A body is started from rest and moves on an arbitrary path in an inviscid isothermal compressible atmosphere. The phase configuration of the internal waves and the gravity-modified acoustic waves which are generated by the body is studied using small amplitude wave theory. When the body moves at supersonic speeds and the background density gradient approaches zero, it is shown how the wave solutions approach the pure acoustic wave solutions of Lilley et al. (1953).

Numerical and analytical solutions to the steady compressible Euler equations corresponding to a compressible analogue of the linear Stuart vortex array are presented. These correspond to a homentropic continuation, to finite Mach number, of the Stuart solution describing a linear vortex array in an incompressible fluid. The appropriate partial differential equations describing the flow correspond to the compressible homentropic Euler equations in two dimensions, with a prescribed vorticity–density–streamfunction relationship. In order to construct a well-posed problem for this continuation, it was found, unexpectedly, to be necessary to introduce an eigenvalue into the vorticity–density–streamfunction equation. In the Rayleigh–Janzen expansion of solutions in even powers of the free-stream Mach number M∞, this eigenvalue is determined by a solvability condition. Accurate numerical solution by both finite-difference and spectral methods are presented for the compressible Stuart vortex, over a range of M∞, and of a parameter corresponding to a confined mass-flow rate. These also confirm the nonlinear eigenvalue character of the governing equations. All solution branches followed numerically were found to terminate when the maximum local Mach number just exceeded unity. For one such branch we present evidence for the existence of a very small range of M∞ over which smooth transonic shock-free flow can occur.

A new lubrication model solution is found for the late-stage draining of a wetting capillary liquid from a linear interior corner. The solution exploits the symmetry of volumetric sink conditions at opposing ends of such a ‘double-drained’ interior corner flow with applications ranging from liquid recovery in microfluidic devices on Earth to liquid fuel scavenging in large fuel tanks aboard spacecraft. At long times $t$, the nominal liquid depth is $h\sim t^{-1}$, the liquid volume is $V\sim t^{-2}$ and the maximum volumetric liquid removal rate is $Q\sim t^{-3}$. The constraints under which the solution is valid are provided. To qualitatively assess the value of the solution, representative experiments are conducted at larger length scales aboard the International Space Station and at microfluidic length scales in a terrestrial laboratory. Both sets of experiments confirm the predicted power-law dependences. We show that the separation of variables solution offers a method to predict maximum drain rates from related geometries where a single drain location provides the required symmetry of the problem.

The present study is aimed at assessing the existing results concerning the stability of canonical shear flows in channels and tubes with deformable walls, in light of consistent formulations of the nonlinear solid constitutive model and linearised interface conditions at the fluid–solid interface. We show that a class of unstable shear-wave modes at low Reynolds number, predicted by previous studies for pressure-driven flows through neo-Hookean tubes and channels, is absent upon use of consistent interfacial conditions. Furthermore, we analyse the consequences of the change in solid model on the stability of the canonical shear flows by using both neo-Hookean and Mooney–Rivlin models. We show that the salient features of the stability of the system are adequately captured by a consistent formulation of the neo-Hookean solid model, thus precluding the need to employ more detailed solid models. The stability analysis of planar flows past a neo-Hookean solid subjected to three-dimensional disturbances showed that two-dimensional disturbances are more unstable than the corresponding three-dimensional disturbances within the consistent formulations. We show that prior inconsistent formulations of the solid constitutive equation predict a physically spurious spanwise instability in disagreement with experiments thereby demonstrating their inapplicability to predict instabilities in flow past deformable solid surfaces. Using the consistent formulation, the present work provides an accurate picture, over a range of Reynolds numbers, of the stability of canonical shear flows through deformable channels and tubes. Importantly, it is shown how inconsistencies in either the bulk constitutive relation or in the linearisation of the interface conditions can separately lead to physically spurious instabilities. The predictions of this work are relevant to experimental studies in flow through deformable tubes and channels in the low and moderate Reynolds number regime.

A quantitative theory of the average features of turbulent flow in a channel is described without the introduction of empirical parameters. The qualitative problem consists of maximizing the dissipation rate of the mean flow subject to the Rayleigh condition that the mean flow has no inflections. The quantitative features result from a boundary stability study which determines a smallest scale of motion in the transport of momentum. The velocity fields satisfying these conditions, the averaged equations and the boundary conditions uniquely determine an entire mean velocity profile at all Reynolds numbers within ten per cent of the data. The maximizing condition for the reproducibility of averages emerges from the Navier–Stokes equations as a consequence of a novel definition of nonlinear instability. The smallest scale of motion results from a theory for a time-dependent re-stabilization of the boundary layer following a disruptive instability. Computer reassessment of the several asymptotic estimates of the critical boundary eigenstructure can establish the limits of validity of the quantitative results.

The nature of a three-dimensional viscous flow along a corner near its junction has been clarified in this paper by constructing a Stokes slow-flow solution. We have further demonstrated that this Stokes solution can be matched onto an inertial-flow solution in principle by establishing an overlap domain along one sector of an inertial-flow region, namely along the flow symmetry line. This Stokes solution reveals a remarkably complex structure of the flow as characterized by a separating streamwise velocity profile in addition to a sequence of Moffatt's viscous eddies in a cross-flow plane.

By use of a virial equation introduced in a recent paper (Benjamin 1987), the main results of a second-order perturbation theory developed by Longuet-Higgins (1989a) are recovered in comparatively simple fashion. Asymmetric capillary vibrations of a gas bubble in an infinite incompressible liquid are confirmed to generate an increase in the volume of the bubble, a lowering of the mean pressure of the gas and a monopole component in the motion of the liquid. It is shown that the second effect remains when the bubble is incompressible.

The plane turbulent wall-jet flow without externally imposed stream is considered. It is assumed that the wall jet does not emerge from a second wall perpendicular to the velocity vector of the initial wall jet. The (kinematic) momentum flux $K(x)$ of the wall jet decreases downstream owing to the shear stress at the wall. This investigation is based on the hypothesis that the total friction force on the wall is smaller than the total inflow momentum flux. In other words, the turbulent wall jet tends to a turbulent ‘half-free jet’ with a non-zero momentum flux $K_{\infty }\;(\text{m}^{3}~\text{s}^{-2})$ far downstream. The fact that the turbulent half-free jet is the asymptotic form of a turbulent wall jet is the basis for a singular perturbation method by which the wall-jet flow is determined. It turns out that the ratio between the wall distance $y_{m}$ of the maximum velocity and the wall distance $y_{0.5}$ of half the maximum velocity decreases downstream to zero. Dimensional analysis leads immediately to a universal function of the dimensionless momentum flux $K(\mathit{Re}_{x})/K_{\infty }$ that depends asymptotically only on the local Reynolds number $\mathit{Re}_{x}=\sqrt{(x-x_{0})K_{\infty }}/{\it\nu}$, where $x_{0}$ denotes the coordinate of the virtual origin. When the values $K$ and ${\it\nu}$ at the position $x-x_{0}$ are known, the asymptotic momentum flux $K_{\infty }$ can be determined. Experimental data on all turbulent plane wall jets (except those emerging from a second plane wall) collapse to a single universal curve. Comparisons between available experimental data and the analysis make the hypothesis $K_{\infty }\neq 0$ plausible. A convincing verification, however, will be possible in the future, preferably by direct numerical simulations.

In experiments on the stability of a submerged axisymmetric jet at Reynolds numbers large compared with unity, Reynolds (1962) observed axisymmetric ‘condensations’ which appear to grow spontaneously, whereas Batchelor & Gill (1962) have shown that infinitesimal disturbances of this type do not grow in an inviscid fluid. Here it is shown that axisymmetric disturbances do not grow in a slightly viscous fluid either, and the solution for which the rate of damping is smallest is found. It is suggested that the growth of small but finite disturbances is responsible for the condensations observed. The order of magnitude of the disturbance velocity at which non-linear effects could produce growth of disturbance is found to depend on the wave-number of the disturbance. The smallest velocity which a ‘finite’ disturbance may have is found to be of order $R^{-\frac{2}{3}$, and corresponds to a disturbance whose wave-number is of order $R^{\frac{1}{3}$, R being the Reynolds number based on the local radius and maximum velocity of the jet. On the assumption that some disturbances whose velocity is of this order will grow, deductions are made as to the size, position, wave-number, and point of appearance of condensations. The deductions appear to agree with the experimental results.

The work presented in this paper represents an experimental investigation into secondary flows, turbulent boundary layers and the interaction of the two as they develop in a zero-pressure-gradient rotating flow field. A duct of intermediate aspect ratio was used to examine secondary flows and determine when they begin to govern the boundary layer development. The aspect ratio (A) was defined as duct height/width at the upstream end of the working section. Measurements were taken at three aspect ratios: A=1, 2 and 4.

A qualitative indication of secondary flow strength was established with mean-cross-stream-plane velocity measurements. A first-order analysis of the secondary flow is presented which provides a reasonable estimation of their strength. Mid-span mean-flow, turbulence and spectra profiles were measured on the duct walls parallel to the axis of rotation. Results are generally presented for A=2 and 1. For A=4 and 2 there were minor effects of secondary flows observed on the mid-span mean flow parameters. The turbulent shear measurements showed some secondary flow effect for A=2. All turbulence and mean-flow quantities were strongly affected by secondary flows for A=1. Spectra results presented for A=2 showed most variation at the low-to-mid wavenumber end. Spectra results for A=1 showed a bodily shift of the whole spectrum towards low wavenumber on the pressure side and high wavenumber on the suction side.

We report on small-scale instabilities in a thermally driven rotating annulus filled with a liquid with moderate Prandtl number. The study is based on direct numerical simulations and an accompanying laboratory experiment. The computations are performed independently with two different flow solvers, that is, first, the non-oscillatory forward-in-time differencing flow solver EULAG and, second, a higher-order finite-difference compact scheme (HOC). Both branches consistently show the occurrence of small-scale patterns at both vertical sidewalls in the Stewartson layers of the annulus. Small-scale flow structures are known to exist at the inner sidewall. In contrast, short-period waves at the outer sidewall have not yet been reported. The physical mechanisms that possibly trigger these patterns are discussed. We also debate whether these small-scale structures are a gravity wave signal.

We study the axisymmetric spreading of drops deposited on a pre-existing horizontal layer of the same viscous fluid. Using a combination of experiments, numerical modelling based on the axisymmetric free-surface Navier–Stokes equations and scaling analyses, we explore the drops’ behaviour in a regime where the flow is driven by gravitational and/or capillary forces while inertial effects are small. We find that during the early stages of the drops’ evolution there are three distinct spreading behaviours depending on the thickness of the liquid layer. For thin layers the fluid ahead of a clearly defined spreading front is at rest and the overall behaviour resembles that of a drop spreading on a dry substrate. For thicker films, the spreading is characterised by an advancing wedge which is sustained by fluid flow from the drop into the layer. Finally, for thick layers the drop sinks into the layer, accompanied by significant flow within the layer. As the drop keeps spreading, the evolution of its shape becomes self-similar, with a power-law behaviour for its radius and its excess height above the undisturbed fluid layer. We employ lubrication theory to analyse the drop’s ultimate long-term behaviour and show that all drops ultimately enter an asymptotic regime which is reached when their excess height falls below the thickness of the undisturbed layer.

Analysis of bioconvection in dilute suspensions of bottom-heavy but randomly swimming micro-organisms is commonly based on a model introduced in 1990. This couples the Navier–Stokes equations, the cell conservation equation and the Fokker–Planck equation (FPE) for the probability density function for a cell’s swimming direction $\boldsymbol{p}$, which balances rotational diffusion against viscous and gravitational torques. The results have shown qualitative agreement with observation, but the model has not been subjected to direct quantitative testing in a controlled experiment. Here, we consider a simple configuration in which the suspension is contained in a circular cylinder of radius $R$, which rotates at angular velocity ${\rm\Omega}$ about a horizontal axis. We solve the FPE and calculate the cells’ mean swimming velocity, which proves to be horizontal when $B{\rm\Omega}\gg 1$, where $B$ is the gyrotactic reorientation time scale. Then we compute the cell concentration distribution, which is non-uniform only in a thin boundary layer near the cylinder wall when ${\it\beta}^{2}={\rm\Omega}R^{2}/D\gg 1$, where $D$ is the cells’ translational diffusivity. The fact that cells are denser than water means that this concentration distribution drives a perturbation to the underlying solid-body rotational flow which can be calculated analytically. The predictions of the theory are evaluated in terms of a proposed experimental realisation of the configuration, using suspensions of the alga Chlamydomonas nivalis or Chlamydomonas reinhardtii or the algal colony Volvox.

The motion of a spherical inclusion released in a vertical solid-body rotation flow is investigated theoretically and experimentally. Solid spheres and bubbles are considered. The particle Reynolds number, the Taylor number, the Weber number and the capillary number are smaller than unity. The motion equations of the inclusion are obtained by revisiting the hydrodynamic equations. The axial (vertical) motion and the horizontal motion are uncoupled, even though they are sensitive to the rotation rate of the flow. Analytical solutions of the particle motion equation are compared to experimental results obtained by releasing a particle in a rotating tank filled with silicone oil. For solid spheres and bubbles, both the terminal velocity and the particle ejection rate (or trapping rate) predicted by the theory agree with experiments, without any empirical adjustment. In particular, the experimental device enables us to check the validity of various theories involving solid or gaseous inclusions with or without inertia or history effects. It is observed that the mobility tensor obtained by writing the fluid motion equations in the rotating frame accurately predicts the horizontal particle trajectory, like the Boussinesq-Basset equation obtained by writing the fluid motion equations in the non-rotating frame and neglecting the horizontal contribution of inertia effects.

An analysis has been made of a two-dimensional turbulent wake which was subjected to a homogeneous strain. This was applied in such a way that the rate of growth of the wake tended to be accelerated but the main stream convective velocity remained constant. An attempt to describe the flow by classical self-preservation techniques was made. However, the results suggested that a detrainment of wake fluid would result. In lieu of this a more realistic approach was sought and it was found that in certain situations where the flow has had an opportunity to develop before the strain is applied, the growth of the wake in the strain field can be described successfully by a simple geometric distortion. It appears likely that in these cases the interaction between the external strain and the turbulent structure within the wake is negligible, the wake remaining essentially passive. Suggestions as to the mechanism of the turbulent mixing and transport under these conditions are made.

The stability of a laminar mixed-convection boundary layer adjacent to a vertical isothermal surface is examined, using linear stability theory and the parallel-flow approximations. The analysis is valid when the imposed forced-convection effects are small compared to natural-convection effects. The stability equations are solved numerically for aiding and opposing forced-convection effects, for Pr = 0.733 (air) and 6.7 (water). For aiding mixed convection in air, a new feature was found. A small, separated region of instability arises upstream of the ‘conventional’, or ‘primary’, neutral curve. In this region, selective amplification of a narrow band of disturbance frequencies occurred, but disturbance growth was small. Further downstream, disturbance growth rates in flows with an aiding free stream are slower than in natural convection. The opposite is true for an opposing free stream in air. Selective disturbance amplification occurred downstream for all conditions, as in natural convection. In water, an aiding flow was destablizing compared to natural convection, and an opposing flow was stabilizing. Evidence of a separated upstream region of instability was also found for aiding mixed convection in water. However, converged solutions could not be obtained in this circumstance.

An experimental and theoretical investigation has been conducted to determine the stability and disturbance-amplification characteristics of the combined forced and free convection flow adjacent to a vertical uniform-heat-flux surface in a uniform free stream. Previous studies of the stability of mixed convection flows have been limited to linear stability analysis of the effect of weak buoyancy on the neutral stability of a stronger forced flow. Here we consider circumstances where forced-convection effects are small compared with buoyancy effects. The flow behaviour is analysed using linear stability theory. The analysis incorporates a new formulation which permits the calculation of amplification contours for a given flow circumstance. The governing equations have been solved numerically to generate stability planes including the neutral curve and constant amplification contours. Stability planes are presented for assisting and opposed flows at Prandtl numbers Pr of 0·733 and 6·7. In air (Pr = 0·733), the presence of a weak free stream is found to cause the disturbance-amplification rates and the filtered frequency to deviate strongly from those found in purely free-convection flow. In water (Pr = 6·7), the effect of a free stream is much weaker. In addition, hot-wire and thermocouple measurements of the filtered frequencies and the disturbance-amplitude distributions are presented for aiding mixed convection flow in air. The measurements are found to be in very good agreement with the calculated results of the stability analysis.

A theory is developed to predict the motion of N neutrally buoyant spheres suspended in laminar flow between parallel plates. The spheres are at large separation yet nearer each other than the duct walls, and the Reynolds number is small. In this parameter range, viscous interactions are larger than inertial effects, and can be represented in terms of a superposition of ‘strainlets’. Several examples are given to show this viscous interaction effect. Near the leading edge of a front of spheres or near the trailing edge significant lateral migration velocities can occur, being at least one order of magnitude larger than inertially induced migration velocities. This phenomenon may have a negative effect on ‘chromatographic’ separation schemes, affecting particle concentration, recovery and resolution.

In the previous paper (Chester 1961) it was shown that, for large values of the Hartmann number, the asymptotic solution for the flow past a body of revolution has a discontinuity on the surface of a cylinder which circumscribes the body. The flow in the region of this discontinuity is now investigated in more detail when the body is a circular disk broadside-on to the flow. It will be shown that there is actually a region of transition whose thickness is O(|x|½/M½), where x is the axial distance from the disk and M is the Hartmann number. This region is thin near the disk, but gradually thickens until it merges into the over-all flow field for x = O(M).

The leading terms in the expression for the drag are given by $\frac{D}{D_s} = \frac{M \pi }{8} \left( 1 + \frac{2}{M} \right) $, where Ds is the Stokes drag.