This paper studies the nonlinear coupling between the volume pulsation, translational motion and shape modes of an oscillating bubble, especially in the context of translational instability, known as ‘dancing motion’, that is demonstrated by bubbles in acoustic standing waves. A set of coupled equations is derived that describes volume pulsations of a bubble, its translational motion and shape oscillations evolving on the bubble surface. The amplitudes of the surface modes and the translational velocity of the bubble are assumed to be small and allowed for in the equations of motion up to only second-order terms. The amplitude of the volume oscillation is not limited. Unlike earlier work on this subject, where only two adjacent shape modes with given natural frequencies are taken into account, we allow for all shape modes and do not impose any limitations on their natural frequencies. As a result, the present analysis reveals additional features, which have not been noted previously, inherent in the mutual interactions of the shape modes as well as in the interaction between the shape modes and the translational motion.

The linear stability of a compressible inviscid axisymmetric and rotating columnar flow of a prefect gas in a finite-length straight circular pipe is investigated. This work extends a previous analysis to include the influence of Mach number on the flow dynamics. A well-posed model of the unsteady motion of a swirling flow, with inlet and outlet conditions that may reflect the physical situation, is formulated. The linearized equations of motion for the evolution of infinitesimal axially symmetric disturbances are derived. A general mode of disturbance, that is not limited to the axial-Fourier mode, is introduced and an eigenvalue problem is developed. It is found that a neutral mode of disturbance exists at ‘the critical swirl ratio for a compressible vortex flow’. The flow changes its stability characteristics as the swirl ratio increases across this critical level. When the swirl ratio is below the critical level (supercritical flow), an asymptotically stable mode is found and, when it is above the critical level (subcritical flow), an unstable mode of disturbance develops. This result cannot be predicted by any of the previous stability criteria. When the characteristic Mach number of the base flow tends to zero, the results are the same as found for incompressible swirling flows in pipes. The growth rate ratio is positive for all Mach numbers, but decreases as Mach number is increased. This ratio vanishes at the limit Mach number at which the critical swirl tends to infinity. The present results also demonstrate that the axisymmetric breakdown of high-Reynolds-number compressible vortex flows may be delayed to higher swirl ratios with the increase of the incoming flow Mach number.

The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number $E$ is characterized by quasi-geostrophic thermal Rossby waves, which vary slowly in the axial direction on the scale of the sphere radius $r_o$ and have short azimuthal length scale $O(E^{1/3}r_o)$. They also confirmed the localization of the convection about some cylinder radius $s\,{=}\,s_M$ roughly $r_o/2$. Their novel contribution concerned the implementation of global stability conditions to determine, for the first time, the correct Rayleigh number, frequency and azimuthal wavenumber. Their analysis also predicted the value of the finite tilt angle of the radially elongated convective rolls to the meridional planes. In this paper, we study small-Ekman-number convection in a spherical shell. When the inner sphere radius $r_i$ is small (certainly less than $s_M$), the Jones et al. (2000) asymptotic theory continues to apply, as we illustrate with the thick shell $r_i\,{=}\,0.35\,r_o$. For a large inner core, convection is localized adjacent to, but outside, its tangent cylinder, as proposed by Busse & Cuong (1977). We develop the asymptotic theory for the radial structure in that convective layer on its relatively long length scale $O(E^{2/9}r_o)$. The leading-order asymptotic results and first-order corrections for the case of stress-free boundaries are obtained for a relatively thin shell $r_i\,{=}\,0.65\,r_o$ and compared with numerical results for the solution of the complete PDEs that govern the full problem at Ekman numbers as small as $10^{-7}$. We undertook the corresponding asymptotic analysis and numerical simulation for the case in which there are no internal heat sources, but instead a temperature difference is maintained between the inner and outer boundaries. Since the temperature gradient increases sharply with decreasing radius, the onset of instability always occurs on the tangent cylinder irrespective of the size of the inner core radius. We investigate the case $r_i\,{=}\,0.35\,r_o$. In every case mentioned, we also apply rigid boundary conditions and determine the $O(E^{1/6})$ corrections due to Ekman suction at the outer boundary. All analytic predictions for both stress-free and rigid (no-slip) boundaries compare favourably with our full numerics (always with Prandtl number unity), despite the fact that very small Ekman numbers are needed to reach a true asymptotic regime.

We present a model for velocity fluctuations of dilute sedimenting spheres at low Reynolds number. The central idea is that a vertical stratification causes the fluctuations to decrease below those of an independent uniform distribution of particles, such a stratification naturally occurring from the broadening of the sedimentation front. We use numerical simulations, scaling arguments, structure factor calculations, and experiments to show that there is a critical stratification above which the characteristics of the density and velocity fluctuations change significantly. For thin cells, the broadening of the sediment front (and the resulting stratification) is small, so the velocity fluctuations are predicted by independent-Poisson-distribution estimates. In very thick cells, the stratification is significant, leading to persistent decay of the velocity fluctuations for the duration of the experiment. Estimated stratifications quantitatively agree with the simulations, and indicate the likelihood that previous experimental measurements were also affected by stratification. The velocity fluctuations in sedimentation are therefore not universal but instead depend on both the cell shape and developing stratification.

We examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. Using the so-called lubrication approximation, we derive an asymptotic equation for three-dimensional motion of the film and use this equation to examine its linear stability. It is demonstrated that: (i) there are infinitely many normal modes (harmonic in the axial variable and time), which are all neutrally stable and their eigenfunctions form a complete set; (ii) but the film is nonetheless unstable with respect to non-harmonic disturbances, which develop singularities in a finite time.

The steady two-dimensional laminar flow past a stationary cylinder is well known to lose stability to a periodic flow at a supercritical Hopf bifurcation point as the flow rate is increased. It is less well known that the critical flow rate at which the instability occurs can be increased by rotating the cylinder about is own axis, and that at a fixed flow rate the vortex street can be eliminated entirely by sufficiently large rotation. We confine the flow between two parallel walls and report the effect of cylinder rotation on the Reynolds number and frequency at the Hopf bifurcation point. Two new, and somewhat surprising, results are that for stationary cylinders at large blockage ratios, the steady symmetric flow can restabilize above a critical Reynolds number, and that steady asymmetric flows exist.

The measurements by Zagarola & Smits (1998) of mean velocity profiles in fully developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the uncertainties due to velocity gradient corrections. A new static pressure correction (McKeon & Smits 2002) is used in analysing all data and leads to significant differences from the Zagarola & Smits conclusions. The results confirm the presence of a power-law region near the wall and, for Reynolds numbers greater than $230\,{\times}\,10^3$ ($R^+\,{>}\,5\,{\times}\,10^3$), a logarithmic region further out, but the limits of these regions and some of the constants differ from those reported by Zagarola & Smits. In particular, the log law is found for $600\,{<}\, y^+\,{<}\,0.12R^+$ (instead of $600\,{<}\,y^+\,{<}\,0.07R^+$), and the von Kármán constant $\kappa$, the additive constant $B$ for the log law using inner flow scaling, and the additive constant $B^*$ for the log law using outer scaling are found to be $0.421 \pm 0.002$, $5.60 \pm 0.08 $ and $1.20 \pm 0.10$, respectively, with 95% confidence level (compared with $0.436 \pm 0.002$, $6.15 \pm 0.08$, and $1.51 \pm 0.03$ found by Zagarola & Smits). The data also confirm that the pipe flow data for Re$_D\,{\le}\,13.6\,{\times}\,10^6$ (as a minimum) are not affected by surface roughness.

Discrete particles released into turbulent carrier flows respond to that turbulence over periods of time which depend on the particle inertia and the initial state of the particle. Due to the turbulence, the velocity of the particles becomes random and they will become dispersed throughout the carrier flow. The level of randomness of the particle velocities can be quantified by the particle kinetic stress and the spreading rate is characterized by the dispersion coefficient. In the idealized case of stationary isotropic turbulence, the kinetic stress and dispersion coefficients approach limiting values long after release. Previous analysis by the author has provided development times for dispersion coefficients and kinetic stress in cases where particles were released from a point source: (i) from an initial state of rest, and (ii) with kinetic stress identical to that found in the steady state. This paper generalizes the analysis by allowing for arbitrary initial particle velocity and displacement distributions. As with the previous analysis, the primary focus is on high-inertia particles. Unlike the previous work, however, the present analysis is applicable to the common practice in direct numerical simulations of setting initial particle velocities equal to local fluid velocities. Several different estimators of the particle dispersion coefficient are considered. Whereas each estimator leads to the same long-time value, the time taken to approach this value can vary dramatically. Expressions for development times are given for the particle kinetic stress and dispersion coefficients. Consequences of the analysis for experimental and computational studies are discussed.

We perform direct numerical simulations of dynamic equations of decaying gravity waves on infinite-depth water. Power-law behaviour of the wave action spectrum and structure functions of the surface elevation is obtained. These power laws agree with the prediction of the weak turbulence theory. The probability density function (p.d.f.) of the surface elevation is close to the Gaussian distribution around the mean value which seems to be consistent with the random phase approximation. However, the p.d.f. deviates weakly from the Gaussian in the tail region. This deviation is significant and can be amplified by taking the Laplacian. In addition, intermittency and breakdown of the weak turbulence theory are discussed.

The effects of resistive forces on unsteady shallow flows over rigid horizontal boundaries are investigated theoretically. The dynamics of this type of motion are driven by the streamwise gradient of the hydrostatic pressure, which balances the inertia of the fluid and the basal resistance. Drag forces are often negligible provided the fluid is sufficiently deep. However, close to the front of some flows where the depth of the moving layer becomes small, it is possible for drag to substantially influence the motion. Here we consider three aspects of unsteady shallow flows. First we consider a regime in which the drag, inertia and buoyancy (pressure gradient) are formally of the same magnitude throughout the entire current and we construct a new class of similarity solutions for the motion. This reveals the range of solution types possible, which includes those with continuous profiles, those with discontinuous profiles and weak shocks and those which are continuous but have critical points of transition at which the gradients may be discontinuous. Next we analyse one-dimensional dam-break flow and calculate how drag slows the motion. There is always a region close to the front in which drag forces are not negligible. We employ matched asymptotic expansions to combine the flow at the front with the flow in the bulk of the domain and derive theoretical predictions that are compared to laboratory measurements of dam-break flows. Finally we investigate a modified form of dam-break flow in which the vertical profile of the horizontal velocity field is no longer assumed to be uniform. It is found that in the absence of drag it is no longer possible to find a kinematically consistent front of the fluid motion. However the inclusion of drag forces within the region close to the front resolves this difficulty. We calculate velocity and depth profiles within the drag-affected region, and obtain the leading-order expression for the rate at which the fluid propagates when the magnitude of the drag force is modelled using Chézy, Newtonian and power-law fluid closures; this compares well with experimental data and provide new insights into dam-break flows.

From hot-wire anemometer measurements in active-grid wind-tunnel turbulence we have determined the Reynolds number dependence of the velocity derivative moments, the mean-squared pressure gradient, $\chi$, and the normalized acceleration variance, $a_0$, over the Reynolds number range $100\,{\leq}\,R_\lambda\,{\leq}\,900$. The values of $\chi$ and $a_0$ were obtained from the fourth-order velocity structure functions. The derivative moments show power-law dependence on Reynolds number and the exponent is the same with or without shear. In particular, we find the derivative kurtosis, $K_{\partial u/\partial x }\,{\sim}\,R_\lambda^{0.39}$, and there is no evidence of the transition that has been observed in this quantity in some recent work. We find that at high Reynolds numbers, $\chi$ and $a_0$ tend to values similar to those obtained by direct particle tracking measurements and by direct numerical simulation. However, at lower Reynolds number our estimates of $\chi$ and $a_0$ appear to be affected by the evaluation technique which imposes strict requirements on local homogeneity and isotropy.

An infinitely diverging channel with a line source of fluid at its vertex is a natural idealization of flow in a finite channel expansion. Motivated by numerical results obtained in an associated geometry (Tutty 1996), we show in this theoretical model that for certain channel semi-angles $\alpha$ and Reynolds numbers $\hbox{\it Re}\,{:=}\,Q/2\nu$ ($Q$ is the volume flux per unit length and $\nu$ the kinematic viscosity) a steady, spatially periodic, two-dimensional wave exists which appears spatially stable and hence plausibly realizable in the physical system. This spatial wave (or limit cycle) is born out of a heteroclinic bifurcation across the subcritical pitchfork arms which originate out of the well known Jeffery-Hamel bifurcation point at $\alpha\,{=}\,\alpha_2(\hbox{\it Re})$. These waves have been found over the range $5\,{\leq}\,\hbox{\it Re}\,{\leq}\,5000$ and, significantly, exist for semi-angles $\alpha$ beyond the point $\alpha_2$ where Jeffery-Hamel theory has been shown to be mute. However, the limit of $\alpha\,{\rightarrow}\, 0$ at finite $Re$ is not reached and so these waves have no relevance to plane Poiseuille flow.

A simple ‘elliptical-pore model’ of the shrinkage of compressible pores in late-stage planar viscous sintering is proposed. The model is in the spirit of matched asymptotics and relies on splitting the flow into an ‘inner’ and ‘outer’ problem. The inner problem in the vicinity of any given pore involves solving for its free-surface evolution exactly using complex-variable methods. The outer flow due to all other pores is assumed to be given by an assembly of point sinks/sources. As a test of the model, the evolution of a singly infinite periodic row of compressible pores is considered in detail. The effectiveness of the simple model is tested by comparison with a full numerical simulation. A novel boundary integral method based on Cauchy potentials and conformal mapping is used. In the case of pores with constant pressure, it is found that pores shrink faster than if in isolation. Compressible pores obeying the ideal gas law are also studied and are found to tend to a quasi-steady non-circular state. A higher-order model is also presented and compared with numerical simulations of the viscous sintering of a doubly periodic array of pores in Stokes flow.

Liquid of viscosity $\mu$ moves slowly through a cylindrical tube of radius $R$ under the action of a pressure gradient. An immiscible force-free drop having viscosity $\lambda\mu$ almost fills the tube; surface tension between the liquids is $\gamma$. The drop moves relative to the tube walls with steady velocity $U$, so that both the capillary number ${\hbox{\it Ca}}\,{=}\,\mu U/\gamma$ and the Reynolds number are small. A thin film of uniform thickness $\epsilon R$ is formed between the drop and the wall. It is shown that Bretherton's (1961) scaling $\epsilon\propto{\hbox{\it Ca}}^{{2}/{3}}$ is appropriate for all values of $\lambda$, but with a coefficient of order unity that depends weakly on both $\lambda$ and ${\hbox{\it Ca}}$. The coefficient is determined using lubrication theory for the thin film coupled to a novel two-dimensional boundary-integral representation of the internal flow. It is found that as $\lambda$ increases from zero, the film thickness increases by a factor $4^{{2}/{3}}$ to a plateau value when ${\hbox{\it Ca}}^{-{1}/{3}}\,{\ll}\,\lambda\,{\ll}\,{\hbox{\it Ca}}^{-{2}/{3}}$ and then falls by a factor $2^{{2}/{3}}$ as $\lambda\,{\rightarrow}\,\infty$. The multi-region asymptotic structure of the flow is also discussed.

Controlled charged capillary jet breakup of conducting liquids with different viscosities and applied voltages are experimentally analysed in this work. Careful measurements of droplets size and charge transported by the main and satellite droplets have been carried out. The experimental results are compared with those obtained by an augmented one-dimensional Lee model (López-Herrera et al. (1999). Theory and experiments show a remarkable agreement, which validates the rather inexpensive one-dimensional models as suitable predicting tools for scientific and engineering applications from electrospraying to charged jet printing over other sophisticated, more expensive three-dimensional models. Our results show that satellite droplets tend to undergo Coulombic rupture even in the case of very moderate electrification levels when the Ohnesorge number is sufficiently large.

The nonlinear dynamics of the flow in a short annulus driven by the rotation of the inner cylinder and bottom endwall is considered. The shortness of the annulus enhances the role of mode competition. For aspect ratios greater than about 3, the flow dynamics are dominated by a centrifugal instability as the rotating inner cylinder imparts angular momentum to the adjacent fluid, resulting in a three-cell state; the cells are analoguous to Taylor–Couette vortices. For aspect ratios less than about 2.8, the dynamics are dominated by the boundary layer on the bottom rotating endwall that is turned by the stationary outer cylinder to produce an internal shear layer that is azimuthally unstable via Hopf bifurcations. For intermediate aspect ratios, the competition between these instability mechanisms leads to very complicated dynamics, including homoclinic and heteroclinic phenomena. The dynamics are organized by a codimension-two fold-Hopf bifurcation, where modes due to both instability mechanisms bifurcate simultaneously. The dynamics are explored using a three-dimensional Navier–Stokes solver, which is also implemented in a number of invariant subspaces in order to follow some unstable solution branches and obtain a fairly complete bifurcation diagram of the mode competitions.

Baroclinic evolution of coastal currents associated with a surface front over variable topography is investigated numerically, employing a two-layer frontal-geostrophic model. In the frontal-geostrophic dynamical limit the frontal layer velocity is geostrophic to leading order; however the advective terms from the momentum equation are accounted for in the leading-order balance, and the mass conservation equation includes both the temporal and advective contributions. In this study, special focus is placed on the role of topography and the development of coherent vortex features. Perturbed axisymmetric currents in an annular domain are seen to develop typical breaking-wave instabilities at the outside edge. In general, the topography is found to be a destabilizing influence; however very steep topography is shown to inhibit cross-front motions, and thus delay subsequent vortex shedding. In simulations involving a continuous source of buoyant fluid, the ambient layer exhibits a marked increase in anticyclonic vorticity, which prevents eddy pinch-off. The results are compared with previous laboratory experiments involving surface currents, and implications for the stability of fronts at a shelf break are discussed.