Magneto-gravity-elliptic instability is addressed here considering an unbounded strained vortex (with constant vorticity $2\varOmega$ and with ellipticity parameter $\varepsilon$) of a perfectly conducting fluid subjected to a uniform axial magnetic field (with Alfvén velocity scaled from the basic magnetic field $B)$ and an axial stratification (with a constant Brunt–Väisälä frequency $N$). Such a simple model allows us to formulate the stability problem as a system of equations for disturbances in terms of Lagrangian Fourier (or Kelvin) modes which is universal for wavelengths of the perturbation sufficiently small with respect to the scale of variation of the basic velocity gradients. It can model localised patches of elliptic streamlines which often appear in some astrophysical flows (stars, planets and accretion discs) that are tidally deformed through gravitational interaction with other bodies. In the limit case where the flow streamlines are exactly circular ($\varepsilon =0),$ there are fast and slow magneto-inertia-gravity waves with frequencies $\omega _{1,2}$ and $\omega _{3,4},$ respectively. Under the effect of finite ellipticity, the resonant cases of these waves, $\omega _i-\omega _j=n\varOmega$ $(i\ne j)$ ($n$ being an integer), can become destabilising. The maximal growth rate of the subharmonic instability (related to the resonance of order $n=2)$ is determined by extending the asymptotic method by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312). The domains of the $(k_0B/\varOmega, N/\varOmega )$ plane for which this instability operates are identified ($1/k_0$ being a characteristic length scale). We demonstrate that the $N\rightarrow 0$ limit is, in fact, singular (discontinuous). The axial stable stratification enhances the subharmonic instability related to the resonance between two slow modes because, at large magnetic field strengths, its maximal growth rate is twice that found in the case without stratification.

Vertical convection is the fluid motion that is induced by the heating and cooling of two opposed vertical boundaries of a rectangular cavity (see e.g. Wang et al., J. Fluid Mech., vol. 917, 2021, A6). We consider the linear stability of the steady two-dimensional flow reached at Rayleigh numbers of O($10^8$). As a function of the Prandtl number, $Pr$, and the height-to-width aspect ratio of the domain, $A$, the base flow of each case is computed numerically and linear simulations are used to obtain the properties of the leading linear instability mode. Flow regimes depend on the presence of a circulation in the entire cavity, detachment of the thermal layer from the boundary or the corner regions and on the oscillation frequency relative to the natural frequency of oscillation in the stably temperature-stratified interior, allowing for the presence of internal waves or not. Accordingly, the regime is called slow or fast, respectively. Either the global circulation or internal waves in the interior may couple the top and bottom buoyancy currents, while their absence implies asymmetry in their perturbation amplitude. Six flow regimes are found in the range of $0.1 \leq Pr \leq 4$ and $0.5 \leq A \leq 2$. For $Pr \lessapprox 0.4$ and $A>1$, the base flow is driven by a large circulation in the entire cavity. For $Pr \gtrapprox 0.7$, the thermal boundary layers are thin and the instability is driven by the motion along the wall and the detached boundary layer. A transition between these regimes is marked by a dramatic change in oscillation frequency at $Pr = 0.55 \pm 0.15$ and $A <2$.

In this paper, we consider two-dimensional steady free-surface flows where gravity is ignored, but the effects of surface tension are included. It is found that the existence of an additional solid boundary can allow for previously unseen limiting configurations as the surface tension tends to infinity. The free surface of these new solutions is formed of straight lines, arcs of circles and a point where the flow turns at a wall. These limiting configurations form endpoints of solution branches of capillary free-surface flows. Other endpoints of these branches include the surface tension free (i.e. free streamline) solution, and a solution whose free surface is composed simply of a straight line. The model we explore is flow incoming along a channel of constant width. One of the walls terminates, where the fluid is forcibly separated from the wall and a free boundary is formed. The other wall meets a second straight boundary with interior angle $\beta$. Far downstream the solution approaches a uniform stream. Making use of Cauchy's integral formula, the unknowns are expressed in terms of values on the boundary. The integral equations are then solved numerically. The solution space relative to the parameter values of the model is discussed.

In this work, a semianalytic solution for the acoustic streaming phenomenon, generated by standing waves in Maxwell fluids through a two-dimensional microchannel (resonator), is derived. The mathematical model is non-dimensionalized and several dimensionless parameters that characterize the phenomenon arise: the ratio between the oscillation amplitude of the resonator and the half-wavelength ($\eta =2A/\lambda _{a}$); the product of the fluid relaxation time times the angular frequency known as the Deborah number ($De=\lambda _{1}\omega$); the aspect ratio between the microchannel height and the wavelength ($\epsilon =2 H_{0}/\lambda _{a}$); and the ratio between half the height of the microchannel and the thickness of the viscous boundary layer ($\alpha =H_{0}/\delta _{\nu }$). In the limit when $\eta \ll 1$, we obtain the hydrodynamic behaviour of the system using a regular perturbation method. In the present work, we show that the acoustic streaming speed is proportional to $\alpha ^{2.65}De^{1.9}$, and the acoustic pressure varies as $\alpha ^{6/5}De^{1/2}$. Also, we have found that the growth of inner vortex is due to convective terms in the Maxwell rheological equation. Furthermore, the velocity antinodes show a high dependency on the Deborah number, highlighting the fluid's viscoelastic properties and the appearance of resonance points. Due to the limitations of perturbation methods, we will only analyse narrow microchannels.

The fluid–structure interaction problem composed of an elongated, finite-length, flexible cylinder falling in a fluid at rest is investigated experimentally. Tomographic reconstruction of the cylinder and three-dimensional particle tracking velocimetry of the surrounding fluid, based on the Shake-The-Box algorithm, are used jointly to capture both solid and fluid motions. Starting from the rectilinear vertical fall characterized by a steady wake, focus is placed on subsequent regimes involving, mainly in the horizontal direction, periodic rigid-body motions (RBM) of weak amplitude or periodic large-amplitude bending oscillations (BO). Two RBM regimes are explored: the TRA regime where the cylinder exhibits translational oscillations in a plane perpendicular to its axis, and the AZI regime in which the body displays an azimuthal oscillation around its centre. The associated unsteady wakes are composed of counter-rotating vortices bending near the body ends to connect with the adjacent vortex rows. Specific organizations of the vortical structures are uncovered, depending on the regime. In particular, in the AZI regime, they present an antisymmetrical distribution relative to the midspan point. For a sufficiently long cylinder, BO regimes emerge, resembling the structural modes of an unsupported beam. The associated wakes exhibit a cellular organization. Within each cell delimited by two deformation nodes, two counter-rotating vortex rows are shed per oscillation cycle. Flow velocity fluctuations are in phase opposition on each side of a deformation node. For both RBM and BO regimes, frequency and phase analyses of cylinder and wake behaviours, along the span, highlight the spatio-temporal synchronization of the unsteady flow and moving body.

The interaction of steady free-surface flows of viscoplastic material with a surface-piercing obstruction of square cross-section on an inclined plane is investigated theoretically. The flow thickness increases upstream of the obstruction and decreases in its lee. The flow depends on two dimensionless parameters: an aspect ratio that relates the flow thickness, the obstruction width and the plane inclination; and a Bingham number that quantifies the magnitude of the yield stress relative to the gravitationally induced stresses. Flows with a non-vanishing yield stress always form a static ‘dead’ zone in a neighbourhood of the upstream and downstream stagnation points. For relatively wide obstructions, a deep ‘ponded’ region develops upstream with a small dead zone, while the deflected flow reconnects over relatively long distances downstream. The depth of the upstream pond increases with both the dimensionless yield stress and width of the obstruction, while the unyielded dead zone varies primarily with the yield stress. Both are predicted asymptotically by balancing the volume flux of fluid into and out of the ponded region. When the obstruction is narrow, the perturbation to the depth of the oncoming flow is reduced. It exhibits fore–aft antisymmetry, while the dead zone is symmetric to leading order. Increasing the yield stress leads to larger dead zones that eventually encompass all of the upstream- and downstream-facing boundaries of the obstruction and fully divert the flow. Results for obstructions with circular and rhomboidal cross-sections are also presented and illustrate the effects of boundary shape on the properties of the steady flow.

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

We determine the globally minimum time $T$ needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined computationally using a steepest descent, where an adjoint formulation is used to compute the gradients. Because the equations governing fluid mechanics for this problem are nonlinear, multiple local minima could exist. Exploiting the quadratic nature of the objective function and the constraining differential equations, we derive and apply a ‘spectral condition’ to show the converged local optimum to be a global one. The condition states that the optimum is global if the Hessian of the Lagrangian in the state variables constructed using the converged adjoint field is positive semi-definite at every instance. The globally optimum kinematics of the plate starts from rest with speed $\propto t^{1/4}$ and comes to rest with speed $\propto (T-t)^{1/4}$ as a function of time $t$. For distances much longer than the plate, the work-minimizing kinematics consists of an optimum startup, a constant-speed cruising, and an optimum stopping.

Wave turbulence provides a conceptual framework for weakly nonlinear interactions in dispersive media. Dating from five decades ago, applications of wave turbulence theory to oceanic internal waves assigned a leading-order role to interactions characterized by a near equivalence between the group velocity of high-frequency internal waves with the phase velocity of near-inertial waves. This scale-separated interaction leads to a Fokker–Planck (generalized diffusion) equation. More recently, starting four decades ago, this scale-separated paradigm has been investigated using ray tracing methods. These ray methods characterize spectral transport of energy by counting the amplitude and net velocity of wave packets in phase space past a high-wavenumber gate prior to ‘breaking’. This explicitly advective characterization is based on an intuitive assignment and lacks theoretical underpinning. When one takes an estimate of the net spectral drift from the wave turbulence derivation and makes the corresponding assessment, one obtains a prediction of spectral transport that is an order of magnitude larger than either observations or reported ray tracing estimates. Motivated by this contradiction, we report two parallel derivations for transport equations describing the refraction of high-frequency internal waves in a sea of random inertial waves. The first uses standard wave turbulence techniques and the second is an ensemble-averaged packet transport equation characterized by the dispersion of wave packets about a mean drift in the spectral domain. The ensemble-averaged transport equation for ray tracing differs in that it contains the intuitively motivated advective term. We conclude that the aforementioned contradiction between theory, numerics and observations needs to be taken at face value and present a pathway for resolving this contradiction.

We consider high Reynolds number supersonic flow over a compression ramp in the triple-deck formulation. Previous studies of compression-ramp stability have shown rapid growth of high-frequency disturbances in initial-value computations; however, no physical or numerical origin has yet been identified robustly. By considering linear perturbations to steady compression-ramp solutions, we show that instabilities observed in previous studies do not have a growth rate that is described by the integral eigenrelation of Tutty & Cowley (J. Fluid Mech., vol. 168, 1986, pp. 431–456) for a (long-wave) Rayleigh instability. We solve both the temporal and spatial instability problems in the limit of asymptotically large wavenumber $K$ (or equivalently frequency) and show that the growth rate of the instability remains $o(K)$, being dominated by higher-order terms in the expansion at moderate ramp angles.

We consider inertial waves propagating in a fluid contained in a non-axisymmetric three-dimensional rotating cavity. We focus on the particular case of a fluid enclosed inside a truncated cone or frustum, which is the volume that lies between two horizontal parallel planes cutting an upright cone. While this geometry has been studied in the past, we generalise it by breaking its axisymmetry and consider the case of a truncated elliptic cone for which the horizontal sections are elliptic instead of circular. The problem is first tackled using ray tracing, where local wave packets are geometrically propagated and reflected within the closed volume without attenuation. We complement these results with a local asymptotic analysis and numerical simulations of the original linear viscous problem. We show that the attractors, well known in two dimensional or axisymmetric domains, can be trapped in a particular plane in three dimensions provided that the axisymmetry of the domain is broken. Contrary to previous examples of attractors in three-dimensional domains, all rays converge towards the same limit cycle regardless of initial conditions, and it is localised in the bulk of the fluid.

Experiments are conducted to explore the rolling of a cylinder over a pool of viscous fluid. The speed, width and loading of the cylinder are varied along with the initial depth and length of the viscous pool. Depending on the conditions, the cylinder will either ride on a lubrication film or remain in solid contact with the underlying substrate. For the former situation, a lubrication theory is presented that describes the pressure underneath the cylinder and the thickness of the film. The theory approximates the flow by the one-dimensional Reynolds equation with the addition of one term, with an adjustable parameter, to account for the flux of fluid to the cylinder sides. Once this parameter is calibrated against experiment, the theory predicts peak lubrication pressures, gap sizes and film thicknesses to within approximately ten per cent. For lubricated rolling, the film splits evenly between the cylinder and substrate downstream of the nip. The printer's instability arises during the splitting process, patterning the residual fluid films on the substrate and cylinder. If the pool length is less than the cylinder circumference, the fluid adhering to the cylinder is rotated back into contact with the substrate, and when there is sufficient adhered fluid a lubrication film forms that can again be modelled by the theory. Conversely, if there is insufficient adhered fluid, no contiguous lubrication film is formed; instead, the pattern from the printer's instability ‘prints’ from the cylinder to the substrate.

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

The classical Gill's problem, focusing on the stability of thermal buoyancy-driven convection in a vertical porous slab with impermeable isothermal boundaries, is studied from a different perspective by considering a triple-diffusive fluid system having different molecular diffusivities. The assessment of stability/instability of the basic flow entails a numerical solution of the governing equations for the disturbances as Gill's proof of linear stability falls short. The updated problem formulation is found to introduce instability in contrast to Gill's original set-up. A systematic examination of neutral stability curves is undertaken for KCl–NaCl–sucrose and heat–KCl–sucrose aqueous systems which are found to exhibit an anomalous behaviour on the stability of base flow. It is found that, in some cases, the KCl–NaCl–sucrose system necessitates the requirement of four critical values of the Darcy–Rayleigh number to specify the linear stability criteria ascribed to the existence of two isolated neutral curves positioned one below the other. Conversely, the heat–KCl–sucrose system demands only two critical values of the Darcy–Rayleigh number to decide the stability of the system. The stability boundaries are presented and the emergence of a travelling-wave mode supported back and forth with stationary modes is observed due to the introduction of a third diffusing component. In addition, some intriguing outcomes not recognized hitherto for double-diffusive fluid systems are manifested.

We study the possibility of having resonant interactions between three Rossby modes on a coast or channel of arbitrary orientation. A Rossby mode comprises two propagating Rossby waves (RWs) to satisfy the no normal flow through the boundary(ies). In each geometry, we state the conditions, degrees of freedom and RWs of the primary two modes that could force a third mode. We discuss differences between zonal and non-zonal orientations. Resonant interactions are only possible if all RWs participate in the zonal case, while only three RWs participate in the non-zonal case. The non-zonality reduces the degrees of freedom of the resonance conditions, and the solutions are more restrictive for more meridional orientations. In particular, there are no solutions if the coast or channel is meridional. For the non-zonal coast, we find a family of solutions for given periods $T_1$ and $T_2$ of the primary modes. Using multiple scales, we obtain a uniformly valid solution of the quasi-geostrophic potential vorticity equation (QGPVE), with the resonant modes exchanging energy in space. There are no degrees of freedom for the non-zonal channel, and we develop a graphical method to seek resonant solutions, finding some. We provide a bounded solution of the QGPVE in case the primary modes excite one RW, not a channel mode, and the modes do not exchange energy either in time or space. Regarding possible oceanographic applications, we show solutions for the Hawaiian Ridge and inquire if there are solutions in the Mozambique Channel, Tasman Sea, Denmark Strait and the English Channel.

We examine asymptotically the dynamics of two-dimensional, steady detonation wave propagation and failure for a strongly confined high explosive (HE), in which the width of the explosive is small relative to the reaction zone length. An energy balance equation is derived, which shows how the longitudinal acceleration of subsonic flow behind the detonation shock is influenced both by chemical reaction and by the effects of HE boundary streamline deflection, specifically via the induced rate of change of mass flux through the detonation wave. The latter serves to either counteract or reinforce the acceleration of longitudinal flow, depending on the sign of the gradient of the boundary streamline deflection at the detonation shock. The analysis is valid for general equations of state and chemical reaction rates in the HE. The asymptotically derived form of the energy equation represents an eigenvalue problem for the determination of the steady detonation propagation speed, solved via a shooting method. We explore specific results for ideal and stiffened equations of state, along with a pressure-dependent reaction rate for which changes in the pressure exponent and reaction order are also studied. We consider the influences of both straight and curved HE boundary streamline shapes. The asymptotic analysis reveals significant physical insights into how detonation propagation and failure are affected by strong confinement.

We investigate with a two-fluid model the convections developing in a gas–liquid two-phase flow, which consists of a horizontal liquid layer subject to the injection of monodisperse gas bubbles at the bottom. The convections develop in either whole- or multi-layered modes, once the gas injection flux exceeds a critical value (Nakamura et al., Phys. Rev. E, vol. 102, 2020, 053102). We determine the nonlinear evolution of these modes of flows with varying injection flux and find that the whole- and multi-layered modes develop through subcritical and supercritical bifurcations, respectively. The formation of gas plumes is observed in both cases when the nonlinearity is significant. Examining energy transfer from base to perturbation flows, we show that the lift forces on bubbles play a key role in the bifurcations. While they impede the convections in both subcritical and supercritical bifurcations at weak nonlinearity, the lift forces turn to driving the convections in the subcritical bifurcation as nonlinearity increases.

The effect of single perforations and their location on the drag and reconfiguration of flexible plates was explored through laboratory experiments and direct numerical simulations. The plates were subjected to uniform flows with negligible turbulence, and the perforations had a square cross-section resulting in a low porosity ratio of $\gamma \approx 0.028$. Rigid plates with and without perforations and flexible plates without perforations served as the baseline cases. The perforated plates exhibited distinct jets through the openings in the wake, significantly impacting the aerodynamic force and plate deformation. The velocity and position of the centre jet velocity in relation to downwind distance were influenced by both the incoming flow and the location of the perforations. The centre jet velocity profiles were normalized using an effective velocity and corrected perforation half-width, revealing their dependence on these factors. A simple first-order formulation was developed to predict the change in drag for various perforated plates under a wide range of incoming velocities. This formulation was supported by numerical simulations across a wider range of Cauchy number to confirm the proposed model and separate the effect of the Cauchy and Reynolds numbers. The results of this study may inform the design of flexible structures, define effective porosity and serve as an initial step towards modelling the complex interaction between flow and structures with low porosity.

Coherent structures in two-dimensional Navier–Stokes turbulence are ubiquitously observed in nature, experiments and numerical simulations. The present study conducts a comparison between several structure detection schemes based on the Okubo–Weiss criterion, the vorticity magnitude and Lagrangian coherent structures (LCSs), focusing on the inverse cascade in two-dimensional hydrodynamic turbulence. A recently introduced vortex scaling phenomenology (Burgess & Scott, J. Fluid Mech., vol. 811, 2017, pp. 742–756) allows the quantification of the respective thresholds required by these methods based on physical properties of the flow. The resulting improved comparability allows us to identify characteristic relative differences in the detection sensitivity between the employed structure detection techniques. With respect to the inverse cascade of energy, coherent structures contribute, as expected, substantially less to the cross-scale flux than the residual incoherent parts of the flow although the energetically dominant coherent structures lead to an important large-scale deformation of the energy spectrum. This cascade inactivity can be understood by an increased misalignment of strain-rate and subgrid stress tensors within coherent structures. At the same time, the structures exhibit strong and localised nonlinear cross-scale interactions that appear to stabilise them. We quantify and interpret the resulting shape preservation of coherent structures in terms of a multi-scale gradient approach (Eyink, J. Fluid Mech., vol. 549, 2006, pp. 191–214) as the depletion of strain rotation and vorticity gradient stretching whereas the dynamics of the residual fluctuations are consistent with the vortex thinning picture.