flow is uniaxial with a constant inertial number fixed by the inclination angle. The streamwise velocity then corresponds to a lateral juxtaposition of Bagnold profiles scaled by the varying flow depth. The correction at first order introduces two counter-rotating vortices in the plane perpendicular to the main flow direction (with downwelling in the centre), and an upward curve of the free surface. These solutions are compared with discrete element method simulations, which they match quantitatively. This result is then used together with laboratory experiments to infer measurements of the second-normal stress difference in dense dry granular flow.

The stability of free jets is one of the fundamental problems that has driven the development of new theoretical and numerical methods in fluid mechanics. Extensive research has focused on the convective instabilities that characterise their elusive dynamics. However, in real-world configurations, free jets are often confined by solid walls which may exhibit different degrees of flexibility. The present paper presents, for the first time, evidence that even slightly flexible nozzles can lead to global instabilities. To show it, we adopted the classical tools of linear stability analysis, solving the fluid–structure interaction (FSI) problem by an arbitrary Lagrangian–Eulerian method, formulating a monolithic three-field problem. The investigation of the base flow properties reveals the effect of the Reynolds number, based on the bulk velocity and channel height, in the range $[50,200]$ and of the plate stiffness on the nozzle deformation and on the jet flow development. Exploiting an idea first proposed by Luchini and Charru, we develop an ad hoc quasi-one-dimensional model capable of predicting the displacement of elastic boundaries even for large displacements. The stability and sensitivity analysis shows that the interaction of the flow with the flexible structure leads to two categories of globally unstable modes: sinuous (in-phase) modes and varicose (out-of-phase) modes. All the results presented have been cross-checked with direct numerical simulations of the nonlinear FSI system, revealing that the instabilities correspond to supercritical bifurcations. This work has significant implications for many natural and industrial phenomena where a jet is produced by a compliant nozzle.