Potential flows of incompressible fluids admit a pressure (Bernoulli) equation when the divergence of the stress is a gradient as in inviscid fluids, viscous fluids, linear viscoelastic fluids and second-order fluids. We show that in potential flow without boundary layers the equation balancing drag and acceleration is the same for all these fluids, independent of the viscosity or any viscoelastic parameter, and that the drag is zero when the flow is steady. But, if the potential flow is viewed as an approximation to the actual flow field, the unsteady drag on bubbles in a viscous (and possibly in a viscoelastic) fluid may be approximated by evaluating the dissipation integral of the approximating potential flow because the neglected dissipation in the vorticity layer at the traction-free boundary of the bubble gets smaller as the Reynolds number is increased. Using the potential flow approximation, the actual drag D on a spherical gas bubble of radius a rising with velocity U(t) in a linear viscoelastic liquid of density ρ and shear modules G(s) is estimated to be \[D = \frac{2}{3}\pi a^3 \rho {\dot U} + 12\pi a \int_{-\infty}^t G(t - \tau)U(\tau){\rm d}\tau\] and, in a second-order fluid, \[D = \pi a\left(\frac{2}{3}a^2 \rho + 12\alpha _1\right ) {\dot U} + 12\pi a\mu U,\] where α1, < 0 is the coefficient of the first normal stress and μ is the viscosity of the fluid. Because α1 is negative, we see from this formula that the unsteady normal stresses oppose inertia; that is, oppose the acceleration reaction. When U(t) is slowly varying, the two formulae coincide. For steady flow, we obtain the approximate drag D = 12πaμU for both viscous and viscoelastic fluids. In the case where the dynamic contribution of the interior flow of the bubble cannot be ignored as in the case of liquid bubbles, the dissipation method gives an estimation of the rate of total kinetic energy of the flows instead of the drag. When the dynamic effect of the interior flow is negligible but the density is important, this formula for the rate of total kinetic energy leads to D = (ρa – ρ) VBg · ex – ρaVB U where ρa is the density of the fluid (or air) inside the bubble and VB is the volume of the bubble.

Classical theorems of vorticity for potential flow of ideal fluids hold equally for second-order fluid. The drag and lift on two-dimensional bodies of arbitrary cross-section in a potential flow of second-order and linear viscoelastic fluids are the same as in potential flow of an inviscid fluid but the moment M in a linear viscoelastic fluid is given by \[M = M_I + 2 \int_{-\infty}^t [G(t - \tau)\Gamma (\tau)]{\rm d}\tau,\] where MI is the inviscid moment and Γ(t) is the circulation, and \[M = M_I + 2 \mu \Gamma + 2\alpha _1 \partial \Gamma /\partial t\] in a second-order fluid. When Γ(t) is slowly varying, the two formulae for M coincide. For steady flow, they reduce to \[M = M_I + 2 \mu \Gamma ,\] which is also the expression for M in both steady and unsteady potential flow of a viscous fluid. Moreover, when there is no stream, this moment reduces to the actual moment M = 2μΓ on a rotating rod.

Potential flows of models of a viscoelastic fluid like Maxwell's are studied. These models do not admit potential flows unless the curl of the divergence of the extra stress vanishes. This leads to an over-determined system of equations for the components of the stress. Special potential flow solutions like uniform flow and simple extension satisfy these extra conditions automatically but other special solutions like the potential vortex can satisfy the equations for some models and not for others.

We examine the stability characteristics of a two-dimensional flow which consists initially of an inflexionally unstable shear layer on an f-plane. Under the action of the primary instability, the vorticity in the shear-layer initially coalesces into two Kelvin–Helmholtz vortices which subsequently merge to form a single coherent vortex. At a sequence of times during this process, we test the stability of the two-dimensional flow to fully three-dimensional perturbations. A somewhat novel approach is developed which removes inconsistencies in the secondary stability analyses which might otherwise arise owing to the time-dependence of the two-dimensional flow.

In the non-rotating case, and before the onset of pairing, we obtain a spectrum of unstable longitudinal modes which is similar to that obtained previously by Pierrehumbert & Widnall (1982) for the Stuart vortex, and by Klaassen & Peltier (1985, 1989, 1991) for more realistic flows. In addition, we demonstrate the existence of a new sequence of three-dimensional subharmonic (and therefore ‘helical’) instabilities. After pairing is complete, the secondary instability spectrum is essentially unaltered except for a doubling of length- and timescales that is consistent with the notion of spatial and temporal self-similarity. Once pairing begins, the spectrum quickly becomes dominated by the unstable modes of the emerging subharmonic Kelvin–Helmholtz vortex, and is therefore similar to that which is characteristic of the post-pairing regime. Also in the context of non-rotating flow, we demonstrate that the direct transfer of energy into the dissipative subrange via secondary instability is possible only if the background flow is stationary, since even slow time-dependence acts to decorrelate small-scale modes and thereby to impose a short-wave cutoff on the spectrum.

The stability of the merged vortex state is assessed for various values of the planetary vorticity f. Slow rotation may either stabilize or destabilize the columnar vortices, depending upon the sign of f, while fast rotation of either sign tends to be stabilizing. When f has opposite sign to the relative vorticity of the two-dimensional basic state, the flow becomes unstable to new mode of instability that has not been previously identified. Modes whose energy is concentrated in the vortex cores are shown to be associated, even at non-zero f, with Pierrehumbert's (1986) elliptical instability. Through detailed consideration of the vortex interaction mechanisms which drive instability, we are able to provide physical explanations for many aspects of the three-dimensionalization process.

The flow field of an airfoil oscillated periodically over a reduced frequency range, 0 ≤ k ≤ 1.6, is studied experimentally at chord Reynolds numbers of Rc = 22000 and 44000. For most of the data, the NACA0012 airfoil is pitched sinusoidally about one quarter chord between angles of attack α of 5° and 25°. The cyclic variation of the near wake flow field is documented through flow visualization and phase-averaged vorticity measurements. In addition to the familiar dynamic stall vortex (DSV), an intense vortex of opposite sign is observed to originate from the trailing edge just when the DSV is shed. The two together take the shape of the cross-section of a large ‘mushroom’ while being convected away from the airfoil. The phase delay in the shedding of the DSV with increasing k, as observed by previous researchers, is documented for the full range of k. It is observed that the sum of the absolute values of all vorticity convected into the wake over a cycle is nearly constant and is independent of the reduced frequency and amplitude of oscillation but dependent on the mean α. The time varying component of the lift is estimated in a novel way from the shed vorticity flux. The analytical foundation of the method and the various approximations are discussed. The estimated lift hysteresis loops are found to be in reasonable agreement with available data from the literature as well as with limited force balance measurements. Comparison of the lift hysteresis loops with the corresponding vorticity fields clearly shows that major features of the lift variation are directly linked to the evolution of the large-scale vortical structures and the phase delay phenomenon.

Mixing produced by Rayleigh–Taylor instability at the interface between two layers is the subject of a comparative study between laboratory and numerical experiments. The laboratory experiments consist of a layer of brine initially at rest on top of a layer of fresh water. When a horizontal barrier separating the two layers is removed, the ensuing motion and the mixing that is produced is studied by a number of diagnostic techniques. This configuration is modelled numerically using a three-dimensional code, which solves the Euler equations on a 1803 grid. A comparison of the numerical results and the experimental results is carried out with the aim of making a careful assessment of the ability of the code to reproduce the experiments. In particular, it is found that the motions are quite sensitive to the presence of large scales produced when the barrier is removed, but the amount and form of the mixing is not very sensitive to the initial conditions. The implications of this comparison for improvements in the experimental and numerical techniques are discussed.

The spin-up from rest of a homogeneous free-surface fluid contained in a rectangular tank with an inclined bottom has been studied in the laboratory. As in the case of a tank without bottom topography, it is found that in the spin-up process leading to the ultimate state of rigid-body rotation a number of stages can be distinguished, these being (i) the starting flow, characterized by zero absolute vorticity, (ii) the viscous generation of cyclonic vorticity at the lateral tank walls, leading to flow separation, and (iii) the formation of cyclonic and anticyclonic flow cells, which show a complicated interaction. When the topography steepness is small, these cells become organized in a regular array similar to what is observed in the non-sloping bottom case. For steeper topography, however, no organization into a regular cellular pattern is observed, and the relative fluid motion remains unsteady and irregular until eventually it has decayed owing to the spin-ip/spin-down mechanism provided by the Ekman layer at the tank bottom. During the first stage of the adjustment process the starting flow takes on the appearance of a large anticyclonic cell that fills the fluid domain entirely. Depending on the ratio of the horizontal and vertical lengthscales of the tank this cell is either symmetric or asymmetric, with a higher density of streamlines in the deeper part of the tank. The coupled vorticity equation, governing the depth-independent part of the starting flow, and the potential equation describing its depth-dependent part have been solved analytically, and the comparison between these results and observational data is generally good.

A dilute dispersion containing small, force-free drops of one fluid dispersed in a second, immiscible in a linear flow field is considered for small Reynolds numbers and large Péclet numbers under isothermal conditions. The emphasis of our analysis is on the effects of pairwise drop interactions on their collision rate, as described by the collision efficiency, using a trajectory analysis. Simple shear flow and uniaxial extensional or compressional flow are considered. For both flows, the collision efficiency decreases with increasing drop viscosity due to the effects of hydrodynamic interactions. It also decreases as the ratio of the smaller drop radius to the larger radius decreases. For uniaxial flow, finite collision rates are predicted in the absence of interdroplet forces for all finite values of the drop size ratio and the ratio of the viscosities of the drop and suspending medium. In contrast, several kinds of relative trajectories exist for a pair of drops in simple shear flow, including open trajectories, collision trajectories, and closed and semi-closed trajectories, in the absence of interdroplet forces. When the ratio of small to large drop diameters is smaller than a critical value, which increases with increasing drop viscosity, all of the relative trajectories that start with the two drops far apart remain open (no collisions), unless in the presence of attractive forces. Attractive van der Walls forces are shown to increase the collision rates.

A Lagrangian framework is used for analysing reactive solute transport by a steady random velocity field, which is associated with flow through a heterogeneous porous formation. The reaction considered is kinetically controlled sorption–desorption. Transport is quantified by the expected values of spatial and temporal moments that are derived as functions of the non-reactive moments and a distribution function which characterizes sorption kinetics. Thus the results of this study generalize the previously obtained results for transport of non-reactive solutes in heterogeneous formations (Dagan 1984; Dagan et al. 1992). The results are illustrated for first-order linear sorption reactions. The general effect of sorption is to retard the solute movement. For short time, the transport process coincides with a non-reactive case, whereas for large time sorption is in equilibrium and solute is simply retarded by a factor R = 1+Kd, where Kd is the partitioning coefficient. Within these limits, the interaction between the heterogeniety and kinetics yields characteristic nonlinearities in the first three spatial moments. Asymmetry in the spatial solute distribution is a typical kinetic effect. Critical parameters that control sorptive transport asymptotically are the ratio εr between a typical reaction length and the longitudinal effective (non-reactive) dispersivity, and Kd. The asymptotic effective dispersivity for equilibrium conditions is derived as a function of parameters εr and Kd. A qualitative agreement with field data is illustrated for the zero- and first-order spatial moments.

We present the results of an experimental and numerical study of the spin-up from rest to solid-body rotation of a fluid-filled torus. In separate experiments, the rotation rate of the container is suddenly increased to a fixed value and the final rotation rate is used to define a non-dimensionalized control parameter, C. At low values of C, the observed flows during the transient phase are axisymmetric and spin-up is achieved through viscous diffusion. This in turn is followed by significant secondary flow and the appearance of ‘fronts’ as C is increased. During the transient phase the fluid motion near the inner wall of the container is dynamically unstable according to Rayleigh's criterion. Thus at higher values of C wave-like structures break the axisymmetry, non-uniqueness in the details of the process is found and finally, an innear wall instability is observed directly. A plot of the spin-up time versus C shows breaks in the slope at transition points between each of the above dynamical regimes but the overall trend is found to be insensitive to the details of the fluid motion. Further elucidation of the dynamical processes is provided by a novel variant of the now standard phase-space reconstruction techniques. The results show a systematic splitting of the phase paths as C is increased.

Finally, in the complementary numerical study, the time-dependent Navier–Stokes equations are solved for axisymmetric flows. Here, the flow is computed using a velocity–streamfunction–vorticity formulation in a two-dimensional plane with a velocity component normal to this plane. The quantitative and qualitative agreement between the numerical and experimental results is excellent for moderate values of the dynamical control parameter C.

The natural frequencies and stability criterion for cylinderical inviscid conducting liquid bridges and jets subjected to axial alternating magnetic fields in the absence of gravity are obtained. For typical conducting materials a frequency greater than 100 Hz is enough for a quasi-steady approximation to be valid. On the other hand, for frequencies greater than 105 Hz an inviscid model may not be justified owing to competition between viscous and magnetic forces in the vicinity of the free surface. The stability is governed by two independent parameters. One is the magnetic Bond number, which measures the relative influence of magnetic and capillary forces, and the other is the relative penetration length, which is given by the ratio of the penetration length of the magnetic field to the radius. The magnetic Bond number is proportional to the squared amplitude of the magnetic field and inversely proportional to the surface tension. The relative penetration length is inversely proportional to square root of the product of the frequency of the applied field and the electrical conductivity of the liquid. It is shown in this work that stability is enhanced by either increasing the magnetic Bond number or decreasing the relative penetration length.

The flow field generated by axial oscillations of a finite-length cylinder in an incompressible viscous fluid is described by the unsteady Stokes equations and computed with a first-kind boundary-integral formulation. Numerical calculations were conducted for particle oscillation periods comparable with the viscous relaxation time and the results are contrasted to those for an oscillating sphere and spheroid. For high-frequency oscillations, a two-term boundary-layer solution is formulated that involves two, sequentially solved, second-kind integral equations. Good agreement is obtained between the boundary-layer solution and fully numerical calculations at moderate oscillation frequencies. The flow field and traction on the cylinder surface display several features that are qualitatively distinct from those found for smooth particles. At the edges, where the base joins the side of the cylinder, the traction on the cylinder surface exhibits a singular behaviour, characteristic of steady two-dimensional viscous flow. The singular traction is manifested by a sharply varying pressure profile in a near-field region. Instantaneous streamline patterns show the formation of three viscous eddies during the decelerating portion of the oscillation cycle that are attached to the side and bases of the cylinder. As deceleration proceeds, the eddies grow, coalesce at the edges of the particle, and thus form a single eddy that encloses the entire particle. Subsequent instantaneous streamline patterns for the remainder of the oscillation cycle are insensitive to particle geometry: the eddy diffuses outwards and vanishes upon particle reversal; a simple streaming flow pattern occurs during particle acceleration. The evolution of the viscous eddies is most apparent at moderate oscillation frequencies. Qualitative results are obtained for the oscillatory flow field past an arbitrary particle. For moderate oscillation frequencies, pathlines are elliptical orbits that are insensitive to particle geometry; pathlines reduce to streamline segments in constant-phase regions close to and far from the particle surface.

Experiments on the stability of vortex pairs are described. The vortices (ratio of length to core diameter L/c of up to 300) were generated at the edge of a flat plate rotating about a horizontal axis in water. The vortex pairs were found to be unstable, displaying two distinct modes of instability. For the first time, as far as it is known to the authors, a long-wave as well as a short-wave mode of instability were observed to develop simultaneously on such a vortex pair. Experiments involving single vortices show that these do not develop any instability whatsoever. The wavelengths of the developing instability modes on the investigated vortex pairs are compared to theoretical predictions. Observed long wavelengths are in good agreement with the classic symmetric long-wave bending mode identified by Crow (1970). The developing short waves, on the other hand, appear to be less accurately described by the theoretical results predicted, for example, by Windnall, Bliss & Tsai (1974).

Two-dimensional space-periodic cabillary–gravity waves at the interface between two fluids of different densities are considered when the second harmonic and the fundamental mode are near resonance. A weakly nonlinear analysis provides the equations (normal form), correct to third order, that relate the wave frequency with the amplitudes of the fundamental mode and of the second harmonic for all waves with small energy. A study of the normal form for waves which are also periodic in time reveals three possible types of space- and time-periodic waves: the well-known travelling and standing waves as well as an unusual class of three-mode mixed waves. Mixed waves are found to provide a connection between standing and travelling waves. The branching behaviour of all types of waves is shown to depend strongly on the density ratio. For travelling waves the weakly nonlinear results are confirmed numerically and extended to finite-amplitude waves. When slow modulations in time of the amplitudes are considered, a powerful geometrical method is used to study the resulting normal form. Finally a discussion on modulational stability suggests that increasing the density ratio has a stabilizing effect.

A series of experimental observations is presented of a flow in which inertial oscillations are excited. The homogeneous fluid is contained in a completely filled right circular cylinder. The cylinder is spun about its axis of symmetry and a small ‘forced precession’ (or coning motion) is impulsively started. The flow is visualized by an electrolytic dyeline method. The mathematical problem for linear inviscid inertial oscillations in this system, although ill-posed in general, admits a solution in terms of wave modes for the specific boundary conditions considered here. The experiments show that while this linear inviscid theory provides some facility for predicting the flow structure at early times, the flow rapidly and irreversibly distorts away from the predicted form. This behaviour is seen as a precursor to some of the more dramatic breakdowns described by previous authors, and it may be pertinent to an understanding of the breakdowns reported in experiments on elliptical flow instabilities.