Novel experimental results are presented of the transient motion of a large spherical-cap bubble and of the displaced liquid motion in a closed cylindrical vertical tube of finite length. This is a fundamental fluid mechanics problem and has direct application to Space flight where liquids in fuel tanks are exposed to large changes in acceleration. The initial spherical cap shape is produced by a thin membrane which can be considered equivalent to a large surface tension. The bubble is released by puncturing the membrane which, subsequently, retracts in a time of order 1 ms. Apart from the generation of shear instabilities of very short wavelength, the membrane withdrawal has a negligible effect on the bubble and liquid motions and can thus be considered instantaneous. The displacements of the bubble, the annular liquid sheet and the geyser front were determined from high-speed video images from which velocities and accelerations were calculated. The results are compared with a theoretical model of the unsteady bubble and sheet velocities.

The unsteady nonlinear potential flow induced by a submerged line source or sink is studied by a vortex sheet method, both to trace the free surface evolution and to explore the possible existence of steady-state solutions. Only steady-state flows have been considered by other investigators, and these flows have been insensitive to whether they are generated by a source or sink, except with respect to the flow direction along the streamlines. The time-dependent solution permits an assessment of the stability of previously found steady solutions, and also reveals differences between source and sink flows: for the infinite-depth case, steady stagnation-point-type solutions are found both for source flows and sink flows, above the critical value reported by other investigators; finally, it is shown that streamline patterns of steady stagnation-point flows are identical for source and sink flows only in the limiting case of infinite depth.

Two-dimensional mixing driven by an instability mechanism which is concentrated near one of the boundaries is considered, with particular application to Langmuir circulations driven by a wave spectrum. The question of how to define the equivalent of the Rayleigh number is attacked using the energy balance equations and simple truncated models of the instability. Given a particular horizontal wavelength for the disturbance, the strength of the forcing on the cells, and thus the growth rate, is determined by a tradeoff between maximizing the depth-averaged forcing and maximizing the depth of penetration. As a result of this tradeoff, long-wavelength cells grow more slowly, but penetrate more deeply and have a larger equivalent Rayleigh number. At finite amplitude, these long-wavelength cells come to dominate the flow field. The depth of penetration of, and density transport accomplished by, Langmuir cells is considered as a function of the mean stratification and diffusion. An application to oceanic mixed layers is considered assuming the Mellor-Yamada 2½-level turbulence closure model to define the background level of turbulent mixing. For many realistic cases, Langmuir cells are predicted to dominate the vertical transport of momentum and density.

Prandtl–Meyer flows with heat addition from homogeneous condensation not exceeding a critical value (subcritical flows) are investigated by an asymptotic method in the double limit of a large nucleation time followed by a small droplet growth time. The physically distinct condensation zones, with detailed analytical structure, are displayed along streamlines and the flow field in each zone is determined utilizing the asymptotic solution of the rate equation along streamlines. In particular the nucleation wave front, which corresponds to states of maximum nucleation along streamlines, is accurately located independently of the particular condensation model employed. Results obtained using the classical nucleation equation together with the Hertz–Knudsen droplet growth law show, despite qualitative agreement, considerable differences between the nucleation wave fronts and measured onset conditions for the experiments of Smith (1971), because of intersecting characteristics in the heat addition zones. This shows the necessity of including an embedded oblique shock wave in the expansion fan of corner expansion flows for the cases investigated.

The stability of eddies with three-layer stratification is examined experimentally. When the difference in density between the upper two layers is much greater (or less) than that between the lower two layers baroclinic instability on two different lengthscales (the Rossby radii associated with the upper and the lower interfaces) is possible. The vortices are created using modifications of two techniques described by Griffiths & Linden (1981) in their study of two-layer eddies.

‘Constant-flux’ eddies are generated by the release of a constant flux of buoyant fluid from a small source positioned at the surface of a two-layer fluid. In a second variation of this experiment, the source is positioned at the interface between two layers and fluid of intermediate density is injected. As the horizontal lengthscale increases, the vortices evolve from a stable to an unstable state. It is showns that the size at which the vortices become unstable may be significantly altered by the presence of a second interface. The results agree qualitatively with the conclusions of a linear stability analysis of quasi-geostrophic three-layer flow in a channel (Smeed 1988), but it is necessary to examine the effects of horizontal shear and Ekman dissipation to explain the experimental results.

‘Constant-volume’ eddies are produced by the release of a volume of buoyant fluid, initially contained within a cylindrical barrier, at the surface of a two-layer fluid. After the barrier is removed, the buoyant fluid spreads a distance of the order of the Rossby radius. Similarly, vortices are created by releasing a volume of fluid of density intermediate between the initial two layers. Within a few rotation periods the vortices become unstable to disturbances similar to those observed in two-layer experiments. Qualitative agreement is found between the observed wavelength and the fastest growing mode predicted by the linear stability theory (Smeed 1988). When the disturbances reach large amplitude a change in lengthscale is often observed.

A more rigorous proof is given of the validity of a generalized Prandtl—Glauert technique for caculating the solution to time-dependent small perturbation flows. The method, first used by Miles (1950a) for the airfoil problem and later applied by Amiet & Sears (1970) to more general problems, allows calculation of the term of first order in frequency. Anomalous behaviour for the two-dimensional problem is examined in detail and found to be limited to those two-dimensional cases which include shed vorticity downstream of the body. This anomaly, which precludes using the method for these cases, results from the need to satisfy a velocity boundary condition on the body. For this purpose the velocity must be calculated from the basic variable, the pressure, through an integrated form of the momentum equation. It is in thus calculating the velocity that the anomaly occurs. The method can be applied to both the two-dimensional case without shed vorticity and the general three-dimensional case.

The jet-stream in a rotating fluid is treated as a thermal boundary layer, but viscous effects are omitted from the first approximation. A theoretical justification for this treatment is presented, and a particular solution of the resulting equations is found. This solution is shown to give a reasonable picture of the flow in the neighbourhood of the stream far from solid boundaries.

A fuel cell is an electrochemical device that converts chemical energy into electrical energy. The essential difference between a fuel cell and a battery is that the fuel and oxidant are continually supplied to a fuel cell whereas a battery has a finite amount of fuel stored within it which eventually becomes exhausted, after which the battery has to be recharged or replaced. We consider here a tubular cell which is run on methane. A mathematical model for the reaction and transport processes in the cell is presented. This takes into account diffusion, advection and reaction as well as the electrochemical flow of oxygen. When the device is running steadily, we perform an analysis, using the method of matched asymptotic expansions to find the flow field, the mass fractions of each chemical species and the electrical power output. These results are confirmed by a numerical technique and compared with experimental results.

The values reported by Willmarth & Wooldridge (1962) for the root-mean-square wall pressure and the power spectra of the wall pressure are in error. G. M. Corcos and M. K. Bull pointed out to me, in April 1963, that the area beneath the power spectra of figure 4 does not agree with the values of $\sqrt {(p^2)}| q \infty$ and $\sqrt {(p^2)}| \tau_w$ reported in table 3 and in §4·1. The measurements for the wall pressure at high speed, U∞ = 206ft./sec, were fortunately recorded on magnetic tape. It was therefore possible to check the measurements for that case.

A generalized theoretical analysis and finite-difference solutions of the Navier-Stokes equations of the initial-value problem are applied to obtain the linear internal wave fields generated by a density perturbation and two rotational velocity perturbations in an inviscid linearly stratified fluid. The velocity perturbations are those due to an axisymmetric swirl and a vortex pair. Solutions obtained correspond to the strong stratification limit.

The theoretical results of the rotational perturbation cases show an oscillating non-propagating disturbance, which is absent in the density-perturbation case. The swirl-flow solution shows an oscillatory behaviour in both the angular momentum deposited in the fluid and in the torque exerted by the external gravitational force field. The vortex-flow solution shows a vertical ray pattern.

The equi-partitioning of energy is reached at about 0.4 of a Brunt-Väisälä (B.V.) period. The potential energy-kinetic energy conversion, or vice versa, takes place between 0.15 and 0.3 B.V. periods.

Considering the miniaturization trend in technical applications, the need of a slender nozzle theory for such conventional, that is ideal-gas-like, fluids, which accounts for a strong boundary-layer interaction with the core region, arises in quite a natural way as the dimensions of the flow device are successively reduced. Moreover, a number of modern technological processes (e.g. organic Rankine cycles) involve fluids with high molecular complexity, some of which are expected to exhibit embedded regions with negative values of the fundamental derivative in the vapour phase commonly termed Bethe–Zel'dovich–Thompson (BZT) fluids. Linked to it, unconventional Laval nozzle geometries are needed to transform subsonic to supersonic internal flows. In the present paper, the transonic flows through nozzles of short length scales located in a channel of constant cross-section so slender that the flow in the inviscid core region is one-dimensional are considered. Rapid streamwise changes of the flow field caused by the nozzle then lead to a local breakdown of the classical hierarchical boundary-layer approach, which is overcome by the triple-deck concept. Consequently, the properties of the inviscid core and the near-wall (laminar) boundary-layer regions have to be calculated simultaneously. The resulting problem is formulated for both regular (ideal-gas-like) fluids and dense gases. Differences and similarities of the resulting flow pattern with respect to the well-known classical Laval nozzle flow are presented for perfect gases, and the regularizing influence of viscous–inviscid interactions, is examined. Furthermore, the analogous problem is considered for BZT fluids in detail as well. The results indicate that the passage through the sonic point in the inviscid core is strongly affected by the combined influence of nozzle geometry and boundary-layer displacement effects suggesting in turn an inverse Laval nozzle design in order to obtain the desired flow behaviour.

The steady, two-dimensional, isothermal flow of an incompressible Newtonian fluid in a semi-infinite channel is modelled using a finite-element method. The flow is driven by injecting two identical jets through symmetrically placed slit-like nozzles into the otherwise closed end of the channel. Multiple steady-state solutions are observed for Reynolds numbers greater than 18.8, where seven solutions have been found. Six of these solutions exist on branches that are not connected to the Stokes flow solution via continuation in the Reynolds number. Further bifurcations of these solutions has led to the discovery of 17 solutions at a Reynolds number of 40. A two-dimensional linear stability analysis of the solution branches shows that for Reynolds numbers in the range of 18.8 to 26.8 there are three stable solutions. One solution is symmetric about the channel centreline while the other two stable solutions are a pair of mirror-image asymmetric flows. For Reynolds numbers in the range 26.8 to 40, there are four known stable solutions consisting of two asymmetric solutions and their mirror-images.

The assumption of low Reynolds number flow, or Stokes flow, is often applied to the understanding of a broad range of microfluidic devices, including micro-reactors, biomedical devices, and membraneless electrochemical cells. However, recent studies have shown that various inertial effects can play a significant role, even in microfluidic systems. In this work, two- and three-dimensional secondary flows are identified in a generic rectangular flow channel design consisting of a secondary channel feeding fluid into a main channel. We identify a scaling argument which is able to predict the occurrence of these secondary flows as a function of system parameters. The impact of these behaviours on the assumption of fully developed colaminar flow is investigated. This work considers a representative geometry, and identifies a set of conditions where inertial effects can play a key role in a microfluidic device.

We investigate the dewetting of a droplet on a smooth horizontal solid surface for different slip lengths and equilibrium contact angles. Specifically, we solve for the axisymmetric Stokes flow using the boundary element method with (i) the Navier-slip boundary condition at the solid/liquid boundary and (ii) a time-independent equilibrium contact angle at the contact line. When decreasing the rescaled slip length $\tilde{b}$ with respect to the initial central height of the droplet, the typical non-sphericity of a droplet first increases, reaches a maximum at a characteristic rescaled slip length $\tilde{b}_{m}\approx O(0.1{-}1)$ and then decreases. Regarding different equilibrium contact angles, two universal rescalings are proposed to describe the behaviour of the non-sphericity for rescaled slip lengths larger or smaller than $\tilde{b}_{m}$. Around $\tilde{b}_{m}$, the early time evolution of the profiles at the rim can be described by similarity solutions. The results are explained in terms of the structure of the flow field governed by different dissipation channels: elongational flows for $\tilde{b}\gg \tilde{b}_{m}$, friction at the substrate for $\tilde{b}\approx \tilde{b}_{m}$ and shear flows for $\tilde{b}\ll \tilde{b}_{m}$. Following the changes between these dominant dissipation mechanisms, our study indicates a crossover to the quasistatic regime when $\tilde{b}$ is many orders of magnitude smaller than $\tilde{b}_{m}$.

The outer cylinder was kept stationary whilst the inner rotated. Weak circulatory motion (‘embryo cells’) was observed in all the fluids investigated at Taylor numbers Ta less than the critical Taylor number Tac. These cells grew with increasing Ta, then burst into vigorous Taylor vortices at Ta = Tac. In solutions of rigid elongated molecules having concentrations above a minimum concentration, an axial oscillation (overstability?) of the embryo cells was observed. The oscillation did not occur in solutions of flexible folded molecules nor in the solvent. The oscillation was observed over a range Ta0 ≤ Ta ≤ Tac. For concentrations above the minimum Ta0/Tac decreased with increasing concentration.

Other phenomena of the flow were also studied. It was found for solvent and solutions that the torque on the outer cylinder increased relatively less rapidly with the speed of rotation of the inner cylinder when \[ Ta/Ta_c > 1.08 \] than it did when \[ 1 < Ta/Ta_c\leqslant 1.08. \] In this latter range there was a torque reduction in the flow of the solutions relative to that in the flow of solvent. At sufficiently high speeds of rotation circumferential waves occurred in the fully developed Taylor cells (at Ta = Taw) and it was found that when R1/R2 = 0·95 then Taw/Tac = 1·5 for the solvent, whereas for the solutions Taw/Tac ≥ 1·5. When R1/R2 = 0·90 then Taw/Tac = 1·25, and for the solutions Taw/Tac > 1·25. No experiment was done at Ta > Taw. The normal-stress coefficients of the solution have been evaluated from the results.

A set of partial differential equations of the Navier-Stokes type is derived for external rarefied gas flows at all Knudsen numbers. Only the expressions of the stress tensor and heat flux vector are different from the customary Navier-Stokes relations and Fourier law. The new expressions are calculated from an approximate distribution function constructed through analysis of the BGK model of the Boltzmann equation so that it retains the qualitative property of the collision term and reproduces accurately the two extreme continuum and free-molecule regimes. Explicit forms of the stress tensor and heat flux components are given for low-speed two-dimensional flows. Solutions for vanishing Mach number and compressibility effects are then discussed. For a nearly isothermal cylinder, the present formulation leads to only one governing equation of the Navier-Stokes type for all flow regimes.

A class of compressible laminar boundary-layer flows subject to adverse pressure gradients of different magnitude is studied using a finite-element–differential method in which the assumed solutions are represented by classical cubic spline functions. The numerical integration process for the reduced initial-value problem has been carried out directly to at least one integration step upstream of the separation point, and very accurate numerical results have been obtained for a large number of integration steps extremely close to separation. The skin-friction and heat-transfer coefficients for nearly zero-heat-transfer, cooled-wall and heated-wall cases, computed under the assumption of constant Prandtl number Pr = 1.0 as well as Pr = 0.72, have clearly exhibited the same distinctive behaviour near separation. It is deduced that Buckmaster's series expansions for the solution near separation, derived on the assumptions of cooled wall and Pr = 1.0, are valid for all the cases considered. By matching the numerical results with Buckmaster's expansions, accurate distributions of skin friction and heat transfer have been obtained up to the separation point. Moreover, the importance of Prandtl number on the solution is evidenced from the numerical results presented.

We present a new model for an axisymmetric vortex ring confined in a tube. The model takes into account the elliptical (elongated) shape of the vortex ring core and thus extends our previous model (Danaila et al. J. Fluid Mech., vol. 774, 2015, pp. 267–297) derived for vortex rings with quasi-circular cores. The new model offers a more accurate description of the deformation of the vortex ring core, induced by the lateral wall, and a better approximation of the translational velocity of the vortex ring, compared with the previous model. The main ingredients of the model are the following: the description of the vorticity distribution in the vortex ring is based on the previous model of unconfined elliptical-core vortex rings (Kaplanski et al. Phys. Fluids, vol. 24, 2012, 033101); Brasseur’s approach (Brasseur, NASA Tech. Rep. JIAA TR-26, 1979) is then applied to derive a wall-induced correction for the Stokes streamfunction of the confined vortex ring flow. We derive closed formulae for the flow streamfunction and vorticity distributions. An asymptotic expression for the long-time evolution of the drift velocity of the vortex ring as a function of the ellipticity parameter is also derived. The predictions of the model are shown to be in agreement with direct numerical simulations of confined vortex rings generated by a piston–cylinder mechanism. The predictions of the model support the recently suggested heuristic relation (Krieg & Mohseni Trans. ASME J. Fluids Engng, vol. 135, 2013, 124501) between the energy and circulation of vortex rings with converging radial velocity. A new procedure for fitting experimental and numerical data with the predictions of the model is described. This opens the way for applying the model to realistic confined vortex rings in various applications including those in internal combustion engines.

We consider the self-induced motions of three-dimensional oblate spheroids of density $\unicode[STIX]{x1D70C}_{s}$ with varying aspect ratios $AR=b/c\leqslant 1$, where $b$ and $c$ are the spheroids’ centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that $y_{s}(t)=A\sin (2\unicode[STIX]{x03C0}ft)$ in a fluid of density $\unicode[STIX]{x1D70C}$ and kinematic viscosity $\unicode[STIX]{x1D708}$. As in strictly two-dimensional flows, above a critical value $Re_{C}$ of the flapping Reynolds number $Re_{A}=2Afc/\unicode[STIX]{x1D708}$, the spheroid ultimately propels itself horizontally as a result of fluid–body interactions. For $Re_{A}$ sufficiently above $Re_{C}$, the spheroid rapidly settles into a terminal state of constant, unidirectional velocity, consistent with the prediction of Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107) that, at sufficiently high $Re_{A}$, such oscillating spheroids manifest $m=1$ asymmetric flow, with characteristic vortical structures conducive to providing unidirectional thrust if the spheroid is free to move horizontally. The speed $U$ of propagation increases linearly with the flapping frequency, resulting in a constant Strouhal number $St(AR)=2Af/U$, characterising the locomotive performance of the oblate spheroid, somewhat larger than the equivalent $St$ for two-dimensional spheroids, demonstrating that the three-dimensional flow is less efficient at driving locomotion. $St$ decreases with increasing aspect ratio for both two-dimensional and three-dimensional flows, although the relative disparity (and hence relative inefficiency of three-dimensional motion) decreases. For flows with $Re_{A}\gtrsim Re_{C}$, we observe two distinct types of inherently three-dimensional motion for different aspect ratios. The first, associated with a disk of aspect ratio $AR=0.1$ at $Re_{A}=45$, consists of a ‘stair-step’ trajectory. This trajectory can be understood through consideration of relatively high azimuthal wavenumber instabilities of interacting vortex rings, characterised by in-phase vortical structures above and below an oscillating spheroid, recently calculated using Floquet analysis by Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107). Such ‘in-phase’ instabilities arise in a relatively narrow band of $Re_{A}\gtrsim Re_{C}$, which band shifts to higher Reynolds number as the aspect ratio increases. (Indeed, for horizontally fixed spheroids with aspect ratio $AR=0.2$, Floquet analysis actually predicts stability at $Re_{A}=45$.) For such a spheroid ($AR=0.2$, $Re_{A}=45$, with sufficiently small mass ratio $m_{s}/m_{f}=\unicode[STIX]{x1D70C}_{s}V_{s}/(\unicode[STIX]{x1D70C}V_{s})$, where $V_{s}$ is the volume of the spheroid) which is free to move horizontally, the second type of three-dimensional motion is observed, initially taking the form of a ‘snaking’ trajectory with long quasi-periodic sweeping oscillations before locking into an approximately elliptical ‘orbit’, apparently manifesting a three-dimensional generalisation of the $QP_{H}$ quasi-periodic symmetry breaking discussed for sufficiently high aspect ratio two-dimensional elliptical foils in Deng & Caulfield (J. Fluid Mech., vol. 787, 2016, pp. 16–49).