In this work the thermocapillary-induced motion of partially engulfed compound drops is considered. This phenomenon occurs in many natural and technological processes in which heat is exchanged between such hybrid drops and the medium around them through the interfaces. Two types of thermal fields and the resulting motions are studied; flow induced by an external temperature gradient and spontaneous thermocapillary motion. For the first flow type, it was found that, in general, the motion is induced in the direction of the temperature gradient. However, under certain physical conditions and drops' configuration a motion against the temperature gradient may be observed. In the second case, spontaneous thermocapillary motion, the compound drop moves due to surface tension gradients which result from the geometric non-uniformity of the system. Results are presented for several parameters such as configuration of the compound drop, viscosity, thermal conductivity ratio, the dependence of the various interfacial tensions on temperature and the volume ratio of the phases within the drop.

A continous two-parameter family of analytical solutions to the Euler equations are presented representing a class of steadily rotating vortex arrays involving $N\,{+}\,1$ interacting vortex patches where $N\,{\ge}\,3$ is an integer. The solutions consist of a central vortex patch surrounded by an $N$-fold symmetric alternating array of satellite point vortices and vortex patches. One of the parameters governs the size of the central patch, the other governs the size of the $N$ satellite patches. In the limit where the areas of the satellite vortex patches tend to zero, the solutions degenerate to the exact solutions of Crowdy (J. Fluid Mech. vol. 469, 2002, p. 209). Limiting states are found in which cusps form only on the central patch, only on the satellite patches, or simultaneously on both central and satellite patches. Contour dynamics simulations are used to check the mathematical solutions and test their robustness. The linear stability of a class of ‘point-vortex models’ (in which the patches are replaced by point vortices) are also studied in order to examine the stability of the distributed-vorticity configurations to pure-displacement modes. On the other hand, a desingularization of all point vortices to Rankine vortices leads to a class of ‘quasi-equilibria’ consisting purely of interacting vortex patches close to hydrodynamic equilibrium.

Recent experiments by Hammack et al. (J. Fluid Mech., vol. 532, 2005, p. 1) on deep-water waves with two-dimensional, periodic surface patterns showed several features, some of which were unsteady. Fuhrman & Madsen (J. Fluid Mech., vol. 559, 2006, p. 391) explained three of these as being the consequence of sinusoidal forcing by the wavemaker array that did not include forced harmonics, either in time or in the direction parallel to the wavemaker. They predicted that neglected third-harmonic terms cause more serious consequences than neglected second-harmonic terms when the generated wavefields have two-dimensional surface patterns. This paper presents experiments that provide strong evidence that their explanation is correct: including the third-harmonic terms in the wavemaker forcing results in wave patterns that propagate with nearly permanent form.

A numerical study on the interaction of two spherical drops in thermocapillary migration in microgravity is presented. Unequal drop sizes in the axisymmetric model lead to strong drop interaction if the leading drop is smaller. The effect of the ratio of the two drop radii, their initial distance apart, and non-dimensional numbers on the interaction is studied in the case of non-merging drops in detail. The Marangoni number adopted in this paper is fairly large (around 100) so as to reveal the phenomena of real flows. As a result, the heat wake behind the leading drop plays an important role in drop interaction, and obviously different final drop distances and transient migration processes are observed for various sets of non-dimensional numbers. The influence of drop deformation on drop interaction is also investigated for relatively large capillary number (up to 0.2). Finally, some simulations are performed to explain the phenomena of drop interaction in previous experiments, and some suggestions for future experiments are also provided.

Measurements have been made of the net force F acting on a bluff rigid body moving with velocity U (relative to a fluid rotating about a vertical axis with uniform angular velocity Ω) in a plane perpendicular to the axis of rotation. The force F is of magnitude 2ΩρVU, where ρ is the density of the fluid and V is a volume which depends on the size and shape of the body. The relative direction of F and U is found to depend on the quantity \[ {\cal S}\equiv \frac{2\Omega L}{U}\bigg(\frac{h}{D}\bigg), \] where L and h are horizontal and vertical lengths characterizing the object and D is the depth of the fluid in which the object is placed.

The steady axisymmetric flow of viscous liquid relative to a gas bubble due to its buoyancy-driven motion in a round tube is computed by solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted mesh. When the bubble is relatively small compared with the tube size (e.g. the volume-equivalent radius of the bubble is less than a quarter of the tube radius R), the bubble exhibits similar behaviour to one moving in an extended liquid, developing a spherical-cap shape with increasing Reynolds number (Re) if the capillary number is not too small. The long-bubble (also known as a Taylor bubble) characteristics can be observed with bubbles of volume-equivalent radius greater than the tube radius, especially when the surface tension effect is relatively weak (e.g. for Weber number We greater than unity). The computed values of Froude number Fr for most cases agree well with the correlation formulae derived from experimental data for long bubbles, and even with (short) bubbles of volume-equivalent radius three-quarters of the tube radius. All of the computed surface profiles of long bubbles exhibit a prolate-like nose shape, yet various tail shapes can be obtained by adjusting the parameter values of Re and We. At large Weber number (e.g. We=10), the bubble tail forms a concave profile with a gas ‘cup’ developed at small Re and a ‘skirt’ at large Re with sharply curved rims. For We≤1, the bubble tail profile appears rounded without large local curvatures, although a slightly concave tail may develop at large Re. non-uniform annular film adjacent to the tube wall is commonly observed when Weber number is small, especially for bubbles of volume <3πR3, suggesting that the surface tension effect can play a complicated role. Nonetheless the computed value of Fr is found to be generally independent of the bubble length for bubbles of volume-equivalent radius greater than the tube radius. If the bubble length reaches about 2.5 tube radii, the value of its frontal radius becomes basically the same as that for long bubbles of much larger volume. An examination of the distribution of the z-component of traction along the bubble surface reveals the basic mechanism for long bubbles rising at a terminal velocity that is independent of bubble volume.

The changes in discharge in pressure-driven flows through channels with longitudinal grooves have been investigated in the laminar flow regime and in the turbulent flow regime with moderate Reynolds numbers ($Re_{2H}\approx 6000$) using both analytical and numerical methodologies. The results demonstrate that the long-wavelength grooves can increase discharge by 20 %–150 %, depending on the groove amplitude and the type of flow, while the short-wavelength grooves reduce the discharge. It has been shown that the reduced geometry model applies to the analysis of turbulent flows and the performance of grooves of arbitrary form is well approximated by the performance of grooves whose shape is represented by the dominant Fourier mode. The flow patterns, the turbulent kinetic energy as well as the Reynolds stresses were examined to identify the mechanisms leading to an increase in discharge. It is shown that the increase in discharge results from the rearrangement of the bulk fluid movement and not from the suppression of turbulence intensity. The turbulent kinetic energy and the Reynolds stresses are rearranged while their volume-averaged intensities remain the same as in the smooth channel. Analysis of the interaction of the groove patterns on both walls demonstrates that the converging–diverging configuration results in the greatest increase in discharge while the wavy channel configuration results in a reduction in discharge.

This work is a parametric study of the fluxes of heat and salt across unsheared and sheared double-diffusive interfaces using one-dimensional-turbulence (ODT) simulations. It is motivated by the need to understand how these fluxes scale with parameters related to the fluid molecular properties and background shear. Comparisons are made throughout with previous models and available measurements. In unsheared interfaces, ODT simulations show that the dimensionless heat flux Nu scales with the stability parameter Rρ, Rayleigh number Ra and Prandtl number Pr as Nu ~ (Ra/Rρ)0.37±0.03 when Pr varies from 3 to 100 and as Nu ~ (Ra/Rρ)0.31Pr0.22±0.04 when Pr varies from 0.01 to 1. Here Ra/Rρ can be seen as the ratio of destabilizing and stabilizing effects. The simulation results also indicate that the ratio of salt and heat fluxes Rf is independent of Pr, scales with the Lewis number Le as Rf ~ Le0.41±0.04 when Rρ is large enough and deviates from this expression for low values of Rρ, when the interface becomes heavily eroded. In sheared interfaces, the simulations show three flow regimes. When the Richardson number Ri ≪ 1, shear-induced mixing dominates, the heat flux scales with the horizontal velocity difference across the interface and Rf = Rρ. Near Ri ~ 1 the heat and salt fluxes are seen to increase abruptly as the shear increases. The flow structure and scaling of the fluxes are similar to those of unsheared interfaces when Ri ≫ 1.

In this study, the Bragg–Hawthorne equation (BHE) is extended in the context of a steady, inviscid and compressible fluid, thus leading to an assortment of partial differential equations that must be solved simultaneously. A solution is pursued by implementing a Rayleigh–Janzen expansion in the square of the reference Mach number. The corresponding formulation is subsequently used to derive a compressible approximation for the Trkalian model of the bidirectional vortex. The approximate solution is compared to a representative computational fluid dynamics simulation in order to validate the modelling assumptions under realistic conditions. The latter is found to exhibit an appreciable steepening of the axial velocity profile, which is accompanied by an axial dependence in the mantle location that is somewhat reminiscent of the radial shifting of mantles reported in some experimental trials and simulations. In this context, flows with a strong swirl intensity do not seem to be significantly affected by the introduction of compressibility. Rather, as the swirl intensity is reduced the effects of compressibility become more noticeable, especially in the axial and radial velocity components. It may also be realized that imparting a progressively larger swirl component stands to promote the axisymmetric distribution of flow field properties, and these include an implicit resistance to dilatational effects in the tangential direction. From a broader perspective, this study provides a viable approximation to the Trkalian motion associated with cyclonic flows, while serving as a limited proof of concept for the compressible Bragg–Hawthorne procedure applied to a steady, axisymmetric and inviscid fluid.

Motivated by the reopening mechanics of strongly collapsed airways, we study the steady propagation of an air finger through a collapsed oil-filled channel with a single compliant wall. In a previous study using fully compliant elastic tubes, a ‘pointed’ air finger was found to propagate at high speed and low pressure, which, if clinically accessible, offers the potential for rapid reopening of highly collapsed airways with minimal tissue damage (Heap & Juel Phys. Fluids, vol. 20 (8), 2008, 081702). The mechanism underlying the selection of that pointed finger, however, remained unexplained. In this paper, we identify the required selection mechanism by conducting an experimental study in a simpler geometry: a rigid rectangular Hele-Shaw channel with an elastic top boundary. The constitutive behaviour of this elasto-rigid channel is nonlinear and broadly similar to that of an elastic tube, but unlike the tube, the channel’s cross-section adopts self-similar shapes from the undeformed state to the point of first near wall contact. The ensuing simplification of the vessel geometry enables the systematic investigation of the reopening dynamics in terms of increasing initial collapse. We find that for low levels of initial collapse, a single centred symmetric finger propagates in the channel and its shape is set by the tip curvature. As the level of collapse increases, the channel cross-section develops a central region of near opposite wall contact, and the finger shape evolves smoothly towards a ‘flat-tipped’ finger whose geometry is set by the strong depth gradient near the channel walls. We show that the flat-tipped mode of reopening is analogous to the pointed finger observed in tubes. Its propagation is sustained by the vessel’s extreme cross-sectional profile at high collapse, while vessel compliance is necessary to stabilise it. A simple scaling argument based on the dissipated power reveals that this reopening mode is preferred at higher propagation speeds when it becomes favourable to displace the elastic channel wall rather than the viscous fluid.

The linear instability mechanisms of horizontal convection in a rectangular cavity forced by a horizontal buoyancy gradient along its surface are investigated using local and global stability analyses for a Prandtl number equal to unity. The results show that the stability of the base flow, a steady circulation characterized by a narrow descending plume and a broad upwelling region, depends on the Rayleigh number, $Ra$. For free-slip boundary conditions at a critical value of $Ra\approx 2\times 10^{7}$, the steady base flow becomes unstable to three-dimensional perturbations, characterized by counter-rotating vortices originating within the plume region. A Wentzel–Kramers–Brillouin (WKB) method applied along closed streamlines demonstrates that this instability is of a Rayleigh–Taylor type and can be used to accurately reconstruct the global instability mode. In the case of no-slip boundary conditions, the base flow also becomes unstable to a self-sustained two-dimensional instability whose critical Rayleigh number is $Ra\approx 1.7\times 10^{8}$. Beyond this critical $Ra$, two-dimensional equilibrium stationary states of the Navier–Stokes equations are computed using the selective frequency damping method. The two-dimensional onset of instability is shown to be characterized by a family of modes also originating within the plume. A local spatio-temporal stability analysis shows that the flow becomes absolutely unstable at the origin of the plume. Taken together, these results illustrate the mechanisms behind the onset of turbulence that has been observed in horizontal convection.

Using high-resolution particle image velocimetry, we measure velocity profiles, the wind Reynolds number and characteristics of turbulent plumes in Taylor–Couette flow for a radius ratio of 0.5 and Taylor number of up to $6.2\times 10^{9}$. The extracted angular velocity profiles follow a log law more closely than the azimuthal velocity profiles due to the strong curvature of this ${\it\eta}=0.5$ set-up. The scaling of the wind Reynolds number with the Taylor number agrees with the theoretically predicted $3/7$ scaling for the classical turbulent regime, which is much more pronounced than for the well-explored ${\it\eta}=0.71$ case, for which the ultimate regime sets in at much lower Taylor number. By measuring at varying axial positions, roll structures are found for counter-rotation while no clear coherent structures are seen for pure inner cylinder rotation. In addition, turbulent plumes coming from the inner and outer cylinders are investigated. For pure inner cylinder rotation, the plumes in the radial velocity move away from the inner cylinder, while the plumes in the azimuthal velocity mainly move away from the outer cylinder. For counter-rotation, the mean radial flow in the roll structures strongly affects the direction and intensity of the turbulent plumes. Furthermore, it is experimentally confirmed that, in regions where plumes are emitted, boundary layer profiles with a logarithmic signature are created.

An approximate analytical solution is derived for the inclined non-inertial sinkage of a flat plate onto a smooth surface, for the case of small angles. The result is a modification to the existing solutions for parallel sinkage, in the form of a polynomial in the dimensionless angle of inclination. Even the smallest angle of inclination of plate to surface produces a large increase in the sinkage rate otherwise obtained. The results are confirmed experimentally.

A model is constructed for perturbations created in the surrounding laminar flow by a turbulent spot in plane Poiseuille flow. The turbulent spot is represented as a distribution of increased Reynolds stress, which travels steadily through the surrounding laminar flow. The Navier-Stokes equations are linearized and are solved by using Fourier transforms in the plane parallel to the channel walls and a finite-difference method in the direction perpendicular to the walls. The travelling Reynolds stress distribution acts as a forcing term in the equations.

Numerical results show that a packet of oblique waves are generated around the disturbance when the force is antisymmetric with respect to the channel centreline, whereas no identifiable wave crests are found when the forcing is symmetric. Furthermore, wavelengths of the typical waves composing the packet are insensitive to the size of the region of Reynolds stress. The dependencies of the flow field on Reynolds number and spot speed are investigated. In the case of symmetric forcing, the flow is forced around the disturbance, causing distortions to the basic velocity profiles. These results are in qualitative agreement with experimental observations.

Time-dependent convection motions in the form of rolls in a thin vertical fluid layer (Hele-Shaw cell) heated from below are investigated numerically. Perpendicular to the convection-roll axis the fluid is bounded by parallel adiabatic rigid sidewalls. Stress-free top, bottom and end boundaries are assumed. The horizontal extension of the convection rolls is described by the wavenumber α. Solutions for the time-dependent behaviour of the convective motion are presented for a range of wavenumbers between α = ½π and 2π. The onset of the oscillation is shifted to higher Rayleigh numbers with increasing wavenumber. The oscillatory Hele-Shaw convection is caused by an instability of the thermal boundary layer, as is evident from the plotted temperature field and streamlines. From the variation of the Nusselt number with time it is found that the oscillatory motion starts with a sinusoidal time dependence and passes into a periodic state with several frequencies as the Rayleigh number is increased. Quantitative and qualitative agreement with previous experimental and numerical results is found.

The nonlinear evolution of an array of pairs of inviscid counter-rotating vortices, subjected to an applied stretching strain field, has been studied numerically using the contour-dynamics method. The array configuration is effectively the Corcos-Lin model of streamwise vortices in the braid region of a nominally two-dimensional mixing layer. For each individual vortex the simulations elucidate the strong interaction between the vortex self-induction, the vorticity amplification of the stretching strain, and the local in-plane strain applied by all other members of the array. When the initial vorticity distribution is modelled by a non-uniform piece-wise-constant vorticity field defined over a nested set of non-intersecting contours, the dynamical evolution reveals fine structure consisting of strong vortex roll-up accompanied by trailing, filament-like spiral vortex sheets, and the presence of tertiary instabilities. It is shown by a particular example that these features are largely absent in an equivalent computation in which array members are modelled by the commonly used uniform-vortex approximation.