When, for an otherwise unbounded fluid, the unique irrotational flow compatible with the instantaneous motion of an immersed body has been calculated, it is straightforward to deduce the pressure field from the unsteady form of Bernoulli's equation if the body is rigid. On the other hand, if the body is flexible, a somewhat subtle analysis is required to determine the time derivative of velocity potential, ∂ϕ/∂t, which occurs in that equation. This is because no simple relationship exists between the instantaneous form of ϕ and its form at a nearby instant.

In the case of two-dimensional flow, however, the two forms of ϕ for a flexible body may be related, not in general by a simple translational and/or rotational mapping as for a rigid-body motion, but by a conformal mapping. The example of a flexible flat plate is used here to illustrate this approach to calculating the pressure field.

In the analysis of balistiform motion by elongated-body theory (Lighthill & Blake 1990), one part of the propulsive force on the fish has magnitude equal to P, the area integral of the pressure field just described. This area integral is shown in §3 below to take a simple form $\overline{U}M$− E in terms of the flow's momentum M and kinetic energy E per unit length and a certain weighted average $\overline{U}$ of the plate's velocity normal to itself. Although, in the case of motile fins attached to a rigid body of much greater depth, M was found (Lighthill & Blake 1990) to take an enhanced value, no such enhancement is found either for the product $\overline{U}M$ or for E, so that P itself is also not enhanced. For the relevance of these findings to the efficiency of balistiform motion, see Lighthill & Blake (1990).

Cross-sectional shapes of many fish bodies are well approximated by ellipses. The simple elongated-body theory of balistiform locomotion was developed by Lighthill & Blake (1990) only in the limiting case when the axis ratio of the cross-section tends to zero. In that case they established that the movements of dorsal and anal fins, if attached to a rigid fish body of far greater depth, create fluid motions with substantially enhanced momentum. In this paper, standard conformal mappings are used to establish that enhancement is substantial also with elliptic cross-sections of arbitrary axis ratio, not only in balistiform locomotion with synchronous movement of two median fins but also in gymnotiform locomotion with movement of just a single (ventral) fin.

Elongated-body theory, used by Lighthill & Blake (1990) to investigate fish locomotion by undulatory movements of median fins, and to demonstrate momentum enhancement in the case when motile fins are attached to a rigid fish body of far greater depth, approximates local fluid motions by solutions of the two-dimensional Laplace equation. Here, a better local approximation (equation (2) below) to the three-dimensional Laplace equation for fluid motions of undulatory type is used to investigate the possibility of short-wavelength limitations on momentum enhancement. In an extreme case (fish bodies of very small width and very large depth) when on elongated-body theory the momentum enhancement factor β is predicted to become very large, short-wavelength considerations are shown to place a specific upper limit on β (see figure 2). In more general cases, this upper limit should perhaps be regarded as coexisting with other upper limits associated with either nonzero width or finite depth of fish body.

Short-wavelength limitations on momentum enhancement are of some biological interest as implying the existence not only of advantages (including a reduction in body drag) but also of some competing disadvantages (limitations in propulsive force) arising from progressive reductions in the wavelength of fin undulations.

An experimental effort is presented here that examines the nonlinear interaction of multiple frequencies in the forced wake of an airfoil. Wakes with one or two distinct frequencies behave in an ordered manner – being either locked or quasi-periodic. When a third incommensurate frequency is added to the system, the flow demonstrates chaotic behaviour. Previously, the existence of the three-frequency route to chaos has been reported only for closed system flows. It is important to note that this chaotic state is obtained at a low Reynolds number. However, the chaotic flow shows localized characteristics similar to those of high Reynolds number turbulent flows. The degree of chaotic behaviour is verified by applying ideas from nonlinear dynamics (such as Lyapunov exponents and Poincaré sections) to the experimental data, thus relating the basic physics of the system to the concepts of mode interaction and chaos. Significant changes to the vortex configuration in the wake and to the r.m.s. velocity profile occur during the transition from order to chaos.

An approximation of the second-order diffraction potential is derived, for water waves of small amplitude incident upon a fixed body in a fluid of large depth. Attention is focused on the second-harmonic component of this potential, in terms of the fundamental incident-wave frequency, and on the particular solution of the inhomogeneous free-surface boundary condition with quadratic forcing by the first-order solution. By considering only the far-field approximation of the forcing function, a simple solution is derived in the near field of the body which is dominant when the submergence of the field point is large. The validity of this approach is confirmed by comparisons with two-dimensional experiments and three-dimensional computations.

We present results from our experiments with the irregular reflection of shock waves in argon. We compare the data with the results we obtained numerically; the assumptions for the computational code were that we had unsteady, two-dimensional, compressible, inviscid, flow of a perfect gas. When precautions were taken to reduce the effects of the gas viscosity on the experimental data, we obtained very good agreement between the numerical and the experimental results for the ramp Mach number and the trajectory path triple-point angle, but there were discrepancies with the wave-angle data. The discrepancies were ascribed to the sensitivity of the data to both viscosity and to a singularity. We show that there are actually two weak irregular wave reflections, namely a classic Mach reflection (MR) and a new type, that we call a von Neumann reflection (NR). The structure of the NR is discussed in some detail, and so are the transition criteria for the various wave systems.

Experiments on single-mode Faraday waves in small rectangular and circular cylinders in which both capillary and viscous effects were significant aie reported. Measurements of threshold forcing (for neutral stability) and steady-state wave amplitudes are compared with theoretical predictions. Theoretical predictions of the resonant frequency of a single mode and of the threshold amplitude for its excitation on the hypothesis of linear boundary-layer damping agree well with the measured data. (The theory must use the measured damping rate to predict these quantities for waves in the rectangular cylinder.) Theoretical predictions of wave amplitudes are in reasonable agreement with those observed in the circular cylinder; however, the theory provides only qualitative predictions of amplitudes for waves in the rectangular cylinder. In experiments in which two modes are theoretically admissible, the one with the smaller damping rate is observed; however, a single-mode calculation proves inadequate for the prediction of the stability boundary.

A review of the literature on natural convection in a horizontal layer heated from below shows the need for reliable data at high Rayleigh number (Ra) to determine the asymptotic Nusselt number (Nu) variation with Rayleigh number. The present study expands the data base by the use of an electrochemical mass transfer technique to determine the asymptotic dependence of the Sherwood number (Sh) on Ra at high Schmidt number (Sc). The results of the present study give Sh = 0.0659 $Ra^{frac{1}{3}}$ for Sc ≈ 2750, 3 × 109 < Ra < 5 × 1012. Using the heat-mass transfer analogy, this indicates the high Prandtl number variation of Nu with Ra.

We investigate the flow induced by a localized insoluble surfactant on a thin film. This problem is intended to model the behaviour of the lung's thin-film lining after an aerosol droplet lands on its surface. The surfactant-induced surface-tension gradients drive convection (Marangoni convection) within the film, disrupting the film surface and causing the surfactant to spread. The surfactant may also spread on the film's surface by surface diffusion without inducing convection. Gravity provides a restoring force that decreases film disturbances.

Lubrication theory is employed to derive equations that describe the evolution of the film thickness and surfactant concentration. A nonlinear surface-tension equation of state describes the relationship between the surfactant concentration and the surface tension. Solutions of the evolution equations are found numerically using the method of lines and analytically under limiting cases of small and large surface diffusivity. The results elucidate the behaviour of the thin-film/surfactant system.

We find that surface-tension-induced convection creates film disturbances that increase the film thickness near the surfactant's leading edge, and thins the film in the central region. Surface diffusion causes more rapid spreading of the surfactant, and decreases the film disturbances. Gravity decreases the film disturbances by creating bi-directional flow in the form of a ring vortex. This behaviour may have implications for the delivery of medications or toxins by aerosol inhalation.

Experimental and theoretical results are presented for the flow in a half-filled annulus between horizontal concentric cylinders with the inner one rotating and the outer one at rest. A laser-Doppler velocimeter was used to measure the mean tangential and/or corresponding turbulent velocity distributions for both the laminar and turbulent regimes. Colour dyes were also used to visualize the flow patterns. The Reynolds number based on the gap width varied from 67 to 3242, which corresponds to a Taylor number ranging from 25 to 1200. From the graph of local moment coefficient versus Taylor number and the visualization results, laminar, transition and turbulent regimes are identified. In the analysis, the governing equations are expressed in stream-function and vorticity forms and expanded in terms of the power series of the annulus aspect ratio. The zero- and first-order solutions are then solved numerically. Various features of the flow, in particular the presence of vortices in the exit end region and their absence from the entry end region, predicted in the analysis, are confirmed by the experimental findings.

The stream wise oscillation of a spring-mounted square-section cylinder set at angles of incidence, α, in the range from 0° to 45° is investigated in the reduced-velocity range 3 < U/ND < 13 and Reynolds-number range from 3.2 × 103 to 1.4. × 104. The mass-damping parameter used for the investigations is 1.6 and this gives vibration amplitude up to 0.12D. For small angles of incidence (i.e. α < 10°), vibration occurs mainly near U/ND = ½S, where S is the Strouhal number for the stationary cylinder. In the neighbourhood of α = 13.5°, which is where one of the separated shear layers is expected to reattach, vibration occurs near U/ND = 1/S. As α approaches 45° the amplitude observed near U/ND = 1/S diminishes and small-amplitude vibration appears near U/ND = ½S.

At α = 0°, vortices help to sustain oscillations by shedding when the cylinder is moving upstream. The mean drag of the oscillating cylinder drops and may reach less than half the stationary-cylinder value. When the amplitude of vibration is small, vortices of opposite sense of rotation are shed alternately and the familiar von Karman vortex street is formed. For moderately high values of amplitude, two vortex patterns fundamentally different from that of the stationary cylinder are observed. Intermittently, pairs of vortices are then shed symmetrically from both sides of the cylinder. When this occurs a pair of contra-rotating vortices forms every cycle of the vibration. When vortices of opposite sign are shed alternately, one vortex from each side of the cylinder forms every two vibration cycles. In this latter case, it appears that each vortex is elongated and split into two parts. Split vortices of opposite sign pair up and then form a vortex street.

In a previous paper, we have considered the weakly nonlinear interaction of a pair of axisymmetric lower branch Tollmien–Schlichting instabilities in cylindrical supersonic flows. Here the possibility that non-axisymmetric modes might also exist is investigated. In fact it is found that such modes do exist and, on the basis of linear theory, it appears that these modes are the most important. The non-axisymmetric modes are found to exist for flows around cylinders with non-dimensional radius a less than some critical value ac. This critical value ac is found to increase monotonically with the azimuthal wavenumber n of the disturbance and it is found that unstable modes always occur in pairs. We show that, in general, instability in the form of lower branch Tollmien—Schlichting waves will occur first for non-axisymmetric modes and that in the unstable regime, the largest growth rates correspond to the latter modes.

A set of equations in the Lagrangian description are derived for the propagation of long gravity waves in two horizontal directions for the purpose of determining the response of harbours with sloping boundaries to long waves. The equations include terms to account for weakly nonlinear and dispersive processes. A finite element formulation for these equations is developed which treats the propagation of transient waves in regions of arbitrary shape with vertical or sloping boundaries. Open boundaries are treated by specifying the wave elevation along the boundary or by applying a radiation boundary condition to absorb the waves leaving the computational domain. Nonlinear aspects of the interaction of long gravity waves with sloping boundaries and frequency dispersion due to non-hydrostatic effects are investigated. Results from the model are then compared with laboratory experiments of the response to long-wave excitation of a narrow rectangular harbour with a depth that decreases linearly from the entrance to the shore line.

The objective of this work is to develop techniques to predict the results of experiments involving chaotic dispersion of passive tracers in two-dimensional low Reynolds number flows. We present the design of a flow apparatus which allows the unobstructed observation of the entire flow region. Whenever possible we compare the experimental results with those of computations. Conventional tracking of the boundaries of the tracer is inefficient and works well only for low stretches (order 102 at most). However, most mixing experiments involve extremely large perturbations from steady flow since this is where the best mixing occurs. The best prediction of widespread mixing and large stretching (order 104–105) is provided by lineal stretching plots; surprisingly the technique also works for relatively low numbers of periods (as low as 2 or 3). The second best prediction is provided by a combination of low-order unstable manifolds – which indicate where the tracer goes, especially for short times – and the eigendirections of low-order hyperbolic periodic points – which indicate the alignment of striations in the flow. On the other hand, Poincaré sections provide only a gross picture of the spreading: they can be used primarily to detect what regions are inaccessible to the dye. Comparison of computations and experiments invariably reveals that bifurcations within islands have little impact in the mixing process.

The shape and stability of liquid menisci attached to a solid edge with dihedral angle 2α is investigated. It is shown that in addition to the family of cylindrical menisci a family of azimuthally modified unduloids exists. A double Fourier series of the latter with respect to their axis (parallel to the extension of the edge) and with respect to the azimuth is derived. The dispersion relation between the axial wavenumber q, the azimuthal wavenumber s and the waviness parameter d is calculated. When the condition of constant contact angle γ along the contact lines with the solid is applied, a one-dimensional family of modified unduloids fitting to the edge is obtained. Their axial wavenumber q becomes independent of the waviness d at the bifurcation with the family of cylindrical menisci, such that this bifurcation limits the stability. The respective stability criteria are derived and evaluated. For α + γ > ½π the cylindrical menisci are convex. They reveal a maximum stable length, which quadratically tends to infinity when α + γ = ½π is approached. The smallest stable extension arises for the free cylindrical column (the Rayleigh jet), which is covered by the present investigations by assuming α = π, γ = ½π. For α + γ < ½π the cylindrical menisci are concave and stable: no bifurcation with the family of modified unduloids arises.

The influence of Reynolds number on the topology of the organized motion in a turbulent boundary layer is studied with the use of an array of eight ×-probes in the plane of mean shear. Instantaneous sectional streamlines and contours of large-scale approximations to the spanwise vorticity and strain rate suggest that the motion is relatively complicated at all Reynolds numbers. Contours, conditioned on the spatially coherent increase in the velocity fluctuation u, extend to larger distances from the wall at the smaller Reynolds numbers. Reasonable correspondence is found between the location of this increase in the near-wall region and the end of a burst. Associated with this increase is an important contribution to the Reynolds shear stress throughout the layer. This contribution is larger at the smaller Reynolds numbers.

The slow viscous motion of a deformable drop moving normal to a planar wall is studied numerically. In particular, a boundary integral technique employing the Green's function appropriate to a no-slip planar wall is used. Beginning with spherical drop shapes far from the wall, highly deformed and ‘dimpled’ drop configurations are obtained as the planar wall is approached. The initial stages of dimpling and their evolution provide information and insight into the basic assumptions of film-drainage theory.

The three-dimensional time-dependent turbulent flow in a neutrally stratified Ekman layer over a smooth surface is computed numerically by directly solving the Navier–Stokes equations. All the relevant scales of motion are included in the simulation so that no turbulence model is needed. Results of the simulations indicate that the horizontal component of the rotation vector has a significant influence on the turbulence; thus the ‘f-plane’ approximation fails. Differences as large as 20% in the geostrophic drag coefficient, u*/G, and 70% in the angle between the freestream velocity and the surface shear stress are found, depending on the latitude and the direction of the geostrophic wind. At 45° latitude, differences of 6 and 30% are noted in the drag coefficient and the shear angle, respectively, owing to the variation of the wind direction alone. Asymptotic similarity theory and a higher-order correction are first tested for the range of low Reynolds numbers simulated, and then used to predict the friction velocity and stress direction at the surface for flows at arbitrary Reynolds number. A model for the variation of these quantities with latitude and wind angle is also proposed which gives an acceptable fit to the simulation results. No large-scale longitudinal vortices are found in the velocity fields, reinforcing the conjecture that unstable thermal stratification, in addition to inflectional instability, is required to produce and maintain the large-scale rolls observed in the Earth's boundary layer. Comparisons of the Ekman layer with a related three-dimensional boundary layer reveal similarities of the mean profiles, as well as qualitative differences.

The free creeping viscous incompressible plane flow of a finite region, bounded by a simple smooth closed curve and driven solely by surface tension, is analysed. The shape evolution is described in terms of a time-dependent mapping function z = Ω (ζ, t) of the unit circle, conformal on |ζ| [les ] 1. An equation giving the time evolution of the Ω (ζ, t) is derived. In practice, it has been necessary to guess a parametric form, i.e. Ω (ζ, t) = Ω[ζ; a1(t), a2(t), …], whose validity must be verified using the shape-evolution equation. Polynomial and proper rational mappings with no repeated factors are apparently always valid in principle. Solutions are given for (i) regions bounded initially by a regular epitrochoid, (ii) the limiting case of a half-plane bounded by a trochoid, and (iii) a class of rosettes whose mapping is rational. The two-lobed rosette gives the exact solution of the coalescence of equal cylinders. All these mappings involve limiting initial shapes having inward-pointing cusps. Useful parameterizations providing regions whose limiting shapes possess corners or outward-pointing cusps have not been found.

Natural convection in a cube of fluid-saturated porous medium having a constant temperature top and bottom is studied numerically. In the first of two special cases considered, the vertical sides are insulated. In this case, the numerical simulations indicate permanently unsteady regularly and irregularly fluctuating convective states at Rayleigh numbers (R*) above 550. The regularly fluctuating convective state defined by simply periodic oscillations in the Nusselt number begins at R* between 550 and 560. The frequency of the oscillations appears to depend on R* approximately as f∝ (R*)3.6. The irregularly fluctuating convective state defined by random variations in the Nusselt number begins at R* between 625 and 640. In the second case, heat is transferred through the vertical sides of the cube. Three distinct flow patterns are identified depending on the rate of heat transfer and the Rayleigh number. For all runs in the range of Rayleigh numbers studied, the transition from the first to the second flow pattern occurs abruptly.