A ring with a cross-section that has a blunt inner and sharper outer edge can attain an equilibrium orientation in a Newtonian fluid subject to a low Reynolds number simple shear flow. This may be contrasted with the continuous rotation exhibited by most rigid bodies. Such rings align along an orientation when the rotation due to fluid vorticity balances the counter-rotation due to the extensional component of the simple shear flow. While the viscous stress on the particle tries to rotate it, the pressure can generate a counter-vorticity torque that aligns the particle. Using boundary integral computations, we demonstrate ways to effectively control this pressure by altering the geometry of the ring cross-section, thus leading to alignment at moderate particle aspect ratios. Aligning rings that lack fore–aft symmetry can migrate indefinitely along the gradient direction. This differs from the periodic spatial trajectories of fore–aft asymmetric axisymmetric particles that rotate in periodic orbits. The mechanism for migration of aligned rings along the gradient direction is elucidated in this work. The migration speed can be controlled by varying the cross-sectional shape and size of the ring. Our results provide new insights into controlling motion of individual particles and thereby open new pathways towards manipulating macroscopic properties of a suspension.

One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.

We undertake a detailed analysis of linear stability of geostrophically balanced double density fronts in the framework of the two-layer rotating shallow-water model on the $f$-plane with topography, the latter being represented by an escarpment beneath the fronts. We use the pseudospectral collocation method to identify and quantify different kinds of instabilities resulting from phase locking and resonances of frontal, Rossby, Poincaré and topographic waves. A swap in the leading long-wave instability from the classical barotropic form, resulting from the resonance of two frontal waves, to a baroclinic form, resulting from the resonance of Rossby and frontal waves, takes place with decreasing depth of the lower layer. Nonlinear development and saturation of these instabilities, and of an instability of topographic origin, resulting from the resonance of frontal and topographic waves, are studied and compared with the help of a new-generation well-balanced finite-volume code for multilayer rotating shallow-water equations. The results of the saturation for different instabilities are shown to produce very different secondary coherent structures. The influence of the topography on these processes is highlighted.

We study the influence of low-frequency vibration on Marangoni instability in a layer of a binary mixture with the Soret effect. A linear stability analysis is performed numerically for perturbations of a finite wavelength (short-wave perturbations). Competition between long-wave and short-wave modes is found: the former ones are critical at smaller absolute values of the Soret number $\chi $, whereas the latter ones lead to instability at higher $\vert \chi \vert $. In both cases the vibration destabilizes the layer. Two variants of calculations are performed: via Floquet theory (linear asymptotic stability) and taking noise into consideration (empirical criterion). It is found that fluctuations substantially reduce the domains of stability. Further, while studying a limiting case within the empirical criterion, we have found a short-wave instability mode overlooked in former investigations of coupled Rayleigh–Marangoni convection in a layer of pure liquid.

When a potential is applied between two electrodes immersed in a flowing chemi-luminescent solution a blue glow appears at the surface of the anode. This glow may be used to observe the behaviour of liquid flows past solid surfaces. An electrochemical system suitable for generating the glow is described. Photographic studies of separated flows past cylinders, flat plates and cones are presented showing the application of the technique of electrochemiluminescence to flow visualization.

We study steady laminar Ekman boundary layers in rotating systems using an averaging method similar to the technique of von Kármán and Pohlhausen. The method allows us to explore nonlinear corrections to the standard Ekman theory even at large Rossby numbers. We consider both the standard self-similar ansatz for the velocity profile, which assumes that a single length scale describes the boundary layer structure, and a new non-self-similar ansatz in which the decay and the oscillations of the boundary layer are described by two different length scales. For both profiles we calculate the up-flow in a vortex core in solid-body rotation analytically. We compare the quantitative predictions of the model with the family of exact similarity solutions due to von Kármán and find that the results for the non-self-similar profile are in almost perfect quantitative agreement with the exact solutions and that it performs markedly better than the self-similar profile.

High-Reynolds-number steady flow in an annular pipe which encounters a shallow axisymmetric expansion or indentation in the walls is studied using interactive boundary-layer theory. The flow upstream of the indentation (x < 0) is fully developed; the ratio of the shear rate on the outer wall to that on the inner wall is denoted by ρ (0 < ρ < 1): similarity solutions are found for the case where the wall perturbations are proportional to $x^{\frac{1}{3}}$. The solution is unique in a constriction, when the pressure gradient (represented by a parameter b) is favourable (b < 0). In an expansion, however, with an adverse pressure gradient, three different solutions are found if b exceeds a critical value bc. When ρ ≠ 1, one of these solutions, representing a flow that is attached on the inner wall and separated (i.e. has negative wall shear) on the outer, is a continuation of the unique doubly attached flow at small b. The other two, one separated on the inner and not the outer wall and the other separated on both walls, arise from a saddle-node bifurcation at b = bc. The doubly separated flow is never stable, as observed in diffusers. In the case of a planar channel (ρ = 1) symmetry is restored, and the non-uniqueness arises through a supercritical pitchfork bifurcation. This agrees with previous computations on channel flow, but not with Jeffery-Hamel flow, for which the bifurcation is subcritical.

In laboratory experiments that impose shear deformation on partially molten aggregates of initially uniform porosity, melt segregates into high-porosity sheets (bands in cross-section). The bands emerge at $15^{\circ }$–$20^{\circ }$ to the shear plane. A model of viscous anisotropy can explain these low angles whereas previous simpler models have failed to do so. The anisotropic model is complex, however, and the reason that it produces low-angle bands has not been understood. Here we show that there are two mechanisms: (i) suppression of the well-known tensile instability, and (ii) creation of a new shear-driven instability. We elucidate these mechanisms using linearised stability analysis in a coordinate system that is aligned with the perturbations. We consider the general case of anisotropy that varies dynamically with deviatoric stress, but approach it by first considering uniform anisotropy that is imposed a priori and showing the difference between static and dynamic cases. We extend the model of viscous anisotropy to include a strengthening in the direction of maximum compressive stress. Our results support the hypothesis that viscous anisotropy is the cause of low band angles in experiments.

The dynamics of a second-order chemical reaction in an enclosed space driven by the mixing produced by a turbulent buoyant plume are studied theoretically, numerically and experimentally. An isolated turbulent buoyant plume source is located in an enclosure with a single external opening. Both the source and the opening are located at the bottom of the enclosure. The enclosure is filled with a fluid of a given density with a fixed initial concentration of a chemical. The source supplies a constant volume flux of fluid of different density containing a different chemical of known and constant concentration. These two chemicals undergo a second-order non-reversible reaction, leading to the creation of a third product chemical. For simplicity, we restrict attention to the situation where the reaction process does not affect the density of the fluids involved. Because of the natural constraint of volume conservation, fluid from the enclosure is continually vented. We study the evolution of the various chemical species as they are advected by the developing ventilated filling box process within the room that is driven by the plume dynamics. In particular, we study both the mean and vertical distributions of the chemical species as a function of time within the room. We compare the results of analogue laboratory experiments with theoretical predictions derived from reduced numerical models, and find excellent agreement. Important parameters for the behaviour of the system are associated with the source volume flux and specific momentum flux relative to the source specific buoyancy flux, the ratio of the initial concentrations of the reacting chemical input in the plume and the reacting chemical in the enclosed space, the reaction rate of the chemicals and the aspect ratio of the room. Although the behaviour of the system depends on all these parameters in a non-trivial way, in general the concentration within the room of the chemical input at the isolated source passes through three distinct phases. Initially, as the source fluid flows into the room, the mean concentration of the input chemical increases due to the inflow, with some loss due to the reaction with the chemical initially within the room. After a finite time, the layer of fluid contaminated by the inflow reaches the opening to the exterior at the base of the room. During an ensuing intermediate phase, the rate of increase in the concentration of the input chemical then drops non-trivially, due to the extra sink for the input chemical of the outflow through the opening. During this intermediate stage, the concentration of the input chemical continues to rise, but at a rate that is reduced due to the reaction with the fluid in the room. Ultimately, all the fluid (and hence the chemical) that was originally within the room is lost, both through reaction and outflow through the opening, and the room approaches its final steady state, being filled completely with source fluid.

Taylor–Couette flows have been widely studied in part due to the enhanced mixing performance from the variety of hydrodynamic flow states accessible. These process improvements have been demonstrated despite the traditionally limited injection mechanisms from the complexity of the Taylor–Couette geometry. In this study, using a newly designed, modified Taylor–Couette cell, axial mass transport behaviour is experimentally determined over two orders of magnitude of Reynolds number. Four different flow states, including laminar and turbulent Taylor vortex flows and laminar and turbulent wavy vortex flows, were studied. Using flow visualization techniques, the measured dispersion coefficient was found to increase with increasing $Re$, and a single, unified regression is found for all vortices studied. In addition to mass transport, the vortex structures’ stability to radial injection is also quantified. A dimensionless stability criterion, the ratio of injection to diffusion time scales, was found to capture the conditions under which vortex structures are stable to injection. Using the stability criterion, global and transitional stability regions are identified as a function of Reynolds number, $Re$.

Squeeze flows in liquid films between a porous disk and an impermeable disk generated by the relative motion of the disks are analysed. Two configurations that differ by the arrangement of (im)permeable external surfaces that bound the porous disk (i.e. not in contact with the liquid film) are considered. Such configurations allow for bearings with tuneable load-bearing characteristics and are also encountered in joint lubrication, adhesion, printing and composite manufacturing. In the present study, flow in the porous disk is governed by Darcy’s law and flow in the liquid film is described using lubrication theory. The present analysis also allows for slip between the liquid film and porous disk. Analytical solutions of the coupled system of equations governing flow in the liquid film and the porous disk are found. Under certain conditions, somewhat unexpected flow patterns are observed in the porous disk. The load-bearing capacity for both configurations is also examined as a function of the permeability and geometry of the permeable disk.

Mach reflection in steady supersonic flow with two incident shock waves is studied. The second incident shock wave is produced by an additional deflection of the wedge lower surface, at some point ensuring that the two incident shock waves would intersect at the reflecting surface in case of normal reflection. Both theory and computational fluid dynamics (CFD) are used to study the flow structure and the influence of the second incident shock wave. The overall flow configuration, in case of Mach reflection, is shown to be composed of a triple shock structure, a shock/shock interaction structure and a shock/slipline reflection structure. Similar phenomenon, triggered by a high downstream pressure, has been observed before numerically, but not studied theoretically. The second incident shock wave reflects over the slipline to deflect the slipline more towards the reflecting surface, increasing thus the Mach stem height, advancing the transition of regular reflection to Mach reflection of the first incident shock wave, and causing an inverted Mach reflection below the usual von Neumann condition. A Mach stem height model built for a weak second incident shock wave is used to study the influence of the second incident shock wave on the Mach stem height. Both theory and CFD predict a maximum of the Mach stem height at some additional wedge deflection angle.

Injecting a less viscous fluid into a more viscous fluid in a Hele-Shaw cell triggers two-dimensional viscous fingering patterns which are characterised by increasingly long fingers undergoing tip splitting and branching events. These complex structures are considered to be a paradigm for interfacial pattern formation in porous media flow and other related phenomena. Over the past five years, there has been a flurry of interest in manipulating these interfacial fingering patterns by altering the physical components of the Hele-Shaw apparatus. In this Focus on Fluids article, we summarise some of this work, concentrating on a very recent study in which the alterations include replacing one of the two bounding plates with an elastic membrane (Ducloué et al., J. Fluid Mech., vol. 826, 2017, R2). The resulting experimental set-up gives rise to a wide variety of novel interfacial patterns including periodic sideways fingers, dendritic-like patterns and short, flat-tipped viscous fingers that appear to resemble molar teeth. These latter fingers are similar to those observed in the printer’s instability and when peeling off a layer of adhesive tape. This delightful work brings together a number of well-studied themes in interfacial fluid mechanics, including how viscous and surface tension forces compete to drive fingering patterns, how interfaces are affected by fluid–solid interactions and, finally, how novel strategies can be implemented to control interfacial instabilities.

We study the heat or mass transfer from a neutrally buoyant spherical drop embedded in an ambient Newtonian medium, undergoing a general shearing flow, in the strong convection limit. The latter limit corresponds to the drop Péclet number being large ($Pe\gg 1$). We consider two families of ambient linear flows: (i) planar linear flows with open streamlines (parametrized by $\unicode[STIX]{x1D6FC}$ with $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 1$, the extremal members being simple shear flow ($\unicode[STIX]{x1D6FC}=0$) and planar extension ($\unicode[STIX]{x1D6FC}=1$)) and (ii) three-dimensional extensional flows (parameterized by $\unicode[STIX]{x1D716}$, with $0\leqslant \unicode[STIX]{x1D716}\leqslant 1$, the extremal members being planar ($\unicode[STIX]{x1D716}=0$) and axisymmetric extension ($\unicode[STIX]{x1D716}=1$)). For the first family, an analysis of the exterior flow field in the inertialess limit (the drop Reynolds number, $Re$, being vanishingly small) shows that there exist two distinct streamline topologies separated by a critical drop-to-medium viscosity ratio ($\unicode[STIX]{x1D706}$) given by $\unicode[STIX]{x1D706}_{c}=2\unicode[STIX]{x1D6FC}/(1-\unicode[STIX]{x1D6FC})$. For $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ all streamlines are open, while the near-field streamlines are closed for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$. For the second family, the exterior streamlines remain open regardless of $\unicode[STIX]{x1D706}$. The two streamline topologies lead to qualitatively different mechanisms of transport for large $Pe$. The transport in the open streamline regime is enhanced in the usual manner via the formation of a boundary layer. In sharp contrast, the closed-streamline regime displays diffusion-limited transport, so there is only a finite enhancement even as $Pe\rightarrow \infty$. For $Re=0$, the drop surface streamlines in a planar linear flow may be regarded as generalized Jeffery orbits with a flow and viscosity dependent aspect ratio Jeffery orbits denote the aspect-ratio-dependent inertialess trajectories of a rigid axisymmetric particle in a simple shear flow; see Jeffery (Proc. R. Soc. Lond. A, vol. 102 (715), 1922, pp. 161–179). A Jeffery-orbit-based non-orthogonal coordinate system thus serves as a natural candidate to tackle the transport problem from a drop, in a planar linear flow, in the limit $Pe\gg 1$. Use of this system allows one to derive a closed-form expression for the dimensionless rate of transport (the Nusselt number $Nu$) from a drop in the open-streamline regime ($\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$). Symmetry arguments point to a Jeffery-orbit-based coordinate system for any linear flow, and a variant of this coordinate system is therefore used to derive the Nusselt number for the family of three-dimensional extensional flows. For both classes of flows considered, the boundary-layer-enhanced transport implies that the Nusselt number takes the form $Nu={\mathcal{F}}(P,\unicode[STIX]{x1D706})Pe^{1/2}$, with the parameter $P$ being $\unicode[STIX]{x1D6FC}$ or $\unicode[STIX]{x1D716}$, and ${\mathcal{F}}(P,\unicode[STIX]{x1D706})$ given as a one and two-dimensional integral, respectively, which is readily evaluated numerically.

A model for calculating the force profile and the moment profile of a floating body in two dimensions with an arbitrary cross-section is proposed. Three types of cross-sections with different contact angles and densities are calculated by using the model to determine the vertical and rotational equilibria and their stabilities. Results show that the model can be applied to convex floating bodies with finitely many sharp edges. The study is then extended to investigate the surface tension effects on the vertical and rotational stabilities by varying the following parameters: the radii of curvature of the solid surface at the contact lines and the size of floating body. In general, the smaller the radii of curvature the better the vertical and rotational stabilities. However, for the contact angle $\unicode[STIX]{x1D703}=0$ (or $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$) the radii of curvature have no effect on the vertical stability of the floating body. By varying the size of the floating body, it is found that the vertical and rotational stabilities of mesoscale floating bodies vary continuously between the stabilities of the macroscale and microscale floating bodies with other parameters remaining unchanged.

The three-plate system in which a vertical plate is located between two spaced parallel plates partially immersed in an infinite water bath in a downward gravity field is considered. With different contact angles and distance between the plates on both sides, the force profiles of the middle plate in this three-plate system are investigated using the Young–Laplace equation in two dimensions, and five non-trivial qualitative force profiles are found to possibly depend on the contact angles and the distance. The study is then extended to the qualitative changes of stability and behaviours in the system, and the striking properties related to the bifurcation theory come to light. Results show that, for different contact angles, there are at most eight possible bifurcation diagrams where the distance between the plates on both sides is chosen as the bifurcation parameter. By analysing the force profile of the middle plate in each of the eight bifurcation diagrams, the stabilities of the equilibria of the plate can be obtained. The number and the stabilities of equilibria will change when the bifurcation parameter passes the critical value.

The radiation resistance (damping coefficient) and virtual mass for a circular disk that executes small, heaving oscillations at the surface of a semi-infinite body of water, originally calculated by MacCamy (1961a) through the numerical solution of an integral equation, are calculated from a systematic hierarchy of variational approximations. The first member of this hierarchy is based on the exact solution of the boundary-value problem for α = 0 and is in error by less than 2% for 0 [les ] α [les ] 1, where α = aσ2/g (a = radius of disk, σ = angular frequency, g = gravity). The second approximation provides a variational interpolation between the limiting results for α = 0 and α = ∞ and appears to be in error by less than 2% for all α except in certain narrow intervals, where pseudoresonances pose difficulties. Those difficulties are overcome by local reference to the third approximation. Numerical results are plotted for 0 [les ] α [les ] 10. Asymptotic results for α ↑ ∞ are developed in an Appendix.

The corresponding formulation and the first variational approximation are developed for pitching oscillations of the disk.

The evolution of long waves generated by a pressure disturbance acting on an initially unperturbed free surface in a channel of finite depth is analysed. Both off-critical and transcritical conditions are considered in the context of the fully nonlinear inviscid problem. The solution is achieved by using an accurate boundary integral approach and a time-stepping procedure for the free-surface dynamics.

The discussion emphasizes the comparison between the present results and those provided by both the Boussinesq and the related Korteweg–de Vries model. For small amplitudes of the forcing, the predictions of the asymptotic theories are essentially confirmed. However, for finite intensities of the disturbance, several new features significantly affect the physical results. In particular, the interaction among different wave components, neglected in the Korteweg–de Vries approximation, is crucial in determining the evolution of the wave system. A substantial difference is indeed observed between the solutions of the Korteweg–de Vries equation and those of both the fully nonlinear and the Boussinesq model. For increasing dispersion and fixed nonlinearity the agreement between the Boussinesq and fully nonlinear description is lost, indicating a regime where dispersion becomes dominant.

Consistently with the long-wave modelling, the transcritical regime is characterized by an unsteady flow and a periodic emission of forward-running waves. However, also in this case, quantitative differences are observed between the three models. For larger amplitudes, wave steepening is almost invariably observed as a precursor of the localized breaking commonly detected in the experiments. The process occurs at the crests of either the trailing or the upstream-emitted wave system for Froude numbers slightly sub- and super-critical respectively.

By specializing to the case of unit Lewis number and Prandtl number equal to ¾, a number of general results for the structure of a plane steady compressible flow field, within which chemical energy is being liberated by a simple Arrhenius type of combustion reaction, can be acquired by the use of elementary arguments. The field is of the semi-infinite variety, with a ‘flameholder’ presumed to exist at the origin of coordinates. In these circumstances it is necessary to specify the velocity gradient at inlet to the system or, equivalently, the pressure difference across the field. These quantities cannot be selected arbitrarily, and the nature and extent of the restrictions upon them is fully explored. Since the Mach number of the stream is hypothesized to be a quantity of order unity, local Damköhler numbers are always small. Therefore the field is shown to consist of relatively long regions within which the combustion activity takes place, with embedded thin domains of rapid, almost chemically inert, gasdynamical adjustment, whose dimension is typically that of the conventional shock wave. When the inlet Mach number is less than unity the gasdynamical adjustment domains are always adjacent to the origin, and this is also true under most supersonic inlet conditions.

However, there are some special circumstances for which the shock is detached from the flameholder and is established in the middle of the combustion activity. A specific example is provided by a shock within the induction domain.

These special circumstances are shown to be ultrasensitive to pressure difference across the whole domain. It is also shown that wholly supersonic combustion does exist, but only under similar conditions of extreme sensitivity to pressure difference.

The general arguments are supported and illuminated by asymptotic analysis based on the large activation energy of the Arrhenius reaction. Space precludes a full asymptotic treatment of the combustion activity but a companion paper that analyses these parts of the general field is being prepared in collaboration with D. R. Kassoy. Analysis of the shock within the induction domain, together with results from the case of subsonic inlet Mach numbers, shows that gasdynamical effects can prevent ignition by channelling combustion energy into kinetic energy of the flow at the expense of thermal energy.

Even mild longitudinal wall curvature is known to produce significant effects on the time-averaged turbulent transport in a boundary layer. The present study was undertaken to study the manner in which the instantaneous structure of turbulence in the boundary layer responds to mild streamline curvature. Both convex and concave boundary layers with a boundary-layer thickness to wall radius ratio of about 0·01 were studied. Attention was directed mainly to two events characterizing the instantaneous turbulence structure. These were the so-called ‘bursting’ and ‘zero-crossing’. Quantitative data on the statistics of these events were obtained using a combination of analog instrumentation and visual counting (from continuous film records). These data were compared with data from flat-wall boundary layers obtained from similar signal-processing techniques. The results indicate that neither the individual nor the joint statistics of these events are significantly affected by curvature in the vicinity of the wall. On the other hand, curvature seems to affect appreciably at least some properties of these events at large distances from the wall. Careful examination of these results shows, however, that neither the process of turbulent production near the wall nor the turbulent dissipative process anywhere in the boundary layer is significantly affected by mild curvature. Apparent curvature effects on the instantaneous structure in the outer part of the boundary layer can be explained as being due to the strong effect of streamline curvature on the turbulent diffusion process. This explanation is consistent with previous observations of the time-averaged structure of the flow. The results of the present study indicate the need to re-examine some of the recent turbulence models for curved flows that involve modification of the production and dissipation terms rather than the diffusion term in the transport equations.