Understanding biolocomotion in fluids has long been a focus of fluid dynamicists. One method to quantify the impact of environmental stresses on locomotion is to systematically change the mechanical properties of the surrounding medium, and measure how that change influences swimming kinematics and energetics. In a recently published investigation, Gagnon et al. (J. Fluid Mech., vol. 758, 2014, R3) employ that approach to investigate the locomotion of the nematode Caenorhabditis elegans in complex fluids. Specifically, they characterize experimentally how the presence of shear-thinning rheology influences the flow around the organism and its swimming ability. Surprisingly, while they measure an important change to the flow structure around the organism, they find no change in its waving motion and the speed at which it is able to swim. While ‘gooeyness’ is a universal feature of natural biological and environmental media, C. elegans seems to find it perfectly bearable.

Instabilities of isolated anticyclonic vortices in the two-layer rotating shallow water model are studied at Rossby numbers up to two, with the main goal to understand the interplay between the classical centrifugal instability and other ageostrophic instabilities. We find that different types of instabilities with low azimuthal wavenumbers exist, and may compete. In a wide range of parameters, an asymmetric version of the standard centrifugal instability has larger growth rate than the latter. The dependence of the instabilities on the parameters of the flow, i.e. Rossby and Burger numbers, vertical shear and the ratios of the layers’ thicknesses and densities, is investigated. The zones of dominance of each instability are determined in the parameter space. Nonlinear saturation of these instabilities is then studied with the help of a high-resolution finite-volume numerical scheme, by using the unstable modes identified from the linear stability analysis as initial conditions. Differences in nonlinear development of the competing centrifugal and ageostrophic barotropic instabilities are evidenced. A nonlinear mechanism of axial symmetry breaking during the saturation of the centrifugal instability is displayed.

Debris flows can spontaneously develop regular large-amplitude surge waves that are interspersed by periods in which the channel fill is completely stationary. These are important because each individual surge is much more destructive than a steady uniform flow with the same mass flux. In this paper small-scale experiments that exhibit similar behaviour are described. The flow consists of carborundum particles that flow down a rough inclined chute covered with a static erodible layer of the same grains. For inflow conditions close to the minimum depth required for steady uniform flows to exist, small disturbances are unstable, creating waves that rapidly coarsen and grow in size. As the waves become sufficiently large, the troughs between the wave crests drop below a critical thickness and come to rest. A series of steadily travelling waves develop which erode the static layer of particles in front of them and deposit grains behind them, to form a layer that is again stationary. This is, in turn, re-eroded and deposited by the next wave. We term these waves granular erosion–deposition waves. Although erosion and deposition problems are notoriously difficult, a simple model is developed which uses a depth-averaged version of the ${\it\mu}(I)$-rheology and Pouliquen and Forterre’s extended friction law. The viscous dissipation combines with dynamic, intermediate and static friction regimes to generate finite-length waves with static and mobile regions. The existence of stationary layers fundamentally distinguishes erosion–deposition waves from granular roll waves, which form in slightly deeper flows and are always completely mobilized. Numerical simulations show that the system of equations is able to model both erosion–deposition waves and granular roll waves. Moreover, the computed wave amplitude, wavespeed and coarsening dynamics are in good quantitative agreement with experiments.

We consider the axisymmetric arrangement of an annular liquid film, coating the inner surface of a narrow cylindrical tube, in interaction with an active core fluid. We introduce a low-dimensional model based on the two-phase weighted residual integral boundary layer (WRIBL) formalism (Dietze & Ruyer-Quil, J. Fluid Mech., vol. 722, 2013, pp. 348–393) which is able to capture the long-wave instabilities characterizing such flows. Our model improves upon existing works by fully representing interfacial coupling and accounting for inertia as well as streamwise viscous diffusion in both phases. We apply this model to gravity-free liquid-film/core-fluid arrangements in narrow capillaries with specific attention to the dynamics leading to flooding, i.e. when the liquid film drains into large-amplitude collars that occlude the tube cross-section. We do this against the background of linear stability calculations and nonlinear two-phase direct numerical simulations (DNS). Due to the improvements of our model, we have found a number of novel/salient physical features of these flows. First, we show that it is essential to account for inertia and full interphase coupling to capture the temporal evolution of flooding for fluid combinations that are not dominated by viscosity, e.g. water/air and water/silicone oil. Second, we elucidate a viscous-blocking mechanism which drastically delays flooding in thin films that are too thick to form unduloids. This mechanism involves buckling of the residual film between two liquid collars, generating two very pronounced film troughs where viscous dissipation is drastically increased and growth effectively arrested. Only at very long times does breaking of symmetry in this region (due to small perturbations) initiate a sliding motion of the liquid film similar to observations by Lister et al. (J. Fluid Mech., vol. 552, 2006, pp. 311–343) in thin non-flooding films. This kickstarts the growth of liquid collars anew and ultimately leads to flooding. We show that streamwise viscous diffusion is essential to this mechanism. Low-frequency core-flow oscillations, such as occur in human pulmonary capillaries, are found to set off this sliding-induced flooding mechanism much earlier.

A high-accuracy numerical solution, coupling one-dimensional shallow water and bed-evolution equations, with, for the first time, a suspended sediment advection equation, thereby including bed and/or suspended load, is used to examine two swash events on an initially plane erodible beach: the event of Peregrine & Williams (J. Fluid Mech., vol. 440, 2001, pp. 391–399) and that of a solitary wave approaching the beach. Equations are solved by the method of characteristics, and the numerical model is verified. Full coupling of suspended load to beach change for Peregrine & Williams (J. Fluid Mech., vol. 440, 2001, pp. 391–399) yields only slightly altered swash flows, depending on beach mobility and sediment response time; a series of similar final beach change patterns results for different beach mobilities. Suspended- and bed-load transport have distinct morphodynamical signatures. For the solitary wave a backwash bore is created (Hibberd & Peregrine, J. Fluid Mech., vol. 95, 1979, pp. 323–345). This morphodynamical bore propagates offshore initially, and leads to the creation of a beach bed step (Larson & Sunamura, J. Sedimentary Petrology, vol. 63, 1993, pp. 495–500), primarily due to bed-load transport. Its height is directly related to bed-load mobility, and also depends strongly on the bed friction coefficient. The shock dynamics of this bed step is explained and illustrated. Bed- and suspended-load mobilities are quantified using field data, and an attempt is made to relate predictions to measurements of single swash events on a natural beach. Average predicted bed change magnitudes across the swash are of the order of 2 mm, with maximum bed changes of up to approximately 10 cm at the bed step.

The thin helical vortex filament is one of the fundamental exact solutions possible under the local induction approximation (LIA). The LIA is itself an approximation to the non-local Biot–Savart dynamics governing the self-induced motion of a vortex filament, and helical filaments have also been considered for the Biot–Savart dynamics, under a variety of configurations and assumptions. We study the motion of such a helical filament in the Cartesian reference frame by determining the curve defining this filament mathematically from the Biot–Savart model. In order to do so, we consider a matched approximation to the Biot–Savart dynamics, with local effects approximated by the LIA in order to avoid the logarithmic singularity inherent in the Biot–Savart formulation. This, in turn, allows us to determine the rotational and translational velocity of the filament in terms of a local contribution (which is exactly that which is found under the LIA) and a non-local contribution, each of which depends on the wavenumber, $k$, and the helix diameter, $A$. Performing our calculations in such a way, we can easily compare our results to those of the LIA. For small $k$, the transverse velocity scales as $k^{2}$, while for large $k$, the transverse velocity scales as $k$. On the other hand, the rotational velocity attains a maximum value at some finite $k$, which corresponds to the wavenumber giving the maximal torsion.

When a layer of particle-laden fresh water is placed above clear, saline water, both double-diffusive and Rayleigh–Taylor instabilities may arise. The present investigation extends the linear stability analysis of Burns & Meiburg (J. Fluid Mech., vol. 691, 2012, pp. 279–314) into the nonlinear regime, by means of two- and three-dimensional direct numerical simulations (DNS). The initial instability growth in the DNS is seen to be consistent with the dominant modes predicted by the linear stability analysis. The subsequent vigorous growth of individual fingers gives rise to a secondary instability, and eventually to the formation of intense plumes that become detached from the interfacial region. The simulations show that the presence of particles with a Stokes settling velocity modifies the traditional double-diffusive fingering by creating an unstable ‘nose region’ in the horizontally averaged profiles, located between the upward-moving salinity and the downward-moving sediment interface. The effective thickness $l_{s}$ ($l_{c}$) of the salinity (sediment) interface grows diffusively, as does the height $H$ of the nose region. The ratio $H/l_{s}$ initially grows and then plateaus, at a value that is determined by the balance between the flux of sediment into the rose region from above, the double-diffusive/Rayleigh–Taylor flux out of the nose region below, and the rate of sediment accumulation within the nose region. For small values of $H/l_{s}\leqslant O(0.1)$, double-diffusive fingering dominates, while for larger values $H/l_{s}\geqslant O(0.1)$ the sediment and salinity interfaces become increasingly separated in space and the dominant instability mode becomes Rayleigh–Taylor like. A scaling analysis based on the results of a parametric study indicates that $H/l_{s}$ is a linear function of a single dimensionless grouping that can be interpreted as the ratio of inflow and outflow of sediment into the nose region. The simulation results furthermore indicate that double-diffusive and Rayleigh–Taylor instability mechanisms cause the effective settling velocity of the sediment to scale with the overall buoyancy velocity of the system, which can be orders of magnitude larger than the Stokes settling velocity. While the power spectra of double-diffusive and Rayleigh–Taylor-dominated flows are qualitatively similar, the difference between flows dominated by fingering and leaking is clearly seen when analysing the spectral phase shift. For leaking-dominated flows a phase-locking mechanism is observed, which intensifies with time. Hence, the leaking mode can be interpreted as a fingering mode which has become phase-locked due to large-scale overturning events in the nose region, as a result of a Rayleigh–Taylor instability.

We examine the basic properties and stability of isolated vortices having uniform potential vorticity (PV) in a non-hydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basic-state stratification for which both the Coriolis and buoyancy frequencies, $f$ and $N$, are constant. Moreover, we take $f/N\ll 1$, as typically observed in the Earth’s atmosphere and oceans. In the small Rossby number ‘quasi-geostrophic’ (QG) limit, when the flow is weak compared to the background rotation, there exist exact solutions for steadily rotating ellipsoidal volumes of uniform PV in an unbounded flow (Zhmur & Shchepetkin, Izv. Akad. Nauk SSSR Atmos. Ocean. Phys., vol. 27, 1991, pp. 492–503; Meacham, Dyn. Atmos. Oceans, vol. 16, 1992, pp. 189–223). Furthermore, a wide range of these solutions are stable as long as the horizontal and vertical aspect ratios ${\it\lambda}$ and ${\it\mu}$ do not depart greatly from unity (Dritschel et al.,J. Fluid Mech., vol. 536, 2005, pp. 401–421). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly $-1$ (anticyclone) to nearly $+1$ (cyclone). That is, QG vortices are more stable than anticyclones at finite negative Rossby number, and generally less stable than cyclones at finite positive Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing ${\it\lambda}$ towards unity or decreasing ${\it\mu}$. An unstable vortex often restabilises by developing a near-circular cross-section, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertia–gravity waves (IGWs) is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities, are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone.

We consider rotating Rayleigh–Bénard convection of a fluid with a Prandtl number of $\mathit{Pr}=0.8$ in a cylindrical cell with an aspect ratio ${\it\Gamma}=1/2$. Direct numerical simulations (DNS) were performed for the Rayleigh number range $10^{5}\leqslant \mathit{Ra}\leqslant 10^{9}$ and the inverse Rossby number range $0\leqslant 1/\mathit{Ro}\leqslant 20$. We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy-dominated regime occurring while the toroidal energy $e_{tor}$ is not affected by rotation and remains equal to that in the non-rotating case, $e_{tor}^{0}$. Second, a rotation-influenced regime, starting at rotation rates where $e_{tor}>e_{tor}^{0}$ and ending at a critical inverse Rossby number $1/\mathit{Ro}_{cr}$ that is determined by the balance of the toroidal and poloidal energy, $e_{tor}=e_{pol}$. Third, a rotation-dominated regime, where the toroidal energy $e_{tor}$ is larger than both $e_{pol}$ and $e_{tor}^{0}$. Fourth, a geostrophic regime for high rotation rates where the toroidal energy drops below the value for non-rotating convection.

Ocean submesoscale baroclinic instability is studied in the framework of the balance equations. These equations are an intermediate model that includes balanced ageostrophic effects with higher accuracy than the quasigeostrophic approximation, but rules out unbalanced wave motions. As such, the balance equations are particularly suited to the study of baroclinic instability in submesoscale ocean dynamics. The linear baroclinic instability problem is developed in generality and then specialized to the case of constant vertical shear. It is found that non-quasigeostrophic effects appear only for perturbations with cross-front variation, and that perturbation energy can be generated through both baroclinic production and shear production. The Eady problem is solved analytically in the balance equation framework. Ageostrophic effects are shown to increase the range of unstable modes and the growth rate of the instability for perturbations with cross-front variation. The increased level of instability is attributed to both ageostrophic baroclinic production and shear production of perturbation energy; these results are verified in the primitive equations. Finally, submesoscale baroclinic instability is examined in a case where the buoyancy frequency increases rapidly near the bottom boundary, mimicking the increase of stratification at the base of the oceanic mixed layer. The qualitative results of the Eady problem are repeated in this case, with increased growth rates attributed to the production of perturbation energy by the ageostrophic velocity. The results show that submesoscale baroclinic instability acts to reduce lateral buoyancy gradients and their associated geostrophic shear simultaneously through lateral buoyancy fluxes and vertical momentum fluxes.

In this study the influence of cell tilting on flow dynamics and heat transport is explored experimentally within a rectangular cell (aspect ratios ${\it\Gamma}_{x}=1$ and ${\it\Gamma}_{y}=0.25$). The measurements are carried out over a wide range of tilt angles ($0\leqslant {\it\beta}\leqslant {\rm\pi}/2\ \text{rad}$) at a constant Prandtl number ($\mathit{Pr}\simeq 6.3$) and Rayleigh number ($\mathit{Ra}\simeq 4.42\times 10^{9}$). The velocity measurements reveal that the large-scale circulation (LSC) is sensitive to the symmetry of the system. In the level case, the high-velocity band of the LSC concentrates at about a quarter of the cell width from the boundary. As the cell is slightly tilted (${\it\beta}\simeq 0.04\ \text{rad}$), the position of the high-velocity band quickly moves towards the boundary. With increasing ${\it\beta}$, the LSC changes gradually from oblique ellipse-like to square-like, and other more complicated patterns. Oscillations have been found in the temperature and velocity fields for almost all ${\it\beta}$, and are strongest at around ${\it\beta}\simeq 0.48\ \text{rad}$. As ${\it\beta}$ increases, the Reynolds number ($\mathit{Re}$) initially also increases, until it reaches its maximum at the transition angle ${\it\beta}=0.15\ \text{rad}$, after which it gradually decreases. The cell tilting causes a pronounced reduction of the Nusselt number ($\mathit{Nu}$). As ${\it\beta}$ increases from 0 to 0.15, 1.05 and ${\rm\pi}/2\ \text{rad}$, the reduction of $\mathit{Nu}$ is approximately 1.4 %, 5 % and 18 %, respectively. Over the ranges of $0\leqslant {\it\beta}\leqslant 0.15\ \text{rad}$, $0.15\leqslant {\it\beta}\leqslant 1.05\ \text{rad}$ and $1.05\leqslant {\it\beta}\leqslant {\rm\pi}/2\ \text{rad}$, the decay slopes are $8.57\times 10^{-2}$, $3.27\times 10^{-2}$ and $0.24\ \text{rad}^{-1}$, respectively.

The deformation of a spherical capsule in oscillating shear flow is presented. The boundary element method is used to simulate the capsule motion under Stokes flow. We show that a capsule at high frequencies follows the deformation given by a leading-order prediction, which is derived from an assumption of small deformation limit. At low frequencies, on the other hand, a capsule shows an overshoot phenomenon where the maximum deformation is larger than that in steady shear flow. A larger overshoot is observed for larger capillary number or viscosity ratio. Using the maximum deformation in start-up shear flow, we evaluate the upper limit of deformation in oscillating shear flow. We also show that the overshoot phenomenon may appear when the quasi-steady orientation angle under steady shear flow is less than $9.0^{\circ }$. We propose an equation to estimate the threshold frequency between the low-frequency range, where the capsule may have an overshoot, and the high-frequency range, where the deformation is given by the leading-order prediction. The equation only includes the viscosity ratio and the Taylor parameter under simple shear flow, so it can be extended to other deformable particles, such as bubbles and drops.

Three-dimensional large-eddy simulations of two-stream mixing layers developing spatially from laminar boundary layers are presented, replicating wind-tunnel experiments carried out in Part 1 of this study. These simulations have been continued through the mixing transition and into the fully turbulent self-similar flow beyond. In agreement with the experiments, the simulations show that the familiar mechanism of growth by vortex amalgamation is replaced at the mixing transition by a previously unrecognised mechanism in which the spanwise-coherent large structures individually undergo continuous linear growth. In the post-transition flow it is this continuous linear growth of the individual structures that produces the self-similar growth of the mixing-layer thickness, the large-structure interactions occurring as a consequence of the growth, not its cause. New information is also presented on the topography of the organised post-transition flow and on its cyclical evolution through the lifetimes of the individual large structures. The dynamic and kinematic implications of these findings are discussed and shown to define quantitatively the growth rate of the homogeneous post-transition mixing layer in its organised state.

When a vertical liquid jet impacts on a solid horizontal surface, the first expectation is to have a circular hydraulic jump. However, in some conditions, for highly viscous fluids, the transition from supercritical to subcritical flow occurs with non-circular shapes such as polygons. Indeed, a quick rotational wave appears on the circular jump before the formation of a polygonal form, which may be related to the Rayleigh–Plateau instability. In this paper, stable polygonal jumps are studied to complete this research. The region of stability is defined for polygonal jumps, and the dependence of this region on the flow governing dimensionless groups is determined experimentally. The results confirm the multistability (hysteresis) of the polygonal jumps, and imply that polygonal jumps with different corner numbers can be created in a certain parameter regime. The size and curvature of the sides of the polygons due to variations of flow rate and downstream obstacle height are also investigated. In addition to the stable ones, our experiments reveal a new type of polygonal jump that has an unstable structure and displays a rotational behaviour with a constant angular velocity, which we call it, ‘rotational hydraulic jump’. It is observed that the angular velocity of this kind of jump depends on the jet flow rate, jet radius and downstream height of the jump. Our observations suggest that the nature of the rotational jump is some kind of surface wave along the jump in clockwise or anticlockwise direction. It seems that the rotational jump has a flow structure that is the same as a type IIb jump. The jump dimensions are studied; the inscribed and circumscribed circular radii of each polygon are measured in order to compare the various polygons together and to find a mean jump radius to compare with Watson’s theory.

The two-dimensional backward-facing step flow is a canonical example of noise amplifier flow: global linear stability analysis predicts that it is stable, but perturbations can undergo large amplification in space and time as a result of non-normal effects. This amplification potential is best captured by optimal transient growth analysis, optimal harmonic forcing, or the response to sustained noise. With a view to reducing disturbance amplification in these globally stable open flows, a variational technique is proposed to evaluate the sensitivity of stochastic amplification to steady control. Existing sensitivity methods are extended in two ways to achieve a realistic representation of incoming noise: (i) perturbations are time-stochastic rather than time-harmonic, (ii) perturbations are localised at the inlet rather than distributed in space. This allows the identification of regions where small-amplitude control is the most effective, without actually computing any controlled flows. In particular, passive control by means of a small cylinder and active control by means of wall blowing/suction are analysed for Reynolds number $\mathit{Re}=500$ and step-to-outlet expansion ratio ${\it\Gamma}=0.5$. Sensitivity maps for noise amplification appear largely similar to sensitivity maps for optimal harmonic amplification at the most amplified frequency. This is observed at other values of $\mathit{Re}$ and ${\it\Gamma}$ too, and suggests that the design of steady control in this noise amplifier flow can be simplified by focusing on the most dangerous perturbation at the most dangerous frequency.

We study a partial dewetting corner flow with a moving contact line at a finite Reynolds number, $0<\mathit{Re}<O(100)$. When the speed of the moving contact line increases, the receding contact line appears with a corner shape that is also observed in a gravity-driven liquid droplet on an incline and on a plate withdrawn from a bath. In the current problem, $\mathit{Re}\,{\it\epsilon}$ is larger than unity, where ${\it\epsilon}$ is the aspect ratio of the flow structure. Therefore, classical lubrication theory is no longer appropriate. We develop a modified three-dimensional lubrication model for the dewetting corner structure at $\mathit{Re}\,{\it\epsilon}>1$ by taking into account the internal flow pattern and by scaling arguments. The key requirement is that the streamlines in the corner are straight and (nearly) parallel. In this case, we can obtain a modified pressure consisting of the capillary pressure and the dynamic pressure. This model describes the three-dimensional dewetting corner structure at the rear of the moving droplets at $\mathit{Re}\,{\it\epsilon}>1$ and furthermore shows that the dynamic pressure effects become dominant at a small half-opening angle. Additionally, this model provides analytical results for the internal flow, which is a self-similar flow pattern. To validate the analytical results, we perform high-speed shadowgraphy and tomographic particle image velocimetry (PIV). We find a good agreement between the theoretical and the experimental results.

The advance of the front of a dense gravity current propagating in a rectangular channel and V-shaped valley both horizontally and up a shallow slope is examined through theory, full-depth lock–release laboratory experiments and hydrostatic numerical simulations. Consistent with theory, experiments and simulations show that the front speed is relatively faster in the valley than in the channel. The front speed measured shortly after release from the lock is 5–22 % smaller than theory, with greater discrepancy found in upsloping V-shaped valleys. By contrast, the simulated speed is approximately 6 % larger than theory, showing no dependence on slope for rise angles up to ${\it\theta}=8^{\circ }$. Unlike gravity currents in a channel, the current head is observed in experiments to be more turbulent when propagating in a V-shaped valley. The turbulence is presumably enhanced due to the lateral flows down the sloping sides of the valley. As a consequence, lateral momentum transport contributes to the observed lower initial speeds. A Wentzel–Kramers–Brillouin like theory predicting the deceleration of the current as it runs upslope agrees remarkably well with simulations and with most experiments, within errors.

Numerical simulation is used to document the statistical structure and better understand energy transfers in a low-Reynolds-number turbulent flow generated by negative axial buoyancy in a long circular tilted pipe under the Boussinesq approximation. The flow is found to exhibit specific features which strikingly contrast with the familiar characteristics of pressure-driven pipe and channel flows. The mean flow, dominated by an axial component exhibiting a uniform shear in the core, also comprises a weak secondary component made of four counter-rotating cells filling the entire cross-section. Within the cross-section, variations of the axial and transverse velocity fluctuations are markedly different, the former reaching its maximum at the edge of the core while the latter two decrease monotonically from the axis to the wall. The negative axial buoyancy component generates long plumes travelling along the pipe, yielding unusually large longitudinal integral length scales. The axial and crosswise mean density variations are shown to be respectively responsible for a quadratic variation of the crosswise shear stress and density flux which both decrease from a maximum on the pipe axis to near-zero values throughout the near-wall region. Although the crosswise buoyancy component is stabilizing everywhere, the crosswise density flux is negative in some peripheral regions, which corresponds to apparent counter-gradient diffusion. Budgets of velocity and density fluctuations variances and of crosswise shear stress and density flux are analysed to explain the above features. A novel two-time algebraic model of the turbulent fluxes is introduced to determine all components of the diffusivity tensor, revealing that they are significantly influenced by axial and crosswise buoyancy effects. The eddy viscosity and eddy diffusivity concepts and the Reynolds analogy are found to work reasonably well within the central part of the section whereas non-local effects cannot be ignored elsewhere.

The free-boundary problem between a liquid region and a mushy layer (a reactive porous medium) must respect both thermodynamic and fluid dynamical considerations. We develop a steady two-dimensional forced-flow configuration to investigate the thermodynamic condition of marginal equilibrium that applies to a solidifying mushy layer with outflow and requires that streamlines are tangent to isotherms at the interface. We show that a ‘two-domain’ approach in which the mushy layer and liquid region are distinct domains is consistent with marginal equilibrium by extending the Stokes equations in a narrow transition region within the mushy layer. We show that the tangential fluid velocity changes rapidly in the transition region to satisfy marginal equilibrium. In convecting mushy layers with liquid channels, a buoyancy gradient can drive this tangential flow. We use asymptotic analysis in the limit of small Darcy number to derive a regime diagram for the existence of steady solutions. Thus we show that marginal equilibrium is a robust boundary condition and can be used without precise knowledge of the fluid flow near the interface.

In this paper, linear stability analysis on a Newtonian fluid film flowing under the effect of gravity over an inclined porous medium saturated with the same fluid in isothermal condition is carried out. The focus is placed on the effect of the anisotropic and inhomogeneous variations in the permeability of the porous medium on the shear mode and surface mode instabilities. The fluid–porous system is modelled by a coupled two-dimensional Navier–Stokes/Darcy problem. The perturbation equations are solved numerically using the Chebyshev collocation method. Detailed stability characteristics as a function of the depth ratio (the ratio of the depth of the fluid layer to that of the porous layer), the anisotropic parameter (the ratio of the permeability in the direction of the basic flow to that in the direction transverse to the basic flow) and the inhomogeneity functions are presented.