Hypersonic flow past axisymmetric power-law bodies $(y \propto x^m; m = 0, { \frac {1}{10}},{ \frac {1}{4}},{ \frac {1}{3}},{ \frac {1}{2}},{ \frac {3}{4}}\; and \; 1)$ has been studied experimentally and a detailed analysis of the dependence of shock-wave shape on body shape made.

Results indicate a continuous relationship between the exponent of the shock-wave shape and that of body shape over the range considered.

This investigation addresses the dynamics of annular viscoelastic films flowing down a flexible tube. The fluid viscoelasticity is assumed to be weak in order to obtain approximate explicit expressions for the stresses. Based on Shkadov’s integral boundary layer method (Fluid Dyn., vol. 2(1), 1967, pp. 29–34), a set of nonlinear evolution equations is derived that is valid for flows with moderate Reynolds numbers. The linear stability property of the system is examined by using normal-mode analysis, which is verified by comparing the results with those resulting from the linearization of the full Navier–Stokes equations. The results indicate that the fluid viscoelasticity plays an unstable role in the stability of the annular film flow. The tube flexibility, which includes wall damping and wall tension, plays a dual role. A bifurcation analysis is performed, and the families of steady travelling waves are catalogued. It is found that the stiffness of the tube tends to stimulate the interfacial capillary ripples. The fluid viscoelasticity acts to strengthen the dispersion of the interfacial waves but weakens the interfacial capillary ripples. The spatio-temporal evolutions of the system are also solved numerically. When the tube radius is small enough, tube closure can be observed due to the Plateau–Rayleigh instability. The fluid viscoelasticity acts to promote tube closure while the tube radius is relatively small. However, it plays a role in postponing the closure of the tube with a large radius.

The acoustic field radiated by a multipole point source positioned near to the surface of a solid sphere is calculated at both low and high frequencies. It is shown that the scattered field at low frequencies is always dipole, but at high frequencies is of the same type as the incident field. The application of the results to the acoustic field radiated by turbulence near a sphere is briefly discussed.

Results of direct numerical simulations of the transitional processes that characterise the evolution of a breaking internal gravity wave to a fully developed and essentially steady turbulent patch are presented. The stationary lee wave was forced by the imposition of an appropriate bottom boundary shape within a density-stratified domain having a uniform upstream velocity and density gradient, and with the ratio of momentum to thermal (or other) diffusivity defined by $\mathit{Pr}=1$. An earlier paper considered the eventual, fully developed turbulent patch arising after the breaking process is complete (Yakovenko et al., J. Fluid Mech., vol. 677, 2011, pp. 103–133); the focus in this paper is on the instabilities in the breaking process itself. The flow is analysed using streamlines, density contours and temporal and spatial spectra, as well as second moments of the velocity and density fluctuations, for a Reynolds number of 4000 based on the height of the bottom topography and the upstream velocity. The computations (on a grid using in excess of $10^{9}$ mesh points) yielded sufficient resolution to capture the fine-scale transition processes as well as the subsequent fully developed turbulence discussed earlier. It is shown that the major instability is of Rayleigh–Taylor type (RTI) with a resulting mixing region depth growing in a manner consistent with more classical RTI studies, despite the much more complicated environment. The resolution was sufficient to capture secondary Kelvin–Helmholtz-type instabilities on the developing RTI structures. Overall evolution towards the fully turbulent state characterised by a significant region of $-\frac{5}{3}$ subrange in both velocity and density spectra is very rapid. It is much faster than the long time scale characterising the subsequent evolution of the turbulent patch; this latter time scale is sufficiently large that the turbulent patch can itself be viewed as essentially steady.

Exact solutions are presented for an arbitrary number of steadily moving bubbles in a Hele-Shaw channel when surface tension is neglected. According to the symmetry displayed by the bubbles, the solutions are classified into two groups: (i) solutions where the bubbles are symmetrical about the channel centreline and (ii) solutions in which the bubbles have fore-and-aft symmetry. The general solutions are expressed in integral form but in some special cases analytic solutions in terms of elliptic integrals are found. The possible relevance of these exact solutions to experiments is also briefly discussed.

The equilibrium shape of an axisymmetric liquid drop or gas bubble in an immiscible supporting liquid held under gravity against a horizontal plane rigid surface is derived. A thin film of supporting liquid remains between the drop/bubble surface and the rigid substrate at equilibrium, of thickness determined by the balance of the disjoining pressure between the drop and the substrate and the internal Laplace pressure of the drop/bubble. The interface is macroscopically flat around the drop axis out to a radius but matches smoothly into the outer shape of radius through a boundary layer region of width where is a small parameter. The outer drop shape is determined by a balance of buoyancy forces and local Laplace pressure and is roughly spherical if , where is the capillary length in the interface with a logarithmic correction due to the action of the disjoining pressure across the flattened region. With the shape determined, we calculate the drag force on this flattened bubble to lowest order in the velocity as it moves across the rigid substrate using a lubrication approximation valid to terms of as an integral over the flattened bubble surface of the hydrodynamic pressure. The lubrication theory of itself is not sufficient to determine the drag due to the divergence of that integral if the outer flow field properties are neglected. By using the known exact result for the drag force on an undistorted bubble, the drag on the flattened bubble can be computed as an integral over the lubrication region alone. We derive the drag as a series expansion in the small parameter by means of a fairly intricate boundary layer analysis. The logarithmic divergence of the translational drag with film thickness for the undistorted bubble is replaced by the stronger divergence to leading order for the flattened bubble case. We present explicit numerical results for the first few terms in the expansion for the case of an exponentially repulsive disjoining pressure, and analytic expressions for these terms in the limit of very short-range disjoining pressure forces. The results of this calculation are compared with recent work (Hodges, Jensen & Rallison, J. Fluid Mech., vol. 512, 2004, p. 95) where disjoining pressure is neglected and hydrodynamic pressure is balanced against buoyancy and Laplace forces. The limits of validity of this linear drag theory are also presented.

The stability of a plane Rossby wave in a homogeneous fluid is considered. When the two-dimensional equation which governs fluid flow on a beta-plane is linearized in the disturbance stream function, a partial differential equation with a periodic coefficient results. Substitution of a solution dictated by the Floquet theory leads to a determinant equation, and it may be shown from its symmetry properties that disturbances to a Rossby wave may be of only two types: (i) neutrally stable modes not necessarily contiguous to a stability boundary and (ii) a pair of temporally unstable waves, one growing and the other decaying.

The determinant is solved numerically for the neutral-stability boundaries and curves of constant disturbance growth rate; two distinct types of instability emerge. The first is the parametric instability, which renders all waves unstable, and is shown to be asymptotic to the classical nonlinear resonant interaction in the limit of vanishing basic-state amplitude. The details of the disturbance frequency bifurcation for zero-amplitude basic-state waves are presented, and calculations for waves with eastward and westward group velocities are made and discussed in the context of Rhines’ (1975) results for waves and turbulence on a beta-plane. In addition, a second type of instability is computed which is separate and distinct from the parametric instability. The very limited evidence presented suggests that this second kind of instability may possess characteristics which are identifiable in part with the shearing of the fluid by the large-amplitude basic state and in part with the overturning of the ambient vorticity gradient.

Bounds for the complex wave velocity c, determined by the Orr-Sommerfeld equation and the boundary conditions for channel flow, have been given by Joseph (1968a,b). In these notes it is shown how two of Joseph's theorems can be uniformly improved.

The fall velocity of a dense large ball in a suspension of neutrally buoyant non-Brownian particles subjected to horizontal oscillatory shear is studied. As the strain amplitude is increased, the velocity increases up to a maximum value before decreasing to the value that it would have in a resting suspension. The higher the frequency is, the stronger the effect is. The falling ball velocity can be largely increased in the presence of the oscillatory cross-shear flow. For instance, for a particle volume fraction of $\varPhi =0.47$ it reaches four times the value it has in the unsheared suspension. At small strain amplitudes, it turns out that the velocity of the falling ball is determined by a balance between the steady drag flow, which drives the apparent suspension viscosity toward a high value, and the oscillatory cross-shear, which lessens it. A simple model is proposed to explain the experimental observations at small strain amplitude. The velocity decrease observed at larger amplitude is not completely understood yet.

The critical value RE of the Reynolds number R is predicted by the application of the energy theory. When R < RE, the Ekman layer is the unique steady solution of the Navier-Stokes equations and the same boundary conditions, and is, further, stable in a slightly weaker sense than asymptotically stable in the mean. The critical value RE is determined by numerically integrating the relevant Euler-Lagrange equations. An analytic lower bound to RE is obtained. Comparisons are made between RE and RL, the critical value of R according to linear theory, in order to demark the region of parameter space, RE < R < RL, in which subcritical instabilities are allowable.

We consider stationary perturbations to Couette–Poiseuille flows. These may be considered to be related to far downstream/upstream entry/end effects in flow inside long cavities and channels. Three distinct classes of basic flow are considered, all of which are exact solutions of the Navier–Stokes equations. We first study the problem in the case of Poiseuille flow, and are able to explain a previous discrepancy between fully numerical results, and asymptotic theory valid for large Reynolds numbers, R. The second case, which may be derived from a combination of an imposed streamwise pressure gradient and sliding of the upper channel wall, is for the particular situation where the flow on the lower surface is on the verge of reversing direction. The third case is relevant to the flow inside a long driven cavity (with closed ends, no imposed streamwise pressure gradient and no net mass flux). The flow is driven exclusively by a sliding top wall and mass conservation demands that the flow is no longer unidirectional.

For low Reynolds numbers, the stationary eigenvalues in all cases considered are complex (and hence are not monotonic in the streamwise direction). Indeed as R → 0 the eigenvalues become completely independent of the base profile. As the Reynolds number is increased, the eigenvalues generally undergo a number of branching processes switching between being complex and real (and vice versa) in nature, and at large Reynolds numbers fall broadly into three distinct categories, namely O(1), O(R−1/7) and O(1/R). In this limit the eigenvalues may be either complex or real (tending to monotonic eigensolutions in the streamwise direction).

Of particular interest are certain of the O(1) eigensolutions for the ‘driven-cavity’ problem, in the high-Reynolds-number limit; these turn out to be highly oscillatory (WKB-type) over much of the cavity section.

In all three cases, we use a combination of numerical and asymptotic techniques, and a thorough comparison between results thus obtained is made.

A two-layer configuration of thermohaline convection is studied, with the principal aim of explaining the observed independence of the buoyancy-flux ratio on the stability parameter when the latter is large. Temperature is destabilizing and salinity is stabilizing, so diffusive interfaces separate the convecting layers. The convection is treated in the single-mode approximation, with a prescribed horizontal planform and wavenumber. Surveys of numerical solutions are presented for a selection of Rayleigh numbers R, stability parameters λ and horizontal wavenumbers a. The solutions yield a buoyancy flux ratio ζ that is insensitive to ζ, in accord with laboratory experiments. However ζ increases with increasing R, in contradiction to laboratory observations.

This investigation is concerned with the numerical calculation of multiple solutions for a mixed-convection flow problem in horizontal rectangular ducts. The numerical results are interpreted in terms of recent observations by Benjamin (1978a) on the bifurcation phenomena for a bounded incompressible fluid. The observed mutations of cellular flows are discussed in terms of dynamic interchange processes. Each cellular flow may be represented by a solution surface in the parametric space of Grashof number Gr and aspect ratio γ, which is delimited by stability boundaries. Such a stability map has been generated for each type of cellular flow by a series of numerical experiments. Once these boundaries are crossed one cellular flow mutates into another via a certain dynamical process. Although the nature of the singular points on this map have not been determined precisely, a plausible general structure of the cellular-flow exchange process emerges from this map with several features in common with the Taylor-Couette flow. The primary modes appear to exchange roles via the formation of tilted cusp. Other salient features such as primary-mode hysteresis and quasi-critical range for cellular development appear to be present. However no anomolous modes have been observed.