This paper focuses on the characteristics of the regime C flow (Tatsuno & Bearman, J. Fluid Mech., vol. 211, 1990, pp. 157–182) around an oscillating circular cylinder in still water. The regime C flow is characterised by the formation of large-scale vortex cores arranged as opposed von Kármán vortex streets, resulting from a regular switching of vortex shedding directions with respect to the axis of oscillation. Both Floquet analysis and direct numerical simulations (DNS) are performed to investigate the two- (2-D) and three-dimensional (3-D) instabilities. The present study reveals that the low-wavenumber 3-D instability can emerge slightly before the 2-D instability in regime C. In total, five spanwise vortex modes were identified: (i) standing-wave pattern, S-mode; (ii) travelling-wave pattern, T-mode; (iii) mixed ST-mode; (iv) X-type vortex pattern, X-mode; and (v) U-type vortex pattern, U-mode. The modal analysis conducted in this study demonstrates that the vortex patterns and the corresponding spatial and temporal modulations of the dynamic loads of the S-, T- and mixed ST-modes are mainly induced by the 3-D instability of a single wavenumber. The characteristics of the X-mode are due to the superposition of the 3-D instabilities of multiple wavenumbers. The U-mode is dominated by a 2-D instability and its interaction with 3-D instabilities. The domain size dependence study demonstrates that the regime C flow is very sensitive to the spanwise length of the computational domain. The subcritical nature of the regime C flow is responsible for the discrepancy in the marginal stability curves obtained by independent Floquet stability analysis, DNS and physical experiments.

Batchelor made his name with research in turbulence in the 1940s and 1950s. He became disillusioned with turbulence at the Marseille meeting in 1961. At the end of the 1960s, he started his second wave of research on low-Reynolds-number suspensions of particles. Ten years after he died, I will describe his key results, what was before and what followed. Eight of his 10 most cited papers are in micro-hydrodynamics.

The rupture of soap films is traditionally described by a law accounting for a balance between momentum and surface tension forces, derived independently by Taylor and Culick in the 1960s. This law is highly relevant to the dynamics of thin liquid films of jets when viscous effects are negligible. However, the minute amounts of surfactant molecules present in soap films play a major role in interfacial rheology, and may result in complex behaviour. Petit et al. (J. Fluid Mech., vol. 774, 2015, R3) challenge standard thin film dynamics via intriguing experiments conducted with highly elastic surfactants. Unexpected structures reminiscent of faults are observed.

The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the bifurcation, but wall shear on the curved wall is found to increase from its undisturbed value downstream.

The non-classical gas dynamics of binary mixtures of organic fluids in the vapour phase is investigated for the first time. A predictive thermodynamic model is used to compute the relevant mixture properties, including its critical point coordinates and the local value of the fundamental derivative of gas dynamics $\Gamma $. The considered model is the improved Peng–Robinson Stryjek–Vera cubic equation of state, complemented by the Wong–Sandler mixing rules. A finite thermodynamic region is found where the nonlinearity parameter $\Gamma $ is negative and therefore non-classical gas dynamics phenomena are admissible. A non-monotone dependence of $\Gamma $ on the mixture composition is observed in the case of binary mixtures of siloxane and perfluorocarbon fluids, with the minimum value of $\Gamma $ in the mixture being always larger than that of its more complex component. The observed dependence indicates that non-ideal mixing has a strong influence on the gas dynamics behaviour, either classical or non-classical, of the mixture. Numerical experiments of the supersonic expansion of a mixture flow around a sharp corner show the transition from the classical configuration, exhibiting an isentropic rarefaction fan centred at the expansion corner, to non-classical ones, including mixed expansion waves and rarefaction shock waves, if the mixture composition is changed.

An analysis is made of the effect of a transverse gravity field on a two-dimensional fully cavitating flow past a flat-plate hydrofoil. Under the assumption that the flow is both irrotational and incompressible, a non-linear method is developed by using conformal mapping and the solution to a mixed-boundary-value problem in an auxiliary half plane. A new cavity model, proposed by Tulin (1964a), is employed. The solution to the gravity-affected case was found by iteration; the non-gravity solution was used as the initial trial of a rapidly convergent process. The theory indicates that the lift and cavity size are reduced by the gravity field. Typical results are presented and compared to Parkin's (1957) linear theory.

We study impacts of a rigid body on a thin elastic sheet floating on a liquid. When struck by a solid object of small size, the elastic sheet deforms and waves propagate in and on the membrane. The impact triggers a longitudinal elastic wave effectively stretching the membrane. The hydroelastic transverse wave that propagates in the stretched domain is similar to capillary waves on a free surface with an equivalent ‘surface tension’ that results from the stretching of the elastic membrane. Two limiting cases, for which a self-similar solution can be computed, corresponding to short and long times are identified. Surprisingly, our study reveals that the fluid–body system behaves as a regular liquid–gas interface, but with an effective surface tension coefficient that scales linearly with the impact velocity.

A micro-mechanical theory is proposed for the prediction of macro-scale properties of flow and dispersion in a current through a periodic array of vertical cylinders standing on a horizontal bed. A two-scale analysis reduces the numerical task to the solution of two canonical boundary value problems in a unit cell. Using measured data on the drag coefficient measured for an array in open channels, the eddy viscosity in the interstitial flow on the micro-scale is calculated for a wide range of Reynolds numbers. The macro-scale relation between the mean velocity and the surface gradient is found in the form of a nonlinear Darcy’s law. The interstitial velocity is then used to derive the macro-scale convection diffusion equation for the solute concentration, also by a two-scale analysis. The Taylor dispersivity and the total effective diffusivity are computed for a wide range of flow rates and solid fractions. Features specific to the periodic geometry are pointed out.

Natural modes of oscillation of a vanishingly thin spherical rotating fluid shell, with frequencies σ less than twice the angular velocity Ω, were found by Haurwitz (1940). Their validity is, however, put in question by the presence of a singularity at critical co-latitudes θc: 2Ω cosθc = σ in the O(ε) term of an expansion in the relative shell thickness ε (Stewartson & Rickard 1969). The problem is investigated here by considering the evolution of flow from a specified initial distribution. The principal features are as follows:

(i) The O(1) natural modes of Haurwitz, decaying on a time scale Ω−1ε−2.

(ii) Corrections to (i), regular and of magnitude O(ε) except near critical latitudes.

(iii) Essentially transient inertial waves of magnitude O(ε).

(iv) Inertial waves of magnitude O(ε½) with the natural-mode frequencies σ and generated by (i) at critical latitudes.

On a time scale Ω−1ε−1, (iii) and (iv) develop vertical and horizontal length scales ε and propagate throughout the ocean. The continuing energy transfer from (i) to (iv), at a rate O(ε2Ω), appears to be the principal respect in which (i) and (ii) fail to constitute a conventional normal mode.

The transport of fluid from a recirculation region adjacent to a free surface is studied using a numerical method validated with experimental flow visualization. The flow is an example of a liquid film coating process, and consists of two counter-rotating rolls placed side-by-side and half-submerged in a bath of fluid. In the gap between the rolls a recirculation zone exists just below the free surface, around which the flow splits into two films. Fluid recirculating for long periods has been identified as a source of coating defects, so this paper considers a possible method of inducing stirring. The flow is modulated by driving one of the rolls through a Hooke's joint, which delivers a well-characterized periodic perturbation to the roll speed. In response to this speed modulation, the free surface undergoes a periodic change in position and shape which drives an exchange of fluid between the recirculation region and the surrounding flow. The amplitude of the free-surface motion is strongly dependent on modulation frequency.

The dynamics of the free surface preclude a quasi-steady approach, even in the small-frequency limit, and so a fully time-dependent analysis based on the finite element method is employed. Trigonometric temporal interpolation of the finite element data is used to make passive tracer advection calculations more efficient, and excellent agreement is seen between simulation and experiment. Computations of the stable and unstable invariant manifolds associated with periodic points on the free surface reveal that the exchange of fluid is governed by a self-intersecting turnstile mechanism, by which most fluid entrained during a modulation cycle is ejected later in the same cycle.

Transport over several cycles is explored by observation of the evacuation of passive tracers initially distributed uniformly in the recirculation zone. Results demonstrate the persistence of unmixed cores whose size is dependent on the modulation frequency. By considering the percentage of tracers remaining after a fixed number of cycles, contours in frequency–amplitude space show that for each modulation amplitude there is a frequency which produces the most effective transport, with up to 80 % of tracers removed by a modulation which produces only a 5 % change in film thickness. Finally it is shown how modulation of both rolls at slightly different phases can reduce the film thickness variation to about 1 % while maintaining the level of transport.

The effect of stretching on the three-dimensional stability of a viscous unsteady vortex is addressed. The basic flow, which satisfies the Navier–Stokes equations, is a vortex with axial flow subjected to a time-dependent strain field oriented along its axis. The linear equations for the three-dimensional perturbations of the stretched vortex are first reduced by using successive changes of variables to equations which are almost identical to those of the unstretched vortex but with time-dependent parameters. These equations are then numerically solved in the particular case of the Batchelor vortex with a strain field which first compresses then stretches the vortex. Through this simulation, it is qualitatively demonstrated how the simultaneous action of stretching and azimuthal vorticity may destabilize a vortex. It is also argued that it provides a possible mechanism for the vortex bursts observed in turbulence experiments.

A cryogenic shock tube has been developed as a tool for research in fluid mechanics and low-temperature physics. The shock tube was designed to operate with the test section immersed in a cryogenic liquid. A unique diaphragm-changing mechanism makes this shock tube an economical and practical device. There are several advantages in operating a shock tube at cryogenic temperatures. Shock waves of very large Mach number can be produced. The flow field can be accurately calculated using ideal-shock-tube/perfect-gas theory. Boundary-layer effects are decreased, so that long test times are possible.

The cases that were studied are test-gas temperatures of 300, 77, 4·2 and 2·3 °K. Helium was used as both test and driver gas. The largest Mach numbers observed range from 2·4 at 300 °K to 32 at 2·3 °K (several runs at 1·46 °K produced Mach 40 shocks). As the temperature of the test gas was decreased, the observed Mach numbers approached those calculated using the ‘shock-tube equation’.

As a laboratory tool, the cryogenic shock tube may be applied in many areas and modified for use in even more. Shock waves with large Mach numbers and flows with large Reynolds numbers can be produced with this device. The rapid increase in temperature and pressure across the shock wave is useful for studies of sublimation, evaporation or chemical reactions. Quantum-mechanical effects in cryogenic materials, superconductors or superfluid helium can also be investigated.

The initial-value problem for the linear, inviscid motion of a contained, rotating stratified fluid is considered in the limit of weak stratification, that is, for small values of the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The limiting flow is of interest because, although the initial-value problem has been studied, both for the case of a homogeneous, rotating fluid and for the case of a stratified, rotating fluid, the exact relationship of the two flows, in the limit of vanishing stratification, is not straightforward. For example, the method of determining, from the initial conditions, the steady geostrophic component of the flow of a rotating, stratified fluid does not in general give a motion that reduces, in the limit S → 0, to the steady component of the flow of a homogeneous fluid. By including a consideration of slow unsteady motions that vary on a time scale dependent on the stratification parameter, the relationship of the limiting flow to the flow of a homogeneous fluid is established.

An analytic solution is obtained for the head-on collision of strong shock waves with supersonically moving, axisymmetric, slender bodies. A new method of the solution is developed using integral transforms. Pressure distributions on the surface of a cone illustrate the effects of shock strength and body speed. The results are compared with those of Blankenship (1965), which were obtained by numerical methods.

It is also shown that the same analysis can, with certain modifications, be adapted for treating the shock-shock interaction on thin two-dimensional aerofoils of arbitrary shape at small incidence.