Rotating convection is considered on the tilted $f$-plane where gravity and rotation are not aligned. For sufficiently large rotation rates, $\Omega$, the Taylor–Proudman effect results in the gyroscopic alignment of anisotropic columnar structures with the rotation axis giving rise to rapidly varying radial length scales that vanishes as $\Omega ^{-1/3}$ for $\Omega \rightarrow \infty$. Compounding this phenomenon is the existence of viscous (Ekman) layers adjacent to the impenetrable bounding surfaces that scale as $\Omega ^{-1/2}$. In this investigation, these constraints are relaxed upon utilising a non-orthogonal coordinate representation of the fluid equations where the upright coordinate aligns with rotation axis. This exposes the problem to asymptotic perturbation methods that permit: (i) relaxation of the constraints of gyroscopic alignment; (ii) the filtering of Ekman layers through the uncovering of parameterised velocity pumping boundary conditions; and (iii) the development of reduced quasi-geostrophic systems valid in the limit $\Omega \rightarrow \infty$. Linear stability investigations reveal excellent quantitative agreement between results from parameterised or unapproximated mechanical boundary conditions. For no-slip boundaries, it is demonstrated that the associated Ekman pumping alters convective onset through an enhanced destabilisation of large spatial scales. The range of unstable modes at a fixed thermal forcing is thus significantly extended with a direct dependence on $\Omega$. This holds true even for geophysical and astrophysical regimes characterised by extreme values of the non-dimensional Ekman number $E$. The nonlinear regime is explored via the global heat and momentum transport of single-mode solutions to the quasi-geostrophic systems which indicate $O(1)$ changes which do not scale with the size of $E$.

A cylindrical liquid thread readily destabilizes into a series of drops due to capillary instability, which is also responsible for undesirable bead-on-fibre structures observed when coating a thin fibre. In this experimental study, we show how a falling liquid thread can be stabilized by internally distorting the cross-sectional shape using two vertically hung fibres. Below a critical flow rate $Q_c$, the dual-fibre system deforms the falling thread into a smooth column with a non-circular cross-section, thereby suppressing instability. Above $Q_{{c}}$, the cylindrical thread is left undeformed by the fibres and destabilizes into beads connected by a stable, non-cylindrical film. An empirical stability threshold is identified showing that flow delays the onset of capillary instability when compared with a marginally stable quasi-static coating. When the flow is unstable $Q>Q_c$, the bead velocity $v$ obeys a simple scaling law that is well supported by our experiments over a large parameter range. This suppression technique can be extended to other slender geometries, such as a ribbon, which shows similar qualitative results but exhibits a different stability threshold due to spontaneous dewetting about its short edge.

We present the results of an experimental study of buoyancy-driven exchange flows in a vertical pipe, where the lower fluid is Newtonian of low viscosity and the upper fluid has a yield stress. The fluids are initially separated by a gate valve, opened at time $\hat {t}=0$. The fluids are miscible, but away from the diffusive limit. For a sufficiently large ratio $Y$, of the yield stress to the buoyancy stress, no sustained fluid motions arise: the flow is stable. For smaller $Y$ numbers an exchange flow results. Commonly, the less dense fluid penetrates upwards in a central finger, displacing the upper fluid downwards around the walls of the pipe. Three regimes are classified: helical finger, disconnected finger and slug flow. The transition between regimes is governed by increasing relevance of inertial to viscous stresses, in balancing buoyancy. The disconnected finger and slug flow regimes are associated with yielded fluid at the interface and early growth of instabilities. Helical fingers are viscous dominated and evolve slowly until late in the experiments. The scenarios studied represent an idealised set-up for the industrial process of plug cementing. The regimes identified are helpful for industrial process design.

Most of the existing theories on electrophoresis are based on the consideration of a weak applied electric field and ions as point charges, which create a mean electric potential and neglect ion–solvent interactions. These theories cannot demonstrate the dependence of electrophoretic mobility on the applied electric field (nonlinear electrophoresis), reversal in mobility with increasing ion concentration and/or surface charge density or counterion saturation in the electric double layer. In this study we consider a modified electrokinetic model to analyse nonlinear electrophoresis by taking into account the finite ion size effects and ion–ion electrostatic correlations. In this approach, the mean-field-based model is extended to capture the many-body phenomena by considering the non-local electrostatic contribution in the ion free energy functional and the ion–ion hydrodynamic steric interactions are incorporated through the volume exclusion effect in the electrochemical potential. The viscosity of the medium is considered to vary with the local ionic volume fraction. Stronger correlations for multivalent counterions create ion layering, charge density oscillation and mobility reversal. Such phenomena are captured by the present continuum model. The ion crowding attenuates the growth of the electrophoretic mobility with the electric field. At a higher range of the imposed electric field, the ion concentration in the electric double layer enhances, which modifies both the overscreening and ion crowding processes.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low-dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes and volume fractions. An upstream shift of the Mach disk in the presence of particles reveals significant two-way coupling even at low mass loading. A semi-analytic model that predicts the extent of the Mach disk shift is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The per cent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio and interphase slip velocity and inversely with the particle diameter.

Porous membranes are thin solid structures that allow the flow to pass through their tiny openings, called pores. Flow inertia may play a significant role in several filtration flows of natural and engineering interest. Here, we develop a predictive macroscopic model to describe solvent and solute flows past thin membranes for non-negligible inertia. We leverage homogenization theory to link the solvent velocity and solute concentration to the jumps of solvent stress and solute flux across the membrane. Within this framework, the membrane acts as a boundary separating two distinct fluid regions. These jump conditions rely on several coefficients, stemming from closure problems at the microscopic pore scale. Two approximations for the advective terms of Navier–Stokes and advection–diffusion equations are introduced to include inertia in the microscopic problem. The approximate inertial terms couple the micro- and macroscopic fields. Here, this coupling is solved numerically using an iterative fixed-point procedure. We compare the resulting models against full-scale simulations, with a good agreement both in terms of averaged values across the membrane and far-field values. Eventually, we develop a strategy based on unsupervised machine learning to improve the computational efficiency of the iterative procedure. The extension of homogenization towards weak-inertia flow configurations as well as the performed data-driven approximation may find application in preliminary analyses as well as optimization procedures towards the design of filtration systems, where inertia effects can be instrumental in broadening the spectrum of permeability and selectivity properties of these filters.

We report an experimental study of Rayleigh–Bénard convection of liquid metal GaInSn in a cuboid cell with an aspect ratio of 0.5 under the effect of a horizontal magnetic field. The Rayleigh number spans a range of $3.8\times 10^5 \leqslant Ra \leqslant 1.1\times 10^7$, while the magnetic field strength reaches up to 0.5 T, corresponding to a maximum Hartmann number to 2041. By combining temperature and velocity measurements, we identify several flow morphologies, including a novel cellular pattern characterized by four stacked vortices that periodically squeeze and induce velocity reversals. Based on the identified flow morphologies, we partition the entire ($Ra, Ha$) parameter space into five distinct flow regimes and systematically investigate the flow characteristics within each regime. The temperature gradient and oscillation frequency exhibit scaling relationships with the combined parameters $Ra$ and $Ha$. Notably, we observe a coupling between flow regime and global transport efficiencies, particularly in a regime dominated by the double-roll structure, which experiences a maximum 36 % decrease in heat transfer efficiency compared with the single-roll structure. The dependencies of heat and momentum transport on $Ra$ and $Ha$ follow scaling laws as $Nu \sim (Ha^{-2/3}RaPr^{-1})^{3/5}$ and $Re \sim (Ha^{-1}RaPr^{-1})^{4/3}$, respectively.

We employ direct numerical simulations to investigate the heat transfer and flow structures in turbulent Rayleigh–Bénard convection in both cylindrical cells and laterally periodic domains, spanning an unprecedentedly wide range of aspect ratios $0.075 \leqslant \varGamma \leqslant 32$. We focus on Prandtl number ${Pr}=1$ and Rayleigh numbers ${{Ra}}=2\times 10^7$ and ${{Ra}}=10^8$. In both cases, with increasing aspect ratio, the heat transfer first increases, then reaches a maximum (which is more pronounced for the cylindrical case due to confinement effects), and then slightly goes down again before it finally saturates at the large aspect ratio limit, which is achieved already at $\varGamma \approx 4$. Already for $\varGamma \gtrsim 0.75$, the heat transfers in both cylindrical and laterally periodic domains become identical. The large-$\varGamma$ limit for the volume-integrated Reynolds number and the boundary layer thicknesses are also reached at $\varGamma \approx 4$. However, while the integral flow properties converge at $\varGamma \approx 4$, the confinement of a cylindrical domain impacts the temperature and velocity variance distributions up to $\varGamma \approx 16$, as thermal superstructures cannot form close to the sidewall.

The dominant mode instability in hypersonic boundary-layer transition is the so-called second-mode instability, which has a peculiar nature strongly coupled with thermoacoustic phenomena. In linear stability theory, the unstable wave is associated with one of the two eigenvalues that originate from the acoustic branches, referred to as slow and fast modes. Interestingly, the unstable mode (slow or fast) reaches its maximum amplification as the other mode (fast or slow) attains a minimum. The phase velocity of the two modes is then very close, and this phenomenon is called synchronization. The aim of the present study is to unravel the physical mechanism that explains the second-mode growth. To that aim, second-order nonlinear equations are written for the disturbances given by linear stability. In this framework, entropy, kinetic energy and temperature energy budgets are obtained up to second order. The budgets are scrutinized for various Mach numbers and for adiabatic and cold-wall thermal conditions. Perturbation entropy budgets clearly show the process is a reversible one. An energy exchange between kinetic energy and temperature energy of the weakly nonlinear modes is driven by pressure–dilatation terms. As underlined in previous studies, the unstable mode experiences an alternate heating and cooling near the wall, which is shown to be a rather nonlinear process. The change in fluctuating thermal energy in the form of a dilatational wave is sustained by pumping disturbance kinetic energy through the pressure–dilatation term, the direction of the conversion being driven by the relative phase between pressure and dilatation. This process is similar for the slow and fast modes, the unstable mode being amplified and the other being damped. No change in the process has been noted at the location of the synchronization, suggesting that the modes have the same nature but evolve independently.

We introduce a continuum framework for the energetics of particle-size segregation in bidisperse granular flows. Building on continuum segregation equations and a recent segregation flux model, the proposed framework offers general analytical expressions to study the physics of granular flows from a mechanical energy perspective. We demonstrate the framework's applicability by examining the energetics of shear-driven granular flows. Numerical experiments with varying frictional coefficients and particle-size ratios reveal two distinct phases in the energetics, marked by the separate onset of particle segregation and diffusive remixing. Furthermore, our numerical simulations alongside previous experimental results show that the bulk Richardson number $Ri$, defined as the potential energy to kinetic energy ratio at steady state, follows the scaling relationship $Ri\equiv \hat {E}^{(s)}_{gp} / \hat {E}^{(s)}_{k} \propto Pe^{-1/2}_{sr}$ for $0.1 \leq Ri\leq ~10^{3}$ and $10^{-4} \leq Pe_{sr} \leq ~300$, the segregation–rheology Péclet number. Finally, we present a Péclet-number-dependent theoretical expression for the degree of mixing (or segregation), validated by the compiled numerical and experimental dataset. Our findings hint that the bulk segregation–mixing state can be predicted and controlled using the segregation Péclet number $Pe$ and $Pe_{sr}$, both determined from known system parameters, providing an instrumental tool for engineering and geophysical applications.

Direct numerical simulations of the turbulence of a Herschel–Bulkley (HB) fluid in a rough channel are performed at a shear Reynolds number $Re_{\tau } \approx 300$ and a Bingham number ${Bn} \approx 0.9$. For the type of rough surface used in this study, the results indicate that Townsend's wall similarity hypothesis also holds for HB fluids. However, there are notable differences compared with the effect of roughness on Newtonian fluids. More specifically, the effect of roughness appears to be slightly stronger for HB fluids, in the sense that the bulk Reynolds number, based on the viscosity at the wall, is reduced further due to the increase in viscosity in the troughs of the roughness surface induced by the low shear. At the same time, for the simulated rough surface, the contribution of form drag to the total pressure drop is reduced from 1/4 to about 1/5 due to the persistence of viscous shear in the boundary layer, reducing its shielding effect. As for the friction factor, due to the nonlinearity of the HB constitutive relation, its use with the wall shear rate from the mean wall shear stress underpredicts the minimum viscosity at the wall by up to 18 %. This inevitably leads to uncertainties in the prediction of the friction factor. Finally, it is observed that the rough surface is unable to break the peculiar near-wall flow structure of HB fluids, which consists of long persistent low-speed streaks occupying the entire domain. This means that the small-scale energy is significantly reduced for HB fluids, even in rough channels, with the energy more concentrated in the lower wavenumber range, implying an increase in the slope of the power spectrum to $-7/2$ in the inertial range, as shown by Mitishita et al. (J. Non-Newtonian Fluid Mech., vol. 293, 2021, 104570).

This paper presents numerical simulations of the free fall of homogenous cylinders of length-to-diameter ratios $2$, $3$ and 5 and solid-to-fluid-density ratios $\rho _s/\rho$ going from 0 to 10 in transitional regimes. The path instabilities are shown to be due to two types of transitional states. The well-known fluttering state is a solid mode, characterised by significant oscillations of the cylinder axis due to a strong interaction between the vortex shedding in the wake and the solid degrees of freedom. Weakly oscillating, mostly irregular trajectories, are fluid modes, associated with purely fluid instabilities in the wake. The interplay of solid and fluid modes leads to a varying scenario in which the length-to-diameter and density ratios play an important role. The description is accompanied by the presentation of the identified transitional states in terms of path characteristics and vorticity structure of the wakes and by bifurcation diagrams showing the evolution of asymptotic states with increasing Galileo numbers. There appears to be a strong difference between the behaviour of cylinders of aspect ratio $L/d=3$ and 5. A similar contrast is stated between light cylinders of density ratios $\rho _s/\rho \le 2$ and dense cylinders of density ratios 5 and 10. Finally, the question of the scatter of values of the drag coefficient and of the frequency of oscillations raised in the literature is addressed. It is shown, that in addition to external parameters (Galileo number, density and aspect ratio) the amplitude of oscillations characterising the instability development is to be taken into account to explain this scatter. Fits of the simulation results to simple correlations are proposed. Namely that of the drag coefficient proves to be accurate (better than 1 % of accuracy) but also that of the Strouhal number (a few per cent of accuracy) may be of practical use.