The initial flow past an impulsively started rotating and translating circular cylinder is asymptotically analysed using a Brinkman penalization method on the Navier–Stokes equation. In our previous study (J. Fluid Mech., vol. 929, 2021, A31), the asymptotic solution was obtained within the second approximation with respect to the small parameter, $\epsilon$, that is of the order of $1 / \lambda$. Here, $\lambda$ is the penalization parameter. In addition, the Reynolds number based on the cylinder radius and the translating velocity is assumed to be of the order of $\epsilon$. The previous study asymptotically analysed the initial flow past an impulsively started translating circular cylinder and investigated the influence of the penalization parameter $\lambda$ on the drag coefficient. It was concluded that the drag coefficient calculated from the integration of the penalization term exhibits a half-value of the results of Bar-Lev & Yang (J. Fluid Mech., vol. 72, 1975, pp. 625–647) as $\lambda \to \infty$. Furthermore, the derivative of vorticity in the normal direction was found to be discontinuous on the cylinder surface, which is caused by the tangential gradient of the pressure on the cylinder surface. The present study hence aims to investigate the variance on the drag coefficient against the result of Bar-Lev & Yang (1975). First, we investigate the problem of an impulsively started rotating circular cylinder. In this situation, the moment coefficient is independent of the pressure on the cylinder surface so that we can elucidate the role of the pressure to the hydrodynamic coefficients. Then, the problem of an impulsively started rotating and translating circular cylinder is investigated. In this situation, the pressure force induced by the unsteady flow far from the cylinder is found to play a key role on the drag force for the agreement with the result of Bar-Lev & Yang (1975), whereas the variance still exists on the lift force. To resolve the variance, an alternative formula to calculate the hydrodynamic force is derived, assuming that there is the pressure jump between the outside and inside of the cylinder surface. The pressure jump is obtained in this analysis asymptotically. Of particular interest is the fact that this pressure jump can cause the variance on the lift force calculated by the integration of the penalization term.

The leakage from a fracture network to a surrounding medium during drainage, or backflow, driven by elastic relaxation, is considered. A network model is extended to include the effects of permeable boundaries, with the permeation through the wall assumed to be proportional to the local pressure. The regimes in which leakage is dominant relative to the parallel flow along the channel are evaluated at different times. Results show that, when the aperture of the channel is large enough, the parallel flow is greater than the permeation through the wall, and the channel thickness decreases in time, $t$, with a $t^{-1/3}$ behaviour, as reported previously. However, when the aperture is small, the channel thickness decreases exponentially in time. An asymptotic investigation of the solution for a single fracture is performed and extended to network systems. The study provides insight into the influence leakage may have on squeezing-induced flows, which is relevant to natural and engineering systems.

The general problem of tracer diffusion in non-equilibrium baths is important in a wide range of systems, from the cellular level to geographical length scales. In this paper, we revisit the archetypical example of such a system: a collection of small passive particles immersed in a dilute suspension of non-interacting dipolar microswimmers, representing bacteria or algae. In particular, we consider the interplay between thermal (Brownian) diffusion and hydrodynamic (active) diffusion due to the persistent advection of tracers by microswimmer flow fields. Previously, it has been argued that even a moderate amount of Brownian diffusion is sufficient to significantly reduce the persistence time of tracer advection, leading to a significantly reduced value of the effective active diffusion coefficient $D_A$ compared to the non-Brownian case. Here, we show by large-scale simulations and kinetic theory that this effect is in fact practically relevant only for microswimmers that effectively remain stationary while still stirring up the surrounding fluid – so-called shakers. In contrast, for moderate and high values of the swimming speed, relevant for biological microswimmer suspensions, the effect of Brownian motion on $D_A$ is negligible, leading to the effects of advection by microswimmers and Brownian motion being additive. This conclusion contrasts with previous results from the literature, and encourages a reinterpretation of recent experimental measurements of $D_A$ for tracer particles of varying size in bacterial suspensions.

Turbulent entrainment at the turbulent/non-turbulent interface (TNTI) plays an important role in understanding the turbulent diffusion. While entrainment in fully developed canonical turbulent flows has been extensively studied, the evolution of entrainment in spatially developing flows remains poorly understood. In this work, characteristics of entrainment and the effect of vortices on entrainment of the shear layer separated from a wall-mounted fence are studied by the experiment in a water channel. The shedding vortex experiences a series of stages, including generation, growth, deformation and breakdown into smaller vortices. With the development of the flow, entrainment varies correspondingly. The prograde vortex near the TNTI is found to suppress entrainment but have little effect on the detrainment process, while the retrograde vortex promotes entrainment and suppresses detrainment as well. Consequently, the local entrainment velocity is decreased by the prograde vortex and increased more significantly by the retrograde vortex. Along the streamwise direction, the time-mean entrainment velocity is smallest where the prograde vortex is strongest in the vortex deformation stage. However, the largest time-mean entrainment velocity is located where the enstrophy gradient near the TNTI is greatest after reattachment, rather than where the retrograde vortex is strongest shortly after the breakdown of the shedding vortex, because the scarcity of retrograde vortices in the vicinity of the TNTI makes their long-time cumulative contribution not as significant as their local enhancement. The present study reveals how entrainment evolves in the separated and reattaching flow, and improves our understanding of the effect of vortices on entrainment.

We present a deep probabilistic convolutional neural network (PCNN) model for predicting local values of small-scale mixing properties in stratified turbulent flows, namely the dissipation rates of turbulent kinetic energy and density variance, $\varepsilon$ and $\chi$. Inputs to the PCNN are vertical columns of velocity and density gradients, motivated by data typically available from microstructure profilers in the ocean. The architecture is designed to enable the model to capture several characteristic features of stratified turbulence, in particular the dependence of small-scale isotropy on the buoyancy Reynolds number $Re_b:=\varepsilon /(\nu N^2)$, where $\nu$ is the kinematic viscosity and $N$ is the background buoyancy frequency, the correlation between suitably locally averaged density gradients and turbulence intensity and the importance of capturing the tails of the probability distribution functions of values of dissipation. Empirically modified versions of commonly used isotropic models for $\varepsilon$ and $\chi$ that depend only on vertical derivatives of density and velocity are proposed based on the asymptotic regimes $Re_b\ll 1$ and $Re_b\gg 1$, and serve as an instructive benchmark for comparison with the data-driven approach. When trained and tested on a simulation of stratified decaying turbulence which accesses a range of turbulent regimes (associated with differing values of $Re_b$), the PCNN outperforms assumptions of isotropy significantly as $Re_b$ decreases, and additionally demonstrates improvements over the fitted empirical models. A differential sensitivity analysis of the PCNN facilitates a comparison with the theoretical models and provides a physical interpretation of the features enabling it to make improved predictions.

While capillary imbibition in tubes or porous materials has been studied extensively in the past, less attention has been paid to imbibition into a swellable porous material. However, swelling is commonly observed when a polymeric network, such as the cellulose composing paper fibres or sponges, absorbs a solvent. The incompressibility of the fluid leads to an elastic expansion of the polymeric matrix. In a porous material, swelling can affect the geometry of the pores, thus affecting the capillary flow. To describe this complex problem, we propose a model experiment, namely the capillary imbibition in a model pore composed of two parallel and stretched elastomeric fibres. In this configuration, one can observe both the progression of a capillary meniscus and the swelling of the fibres. We show that swelling enables a capillary imbibition for fibres placed further apart than the critical distance existing for non-swelling fibres. In this swelling-dominated regime, we identify a new imbibition dynamic at constant velocity which we rationalize using a linear poro-elastic theory. Finally, we describe the elastocapillary collapse of our model pore which is observed when capillary forces overcome the restoring tension force within the fibres.

The reflection of a rightward-moving oblique shock (RMOS) belonging to the first family, over an initially steady oblique shock wave (SOSW) produced by a wedge, is studied in this paper. To cover all possibilities, the problem is divided into a pre-shock reflection problem, for which the incident shock is assumed to reflect over the pre-interaction part of the SOSW, and a post-shock reflection problem, for which the incident shock is assumed to reflect over the post-interaction part. Such division, together with the definition of the equivalent problem defined on the reference frame co-moving with the nominal intersection point of the two shock waves, allows us to connect the reflection patterns with the six types of shock interference of Edney, which include type I–VI shock interferences depending on how an upstream oblique shock intersects a bow shock (types I and II are regular and Mach reflections of two shocks from the opposite sides; type III and type IV have two triple points or two Mach reflection configuration; type V and type VI are irregular and regular reflections of two shocks from the same side). We are thus able to identify all possible shock reflection types and find their transition conditions. Pre-shock reflection may yield IV, V and VI (of Edney's six types) shock interferences and post-shock reflection may yield I, II and III shock interferences. Pre- and post-shock reflections possibly occur at two different parts of the SOSW, and the complete reflection configuration may have one or both of them. Both transition condition study and numerical simulation are used to show how pre-shock reflection and post-shock reflection exist alone or coexist, leading to various types of combined pre-shock and post-shock reflections.