The present paper concerns the linear fate of transverse perturbations in a gravity-driven, thin-film flow over a soluble substrate. We propose a reduced-order model, based on a boundary-layer treatment of the solute transport and a depth-integration of the Stokes equations, using two extended lubrication methodologies found in the literature. We obtain a closed-form dispersion relation, which we compare to a previous, fully resolved analytical investigation (Bertagni and Camporeale, J. Fluid Mech., vol. 913, 2021, A34). The results allow us to distil the essential physical mechanisms behind the instability.

The Earth's landscape hosts a variety of patterns resulting from the interaction between a sediment-carrying fluid and an erodible boundary. Here, the morphodynamics of river bifurcations is interpreted as a second-order phase transition. A consolidated one-dimensional bifurcation model is re-examined in the light of classical Landau theory of critical phenomena. The transition from a balanced to an unbalanced flux partition is described in terms of an order parameter. The equilibrium states of the system are shown to be minima of a morphodynamic potential function. Finally, the role of a generic external forcing is investigated. A threshold value of the forcing is shown to set bounds separating two different morphodynamic responses to allogenic and autogenic dynamics.

In karst and alpine areas, the interactions between water and rocks give rise to a large variety of marvellous patterns. In this work, we provide a hydrodynamic model for the formation of dissolutional patterns made of parallel longitudinal channels, commonly referred to as linear karren forms. The model addresses a laminar film of water flowing on a rock that is dissolving. The results show that a transverse instability of the water–rock system leads to a longitudinal channelization responsible for the pattern formation. The instability arises because of a positive feedback within the channels between the higher water flow and the enhanced chemical dissolution. The spatial scales predicted by the linear stability analysis span different orders of magnitude depending on the Reynolds number. This may explain why similar patterns of different sizes are observed on natural rocks. Results also show that the rock solubility affects just the temporal scale of the instability and the rock inclination plays a minor role in the pattern formation. It is eventually discussed how rain is not strictly necessary for the appearance of linear karren patterns, but it may affect some of their features.

A model of the growth of an ice lens in a saline porous medium is developed. At high lens growth rates the pore fluid becomes supercooled relative to its equilibrium Clapeyron temperature. Instability occurs when the supercooling increases with distance away from the ice lens. Solute diffusion in the pore fluid significantly enhances the instability. An expression for the segregation potential of the soil is obtained from the condition for marginal stability of the ice lens. The model is applied to a clayey silt and a glass powder medium, indicating parameter regimes where the ice lens stability is controlled by viscous flow or by solute diffusion. A mushy layer, composed of vertical ice veins and horizontal ice lenses, forms in the soil in response to the instability. A marginal equilibrium condition is used to estimate the segregated ice fraction in the mushy layer as a function of the freezing rate and salinity.

Lipid membranes are examples of fluid deformable surfaces, which can be viewed as two-dimensional viscous fluids with bending elasticity. With this solid–fluid duality any shape change contributes to tangential flow and vice versa any tangential flow on a curved surface induces shape deformations. This tight coupling between shape and flow makes curvature a natural element of the governing equations. The modelling and numerical tools outlined in Torres-Sánchez et al. (J. Fluid Mech., vol. 872, 2019, pp. 218–271) open a new field of study by enabling the exploration of the role of curvature in this context.

Ice scallops are a small-scale (5–20 cm) quasi-periodic ripple pattern that occurs at the ice–water interface. Previous work has suggested that scallops form due to a self-reinforcing interaction between an evolving ice-surface geometry, an adjacent turbulent flow field and the resulting differential melt rates that occur along the interface. In this study, we perform a series of laboratory experiments in a refrigerated flume to quantitatively investigate the mechanisms of scallop formation and evolution in high resolution. Using particle image velocimetry, we probe an evolving ice–water boundary layer at sub-millimetre scales and 15 Hz frequency. Our data reveal three distinct regimes of ice–water interface evolution: a transition from flat to scalloped ice; an equilibrium scallop geometry; and an adjusting scallop interface. We find that scalloped-ice geometry produces a clear modification to the ice–water boundary layer, characterized by a time-mean recirculating eddy feature that forms in the scallop trough. Our primary finding is that scallops form due to a self-reinforcing feedback between the ice-interface geometry and shear production of turbulent kinetic energy in the flow interior. The length of this shear production zone is therefore hypothesized to set the scallop wavelength.

We study the effects of externally imposed shear and buoyancy driven flows on the stability of a solid–liquid interface. A linear stability analysis of shear and buoyancy-driven flow of a melt over its solid phase shows that buoyancy is the only destabilizing factor and that the regime of shear flow here, by inhibiting vertical motions and hence the upward heat flux, stabilizes the system. It is also shown that all perturbations to the solid–liquid interface decay at a very modest shear flow strength. However, at much larger shear-flow strength, where flow instabilities coupled with buoyancy might enhance vertical motions, a re-entrant instability may arise.

Currently, the most important preparation process for porous polymer membranes is the phase inversion process. While applied for several decades in industry, the mechanism that leads to diverse morphology is not fully understood today. In this work, we present time resolved experiments using light microscopy that indicate viscous fingering during the early stage of pore formation in porous polymer membranes. Numerical simulations using the smoothed particle hydrodynamics method are also performed based on Cahn–Hilliard and Navier–Stokes equations to investigate the formation of viscous fingers in miscible and immiscible systems. The comparison of pore formation characteristics in the experiment and simulation shows that immiscible viscous fingering is present; however, it is only relevant in specific preparation set-ups similar to Hele-Shaw cells. In experiments, we also observe the formation of Liesegang rings. Enabling diffusive mass transport across the immiscible interface leads to Liesegang rings in the simulation. We conclude that further investigations of Liesegang pattern as a relevant mechanism in the formation of morphology in porous polymer membranes are necessary.

We study static distributions of ferrofluid submitted to non-uniform magnetic fields. We show how the normal-field instability is modified in the presence of a weak magnetic field gradient. Then we consider a ferrofluid droplet and show how the gradient affects its shape. A rich phase transition phenomenology is found. We also investigate the creation of droplets by successive splits when a magnet is vertically approached from below and derive theoretical expressions which are solved numerically to obtain the number of droplets and their aspect ratio as a function of the field configuration. A quantitative comparison is performed with previous experimental results, as well as with our own experiments, and yields good agreement with the theoretical modelling.

Flowing air and water are persistent sculptors, gradually working stone, clay, sand and ice into landforms and landscapes. The evolution of shape results from a complex fluid–solid coupling that tends to produce stereotyped forms, and this morphology offers important clues to the history of a landscape and its development. Claudin et al. (J. Fluid Mech., vol. 832, 2017, R2) shed light on how we might read the rippled and scalloped patterns written into dissolving or melting solid surfaces by a flowing fluid. By better understanding the genesis of these patterns, we may explain why they appear in different natural settings, such as the walls of mineral caves dissolving in flowing water, ice caves in wind, and melting icebergs.

We theoretically investigate the pattern formation observed when a fluid flows over a solid substrate that can dissolve or melt. We use a turbulent mixing description that includes the effect of the bed roughness. We show that the dissolution instability at the origin of the pattern is associated with an anomaly at the transition from a laminar to a turbulent hydrodynamic response with respect to the bed elevation. This anomaly, and therefore the instability, disappears when the bed becomes hydrodynamically rough, above a threshold roughness-based Reynolds number. This suggests a possible mechanism for the selection of the pattern amplitude. The most unstable wavelength scales as observed in nature on the thickness of the viscous sublayer, with a multiplicative factor that depends on the dimensionless parameters of the problem.

A novel linear stability analysis for the pattern formation of ripple-like corrugations, over the surface of stalactites due to calcite-laden falling films, is presented by using a modified version of the gradient expansion technique. A fully analytical theory combining thin film hydrodynamics, Kármán–Pohlhausen solution of the advection–diffusion equation for calcium concentration and an evolution equation for the shape of the liquid–solid interface is developed. Analytical formulas for the selected wavelength and the corresponding phase velocity are provided, which match previous numerical solutions. The obtained easy-to-use mathematical results have a potential for developing a novel set of paleo-reconstruction proxies based on the link between morphological patterns and paleo-flows.

We study a simplified model of ice–ocean interaction beneath a floating ice shelf, and investigate the possibility for channels to form in the ice shelf base due to spatial variations in conditions at the grounding line. The model combines an extensional thin-film description of viscous ice flow in the shelf, with melting at its base driven by a turbulent ocean plume. Small transverse perturbations to the one-dimensional steady state are considered, driven either by ice thickness or subglacial discharge variations across the grounding line. Either forcing leads to the growth of channels downstream, with melting driven by locally enhanced ocean velocities, and thus heat transfer. Narrow channels are smoothed out due to turbulent mixing in the ocean plume, leading to a preferred wavelength for channel growth. In the absence of perturbations at the grounding line, linear stability analysis suggests that the one-dimensional state is stable to initial perturbations, chiefly due to the background ice advection.

Two of the most widespread and fascinating patterns observed on cave walls and icefalls – karst and ice flutings – are demonstrated to share the same morphogenesis, whose core is a water film-induced locking mechanism. Creeping flow-based parallel and non-parallel stability analyses are developed through a numerical and analytical approach. These instabilities are shown to develop at inverted overhung conditions. A sharp transition between fluting and ripple-like patterns is presented. The non-parallel problem is solved with the use of Papkovich–Neuber solutions in order to obtain a finite wavelength selection close to the critical conditions. The method and results can be extended to similar problems where the temporal evolution of the interface is linearly related to the film depth.

A free-surface-induced morphological instability is studied in the laminar regime at large Reynolds numbers () and on sub-horizontal walls (). We analytically and numerically develop the stability analysis of an inclined melting–freezing interface bounding a free-surface laminar flow. The complete solution of both the linearized flow field and the heat conservation equations allows the exact derivation of the upper and lower temperature gradients at the interface, as required by the Stefan condition, from which the dispersion relationship is obtained. The eigenstructure is obtained and discussed. Free-surface dynamics appears to be crucial for the triggering of upstream propagating ice ripples, which grow at the liquid–solid interface. The kinematic and the dynamic conditions play a key role in controlling the formation of the free-surface fluctuations; these latter induce a streamline distortion with an increment of the wall-normal velocities and a destabilizing phase shift in the net heat transfer to the interface. Three-dimensional effects appear to be crucial at high Reynolds numbers. The role of inertia forces, vorticity, and thermal boundary conditions are also discussed.

We consider weakly nonlinear convection in a fluid layer with a melting top boundary. This leads us to derive a new set of non-autonomous envelope equations as a dynamic generalization to the well-known Ginzburg–Landau equation. However, this new system possesses a number of interesting properties not found in systems close to a traditional dynamic bifurcation, because it involves the interaction of two destabilizing mechanisms. We investigate the system both analytically and numerically; specifically, we find the robust ‘locking in’ of spatially complex patterns, and show this is a general feature of systems of this nature.

In many tidal embayments, complex patterns of channels and shoals are observed. To gain a better understanding of these features, an idealized model, that describes the interaction of water motion, sediment transport and bed evolution in a semi-enclosed, rectangular basin, is developed and analysed. To explain the initial formation of channels and shoals, two-dimensional perturbations superposed on a laterally uniform equilibrium bottom are studied. These perturbations evolve due to convergences of various residual suspended sediment fluxes: a diffusive flux, a flux related to the bed topography, an advective flux resulting from internally generated overtides and an advective flux due to externally prescribed overtides. For most combinations of these fluxes, perturbations start to grow if the bottom friction is strong enough. Their growth is mainly a result of convergences of diffusive and topographically induced sediment fluxes. Advective contributions due to internally generated overtides enhance this growth. If only diffusive sediment fluxes are considered, the underlying equilibrium is always unstable. This can be traced back to the depth dependence of the deposition parameter. Contrary to the results of previous idealized models, the channels and shoals always initiate in the shallow, landward areas. This is explained by the enhanced generation (compared to that in previous models) of frictional torques in shallow regions. The resulting initial channel–shoal formation compares well with results found in complex numerical model studies. The instability mechanism and the location of the initial formation of bottom patterns do not change qualitatively when varying parameters. Changes are mainly related to differences in the underlying equilibrium profile due to parameter variations.

A linear stability analysis is performed on the interface that forms during directional solidification of a dilute binary alloy in the presence of time-periodic growth rates. The basic state, in which the flat crystal-melt interface advances at a steady rate with an oscillation superimposed, is solved analytically by expanding the governing equations in terms of the assumed-small amplitude of modulation. We find that there is a frequency window of stabilization, in which the Mullins-Sekerka instability can be stabilized synchronously. Outside of the window, large input frequencies may destabilize the Mullins-Sekerka mode. The subharmonic mode, which occurs with small wave numbers, is stabilized with increasing the frequency. As for the modulation amplitude, larger amplitude tends to reduce the synchronous mode while enhance the subharmonic mode.

We present a mathematical model of the unidirectional solidification of a suspension of hard-sphere colloids. Similarity solutions are obtained for the volume fraction and temperature profiles ahead of a planar solidification front. The highly nonlinear functional dependence of the diffusion coefficient on the volume fraction gives rise to a range of behaviours. For small particles, Brownian diffusion dominates and the system behaviour is reminiscent of binary-alloy solidification. Constitutional supercooling occurs at the interface under certain conditions, leading potentially to an instability in the shape of the interface. For larger particles, Brownian diffusion is weak and the particles form a porous layer above the interface. In this case constitutional supercooling reaches a maximum near the surface of the layer, and the porous medium itself is potentially unstable. In stable systems there exists the possibility of secondary nucleation of ice.