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.

When suspended in a liquid solution, chemically active colloids may self-propel due to an asymmetry in either particle shape or the interfacial distribution of solute absorption. We here consider a chemically homogeneous spherical particle which undergoes self-diffusiophoresis due to the presence of nearby inert wall. In particular, we focus upon the near-contact limit where it was recently observed (Yariv, Phys. Rev. Fluids, vol. 1 (3), 2016, 032101) that the solute-concentration profile within the narrow gap separating the particle and the wall cannot be uniquely determined by a gap-scale analysis. We here revisit this near-contact limit using matched asymptotic expansions, the inner region being the gap domain and the outer region being on the particle scale. Asymptotic matching with the Hankel-transform representation of the outer distribution of solute concentration serves to determine both the scaling and magnitude of the corresponding inner profile. The ensuing gap-scale pressure field, set by a lubrication mechanism, gives rise to an anomalous particle–wall interaction, scaling as an irrational power of the gap clearance.

Previous studies have shown that the behaviour of Rayleigh–Taylor (RT) unstable flames depends on the boundary conditions. If the boundary conditions at the domain walls are impermeable/adiabatic or reflecting then the flame assumes a stable parabolic shape. On the other hand, periodic boundary conditions can produce unstable pulsating solutions. In this paper, we explore why periodic boundary conditions allow unstable solutions by showing the results of two-dimensional direct numerical simulations of model flames. We show that RT unstable premixed model flames pulsate at low gravity because of a shear instability of the vorticity layers behind the flame front. The resulting vortex shedding is controlled by a region of absolute-like instability which moves closer to the flame front as gravity is increased, ultimately disturbing the flame and leading to pulsations. We demonstrate that this region is ‘absolutely unstable’ by showing that the wake is dominated by pure frequency oscillations. In addition, the shear instability can be described by the Landau equation and can be represented dynamically by a Hopf bifurcation. The applicability of the Landau equation allows the apparently complex spatio-temporal behaviour of the vortex shedding to be described by a simple temporal model with only a secondary spatial dependence. We show that the flame behaviour is analogous to the initial instability downstream of a circular cylinder, which leads to the von Kármán vortex street for large enough values of the Reynolds number.

For flows of jet type, the assumption of a coefficient of eddy kinematic viscosity in turbulent flow leads to the possibility of combining in one the equations for laminar and turbulent motion. An approximation to the solution of these equations is found for the flow of compressible fluid issuing from a narrow slit, far from the slit. The stream function is expanded in a power series in squares of the Mach number. Bickley's solution (1937) for the corresponding problem in incompressible flow is used to start the iterative process by which successive terms of the power series are obtained. In order to find an analytical form for the second term of the series, it has been assumed that the Prandtl number is unity, that the viscosity varies as the nth power of the absolute temperature, and that the stagnation temperature of the jet is the same as that of the surrounding gas. The solution found differs only slightly from that of Howarth (1948) and Illingworth (1949) when laminar flow is considered; only the ‘change of scale’ effect (arising from a distortion of the coordinates in Bickley's solution) is of importance. In turbulent flow the effect of the second term of the series is as important as the ‘change of scale’ effect. The effect of compressibility on the width of the mixing region is discussed for both laminar and turbulent jet flow far from the orifice.

A previous study by Forbes (ANZIAM J., vol. 53, 2011, pp. 87–121) has argued that, when a light fluid is injected from a point source into a heavier ambient fluid, the interface between them is most unstable to perturbations at the lowest spherical mode. This means that, regardless of initial conditions, the outflow from a point source eventually becomes a one-sided jet. However, two-sided (bi-polar) outflows are nevertheless often observed in astrophysics, in apparent contradiction to this prediction. While there are many possible explanations for this fact, the present paper considers the effect of a straining flow in the ambient fluid. In addition, solid-body rotation in the inner fluid is also accounted for, in a Boussinesq viscous model. It is shown analytically that there are circumstances under which straining flow alone is sufficient to convert the one-sided jet into a genuine bi-polar outflow, in linearized theory. This is confirmed in a numerical solution of a viscous model of the flow, based on a spectral solution technique that accounts for nonlinear effects. Rotation can also generate flows that are two-sided, and this is likewise revealed through an asymptotic analysis and numerical solutions of the nonlinear equations.

For a viscous incompressible fluid the low Reynolds number description of the flow generated by a three-dimensional point source of oscillating strength situated on a wall is approximately quasi-steady in the neighbourhood of the singularity. This quasi-steady solution contains a number of unacceptable features, the principal one being that it is not a uniformly valid approximation within a small region surrounding the source point. In addition to this the vorticity in this region is predicted to be zero everywhere on the wall except at the singular point where it is infinite, which does not seem to be a physically reasonable distribution. When account is taken of the finite radius of the hole through which the fluid is driven and the finite width of the wall, the above difficulties are resolved yielding results that are quite realistic and informative.

The physical basis for parameterizing topographic stress due to unresolved eddies is examined in a quasi-geostrophic barotropic model. Topographic stress parameterization is shown to represent two effects of eddies: attraction of the flow to a statistical equilibrium featuring topographically correlated mean currents, and dissipation of potential enstrophy. Performance is evaluated by comparing parameterized low-resolution models with explicit high-resolution models.

An ensemble average of the equations of motion for a Newtonian fluid over particle configurations in a dilute fixed bed of spheres or cylinders yields Brinkman’s equations of motion, where the disturbance velocity produced by a test particle is influenced by the Newtonian fluid stress and a body force representing the linear drag on the surrounding particles. We consider a similar analysis for a power-law fluid where the stress $\boldsymbol{\tau} $ is related to the rate of strain $ \mathbisf{e} $ by $\boldsymbol{\tau} = 2m \mathop{ \vert \mathbisf{e} \vert }\nolimits ^{n\ensuremath{-} 1} \mathbisf{e} $, where $m$ and $n$ are constants. In this case, the ensemble-averaged momentum equation includes a body force resulting from the nonlinear drag exerted on the surrounding particles, a power-law stress associated with the disturbance velocity of the test particle, and a stress term that is linear with respect to the test particle’s disturbance velocity. The latter term results from the interaction of the test particle’s velocity disturbance with the random straining motions produced by the neighbouring particles and is important only in shear-thickening fluids where the velocity disturbances of the particles are long-ranged. The solutions to these equations using scaling analyses for dilute beds and numerical simulations using the finite element method are presented. We show that the drag force acting on a particle in a fixed bed can be written as a function of a particle-concentration-dependent length scale at which the fluid velocity disturbance produced by a particle is modified by hydrodynamic interactions with its neighbours. This is also true of the drag on a particle in a periodic array where the length scale is the lattice spacing. The effects of particle interactions on the drag in dilute arrays (periodic or random) of cylinders and spheres in shear-thickening fluids is dramatic, where it arrests the algebraic growth of the disturbance velocity with radial position when $n\geq 1$ for cylinders and $n\geq 2$ for spheres. For concentrated random arrays of particles, we adopt an effective medium theory in which the drag force per unit volume in the medium surrounding a test particle is assumed to be proportional to the local volume fraction of the neighbouring particles, which is derived from the hard-particle packing. The predictions of the averaged equations of motion are validated by comparison with simulations of randomly distributed hydrodynamically interacting cylinders.

We investigate electrophoresis of a tightly fitting sphere of radius $R-h_{0}$ on the axis of a circular tube of radius $R$, using lubrication theory and ideas due to Schnitzer & Yariv (Phys. Fluids, vol. 26, 2014, 122002). The electrical charge clouds on both the cylindrical wall and the surface of the sphere are assumed thin compared to the gap between the sphere and cylinder, so that charge clouds do not overlap and ion exclusion effects are minimal. Nevertheless, non-uniform pumping of counter-ions within the charge clouds leads to a change in the ionic concentration outside the charge clouds in the narrow gap between sphere and cylinder. The electro-osmotic slip velocities at the two surfaces are modified, leading to a decrease in the electrophoretic velocity of the sphere at low Péclet numbers and an increase in the velocity at high Péclet numbers. When the field strength $E_{0}$ is low, it is known that the electrophoretic velocity $U_{0}$ is proportional to $E_{0}(\unicode[STIX]{x1D701}_{s}-\unicode[STIX]{x1D701}_{c})$ which is zero when the zeta potential $\unicode[STIX]{x1D701}_{s}$ on the sphere surface is equal to the zeta potential $\unicode[STIX]{x1D701}_{c}$ on the cylinder. The perturbation to the above low field strength electrophoretic velocity (at high Péclet number) is predicted to be proportional to $E_{0}^{3}(\unicode[STIX]{x1D701}_{s}+\unicode[STIX]{x1D701}_{c})^{2}(\unicode[STIX]{x1D70E}_{s}+\unicode[STIX]{x1D70E}_{c})$, where $\unicode[STIX]{x1D70E}_{s}$ and $\unicode[STIX]{x1D70E}_{c}$ are the surface charge densities on the sphere and cylinder. The choice of materials with similar or identical zeta potentials (and surface charge densities) for the cylinder and sphere should therefore facilitate the observation of velocities nonlinear in the field strength $E_{0}$, since the reference linear electrophoretic velocity will be small.

Well-resolved measurements of the small-scale dissipation statistics within turbulent channel flow are reported for a range of Reynolds numbers from $Re_{{\it\tau}}\approx 500$ to 4000. In this flow, the local large-scale Reynolds number based on the longitudinal integral length scale is found to poorly describe the Reynolds number dependence of the small-scale statistics. When a length scale based on Townsend’s attached-eddy hypothesis is used to define the local large-scale Reynolds number, the Reynolds number scaling behaviour was found to be more consistent with that observed in homogeneous, isotropic turbulence. The Reynolds number scaling of the dissipation moments up to the sixth moment was examined and the results were found to be in good agreement with predicted scaling behaviour (Schumacher et al., Proc. Natl Acad. Sci. USA, vol. 111, 2014, pp. 10961–10965). The probability density functions of the local dissipation scales (Yakhot, Physica D, vol. 215 (2), 2006, pp. 166–174) were also determined and, when the revised local large-scale Reynolds number is used for normalization, provide support for the existence of a universal distribution which scales differently for inner and outer regions.

When raindrops with a diameter of the order of 1 mm hit a plane water surface they entrain air bubbles that radiate noise in the course of volume oscillations. The paper presents a model of the underwater noise of rain produced by this process. The depth of submergence, radius, and initial energy of the entrained bubbles are obtained numerically for a number of drop sizes. The bubbles are assumed to radiate as dipoles, and the total underwater noise is calculated by integrating over the size of the entraining rain drops. The results are compared both with laboratory experiments of single-drop impacts and field data of rain noise. It is found that the model gives somewhat larger bubbles than are observed experimentally. As a consequence, the characteristic spectral peak of rain is predicted to occur at a somewhat lower frequency than found in experiment. However the level of the peak is in reasonable agreement with data. The amount of noise due to the process of drop impact itself is also estimated and found to be several orders of magnitude lower than the data. Therefore, in spite of some deficiencies of the model and of the computational results, the proposed mechanism for the underwater noise of rain is strongly supported by this study.

We present the results of an experimental investigation of the motion of a light, solid sphere in a horizontal rotating cylinder filled with viscous fluid. At high rotation rates, the sphere sits near the axis of the cylinder. At lower rotation rates, a set of off-axis fixed points are observed for a range of sphere radii. The locations of these fixed points are in quantitative agreement with the predictions of a model based on available theory. The fixed points are observed to become unstable to periodic orbits below a critical Reynolds number $Re_{c}$. The radius of the observed orbits increases with Reynolds number more slowly than a typical Hopf bifurcation, in this case, growing as $1/Re^{2}$.

Dynamics of fluid tori in slow viscous flow is studied. Such tori are of interest as future carriers of biological and medicinal substances and are also viewed as potential building blocks towards more complex particles. In this study the immiscible ambient fluid is subject to a compressional flow (i.e., bi-extensional flow), and it comprises a generalization of our earlier report on the particular case with viscosity ratio $\unicode[STIX]{x1D706}=1$ (see Zabarankin et al., J. Fluid Mech., vol. 785, 2015, pp. 372–400), where $\unicode[STIX]{x1D706}$ is the ratio between the torus viscosity and that of the ambient fluid. It is found that, for all viscosity ratios, the torus either collapses towards the axis of symmetry or expands indefinitely, depending on the initial conditions and the capillary number, Ca. During these dynamic patterns the cross-sections exhibit various forms of deformation. The collapse and expansion dynamic modes are separated by a limited deformation into a deformed stationary state which appears to exist in a finite interval of the capillary number, $0<Ca<Ca_{cr}(\unicode[STIX]{x1D706})$, and is unstable to axisymmetric disturbances, which eventually cause the torus either to collapse or to expand indefinitely. The characteristic dimensions and shapes of these unstable stationary tori and their dependence on the physical parameters Ca and $\unicode[STIX]{x1D706}$ are reported.