ALBERT

Description: The stability of viscous shear is studied for flows that consist of predominantly linear shear, but contain localized regions over which the vorticity varies rapidly. Matched asymptotic expansion simplifies the governing equations for the dynamics of such 'vorticity defects'. The normal modes satisfy explicit dispersion relations. Nyquist methods are used to find and classify the possible instabilities. The defect equations are analysed in the inviscid limit to establish the connection with inviscid theory. Finally, the defect approximation is used to study nonlinear stability using weakly nonlinear techniques, and the initial value problem using Laplace transforms.

Description: We consider double-diffusive convection between two parallel plates and compute bounds on the flux of the unstably stratified species using the background method. The bound on the heat flux for Rayleigh-Bénard convection also serves as a bound on the double-diffusive problem (with the thermal Rayleigh number equal to that of the unstably stratified component). In order to incorporate a dependence of the bound on the stably stratified component, an additional constraint must be included, like that used by Joseph (Stability of Fluid Motion, 1976, Springer) to improve the energy stability analysis of this system. Our bound extends Joseph's result beyond his energy stability boundary. At large Rayleigh number, the bound is found to behave like RT1/2 for fixed ratio RS/RT, where RT and RS are the Rayleigh numbers of the unstably and stably stratified components, respectively. © 2006 Cambridge University Press.

Description: The stability of fluid flow through a narrow conduit with elastic walls is explored, treating the fluid as incompressible and viscous, and the walls as semi-infinite linear Hookean solids. Instabilities analogous to roll waves occur in this system; we map out the physical regime in which they are excited. For elastic wave speeds much higher than the fluid speed, a critical Reynolds number is required for instability. However, that critical value depends linearly on wavenumber, and so can be made arbitrarily small for long waves. For smaller elastic wave speeds, the critical Reynolds number is reduced still further, and Rayleigh waves can be destabilized by the fluid even at zero Reynolds number. A brief discussion is given of the nonlinear dynamics of the instabilities for large elastic wave speed, and the significance of the results to the phenomenon of volcanic tremor is presented. Although magma itself seems unlikely to generate flow-induced vibrations, the rapid flow through fractured rock of low-viscosity fluids exsolved from magma appears to be a viable mechanism for volcanic tremor. © 2005 Cambridge University Press.

Description: In this study we investigate the Kolmogorov flow (a shear flow with a sinusoidal velocity profile) in a weakly stratified, two-dimensional fluid. We derive amplitude equations for this system in the neighbourhood of the initial bifurcation to instability for both low and high Péclet numbers (strong and weak thermal diffusion, respectively). We solve amplitude equations numerically and find that, for low Péclet number, the stratification halts the cascade of energy from small to large scales at an intermediate wavenumber. For high Péclet number, we discover diffusively spreading, thermal boundary layers in which the stratification temporarily impedes, but does not saturate, the growth of the instability; the instability eventually mixes the temperature inside the boundary layers, so releasing itself from the stabilizing stratification there, and thereby grows more quickly. We solve the governing fluid equations numerically to compare with the asymptotic results, and to extend the exploration well beyond onset. We find that the arrest of the inverse cascade by stratification is a robust feature of the system, occurring at higher Reynolds, Richards and Péclet numbers – the flow patterns are invariably smaller than the domain size. At higher Péclet number, though the system creates slender regions in which the temperature gradient is concentrated within a more homogeneous background, there are no signs of the horizontally mixed layers separated by diffusive interfaces familiar from doubly diffusive systems.

Description: By combining Biot's theory of poro-elasticity with standard shallow-layer scalings, a theoretical model is developed to describe axisymmetric gravity-driven flow through a shallow deformable porous medium. Motivated in part by observations of surface uplift around CO〈inf〉2〈/inf〉 sequestration sites, the model is used to explore the injection of a dense fluid into a horizontal, deformable porous layer that is initially saturated with another, less dense, fluid. The layer lies between a rigid base and a flexible overburden, both of which are impermeable. As the injected fluid spreads under gravity, the matrix deforms and the overburden lifts up. The coupled model predicts the location of the injected fluid as it spreads and the resulting uplift of the overburden due to deformation of the solid matrix. In general, the uplift spreads diffusively far ahead of the injected fluid. If fluid is injected with a constant flux and the medium is unbounded, both the uplift and the injected fluid spread in a self-similar fashion with the same similarity variable ∞r/t1/2. The asymptotic form of this spreading is established. Results from a series of laboratory experiments, using polyacrylamide hydrogel particles to create a soft poro-elastic material, are compared qualitatively with the predictions of the model. © 2015 Cambridge University Press.

Description: Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between normal modes of the internal or Kelvin variety in three-dimensional settings in which the stratification and rotation are orthogonal to both the background flow and its shear. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by certain singular critical levels. These levels are not classical critical levels, where the phase speed c of a normal mode matches the mean flow speed U, but are a different type of singularity where (c - U) matches a characteristic gravity-wave speed ±N/k, based on the buoyancy frequency N and streamwise horizontal wavenumber k. Instead, it is shown that a variant of SRI can occur due to the coupling of a Kelvin or internal wave to such 'baroclinic' critical levels. Two characteristic situations are identified and explored, and the conservation law for pseudo-momentum is used to rationalize the physical mechanism of instability. The critical level coupling removes the requirement for resonance near specific wavenumbers k, resulting in an extensive continuous band of unstable modes. © 2018 Cambridge University Press.

Description: We present a model for thixotropic gravity currents flowing down an inclined plane that combines lubrication theory for shallow flow with a rheological constitutive law describing the degree of microscopic structure. The model is solved numerically for a finite volume of fluid in both two and three dimensions. The results illustrate the importance of the degree of initial ageing and the spatio-temporal variations of the microstructure during flow. The fluid does not flow unless the plane is inclined beyond a critical angle that depends on the ageing time. Above that critical angle and for relatively long ageing times, the fluid dramatically avalanches downslope, with the current becoming characterized by a structured horseshoe-shaped remnant of fluid at the back and a raised nose at the advancing front. The flow is prone to a weak interfacial instability that occurs along the border between structured and de-structured fluid. Experiments with bentonite clay show broadly similar phenomenological behaviour to that predicted by the model. Differences between the experiments and the model are discussed. © 2013 Cambridge University Press.

Description: We present a modelling study of locomotion over a layer of viscoplastic fluid motivated by the self-propulsion of marine and terrestrial gastropods. Our model comprises a layer of viscoplastic mucus lying beneath a fluid-filled foot that is laced internally by muscular fibres under tension and overlain by the main body of the locomotor, which is assumed to be rigid. The mucus is described using lubrication theory and the Bingham constitutive law, and the foot using a continuum approximation for the action of the muscle fibres. The model is first used to study the retrograde strategy of locomotion employed by marine gastropods, wherein the muscle fibres create a backwards-travelling wave of predominantly normal displacements along the surface of the foot. Once such a retrograde forcing pattern is switched on, the system is shown to converge towards a steady state of locomotion in a frame moving with the wave. The steady speed of locomotion decreases with the yield stress, until it vanishes altogether above a critical yield stress. Despite the absence of locomotion above this threshold, waves still propagate along the foot, peristaltically pumping mucus in the direction of the wave. The model is next used to study the prograde strategy employed by terrestrial gastropods, wherein the muscle fibres create a forwards-travelling wave of predominantly tangential displacements of the foot surface. In this case, a finite yield stress is shown to be necessary for locomotion, with the speed of locomotion initially increasing with the yield stress. Beyond a critical yield stress, localized rigid plugs form across the depth of the mucus layer, adhering parts of the foot to the base. These stop any transport of mucus, but foot motions elsewhere still drive locomotion. As the yield stress is increased further, the rigid plugs widen horizontally, increasing the viscous drag and eventually reducing the speed of locomotion, which is therefore maximized for an intermediate value of the yield stress. © 2013 Cambridge University Press.

Description: Calculations are presented of the rate of energy conversion of the barotropic tide into internal gravity waves above topography on the ocean floor. The ocean is treated as infinitely deep, and the topography consists of periodic obstructions; a Green function method is used to construct the scattered wavefield. The calculations extend the previous results of Balmforth et al. for subcritical topography (wherein waves propagate along rays whose slopes exceed that of the topography everywhere), by allowing the obstacles to be arbitrarily steep or supercritical (so waves propagate at shallower angles than the topographic slopes and are scattered both up and down). A complicated pattern is found for the dependence of energy conversion on ε, the ratio of maximum topographic slope to wave slope, and the ratio of obstacle amplitude and separation. This results from a sequence of constructive and destructive interferences between scattered waves that has implications for computing tidal conversion rates for the global ocean.