Bellhouse et al. (1973) have developed a high-efficiency membrane oxygenator which utilizes pulsatile flow through furrowed channels to achieve high mass transfer rates. We present numerical solutions of the time-dependent two-dimensional Navier–Stokes equations in order to show the structure of the flow. Experimental observations which support this work are presented in a companion paper (Stephanoff, Sobey & Bellhouse 1980).

Steady flow through a furrowed channel will separate provided the Reynolds number is sufficiently large. The effect of varying the Reynolds number and the geometric parameters is given and comparisons with solutions calculated using the modern boundary-layer theory of Smith (1976) show excellent agreement. Unsteady flow solutions are given as the physical and geometric parameters are varied. The structure of the flow patterns leads to an explanation of the high efficiency of the devices of Bellhouse.

Observations of flow in furrowed channels support the calculations of part 1 (Sobey 1980). If the mainstream flow is steady there is a critical Reynolds number below which separation does not occur. Above that Reynolds number vortices form and fill the furrow. When the mainstream is oscillatory, the flow may separate during the acceleration to form strong vortices. During the deceleration the vortices grow to fill the furrow and channel. As the mainstream reverses the vortices are ejected from the furrows as the fluid flows between the wall and the vortex. Photographs show that this pattern occurs for sinusoidally varying walls, furrows that are arcs of circles and rectangular hollows.

The Lagally theorem for unsteady flow expresses the forces and moments acting on a rigid body moving in an inviscid and incompressible fluid in terms of the singularities of the analytically continued flow within the body. Previous generalizations of the Lagally theorem, originally given by Lagally (1922) for steady flows, are due to Cummins (1957) and Landweber & Yih (1956), who consider the effect of flow unsteadiness on the forces and moments. In these, the system of image singularities within the body was assumed to consist of isolated or continuous (surface or volume) distributions of sources and doublets. A further extension of Lagally's theorem ie due to Landweber (1967), who derived expressions for the steady forces and moments acting on a rigid body generated by isolated or a continuous distribution of multipoles. The purpose of the present paper is to generalize the Lagally theorem so as to include the effects of multipoles in unsteady flow, and deformability of the body, as well as to present a briefer derivation of the resulting formulae. Two examples, illustrating the application of the force and moment formulae, will be presented.

We show how trains of nonlinear, dispersive wavesIn some of the caaes to be described these are in fact sequences of solitary waves which are ordered by amplitude and which separate in space, as they propagate. In other cases the wave amplitude decreases as the wave propagates, but since the essential balance is between nonlinear steepening and frequency dispersion we feel justified in using the adjective ‘solitary’ to describe them though they violate the clessical description of such waves. can be produced by allowing a region of mixed fluid, with a potential energy greater than its surroundings, to collapse towards its equilibrium state. The number of waves and their amplitude depend on the properties of the mixed region and of the ambient stratification. Three different geometrical configurations have been chosen and while each gives qualitatively the same results the form taken by the generated waves and the final equilibrium shape of the mixed region depend critically on these geometrical factors. We relate the internal waves produced by this mechanism to waves produced in natural systems and show that our observations support at least one possible explanation for those found in the oceans and planetary atmospheres.

A theoretical and experimental investigation of the collapse of a mixed region into a linearly stratified fluid is presented. An analytical model is developed for the motion of the nose of the mixed region, valid for small dimensionless times and it compares well with the experimental results of the present study and previous investigators. For longer times the length of the mixed region did not increase as the square root of the dimensionless time, as previously reported (Wu 1969), for an aspect ratio (half the initial vertical extent divided by the initial horizontal extent) equal to one. In our experiments the length did increase as a power of the dimensionless time also, but with values of the exponent between 0·89 and 0·38, depending on the values of the various parameters. It is concluded that the motion cannot be described by a simple quasi-steady buoyancy-inertia balance only. We argue that the production of internal solitary waves and their interaction with the mixed region are critical to a complete understanding of the experimental results.

Matched asymptotic expansions, are used to describe turbulent Couette–Poiseuille flow (plane duct flow with a pressure gradient and a moving wall). A special modification of conventional eddy-diffusivity closure accounts for the experimentally observed non-coincidence of the locations of zero shear stress and maximum velocity. An asymptotic solution is presented which is valid as the Reynolds number tends to infinity for the whole family of Couette–Poiseuille flows (adverse, favourable, and zero pressure gradients in combination with a moving wall). It is shown that plane Poiseuille flow is a limiting case of Couette–Poiseuille flow. The solution agrees with experimental data for plane Couette flow, for the limiting plane Poiseuille flow, and for a special case having zero net flow and an adverse pressure gradient. The asymptotic analysis shows that conventional eddy diffusivity closures are inadequate for general Couette–Poiseuille flows.

The coupling of Brownian displacements and shear-induced convection of spherical colloidal particles in dilute suspensions is examined using solutions of appropriate convective diffusion equations for the time-dependent probability density and also by calculation of relevant statistical quantities for an ensemble of diffusing particles from Langevin equations. Based on a fundamental solution for convective diffusion from a point in a general linear field, analytical expressions for the probability density fα(r; t) are given for the case of an arbitrary, two-dimensional linear flow field. The parameter α, which characterizes the flow, may range from − 1 (pure rotation), through zero (simple shear), to + 1 (pure elongation). The Langevin approach offers interesting insights into the physical mechanism of diffusive–convective coupling, and may also be used to obtain rigorous expressions for moments of the probability density appropriate to a particle diffusing in an unbounded quadratic (Poiseuille) flow. Preliminary experiments are described which qualitatively verify the theoretical predictions for Poiseuille flow, and which suggest a simple, direct method for measuring particle diffusivities. Finally the effect of bounding walls on convective diffusion is considered by means of Monte Carlo calculations. Results show that particle-wall interactions significantly affect the average behaviour of particles located initially within distances of a few particle radii of the wall, since the frictional force is no longer isotropic.

An experimental study of the motion of small water droplets in a shock tube is reported. Droplet displacement data were obtained by means of reflected-light stroboscopic illumination for droplet diameters in the range 87–575 μm, and for shock strengths, ΔP/P1, in the range 0·0018–0·3. The displacement data are fitted by means of best-fit polynomials in time, which are used to compute droplet velocities, accelerations, and drag coefficients. All of our drag coefficient data have values which are larger than the steady drag at the same Reynolds numbers. The differences are attributed to time changes of the relative fluid velocity Ur. This may affect the size of the recirculating region and, therefore, the drag. In particular, it is argued that the drag is larger or smaller than the steady drag, depending on whether the dUr/dt is negative or positive, respectively. Our experiments, which were performed for dUr/dt < 0, confirm this expectation. Furthermore, it is shown that the difference between steady and transient drag coefficients, at the same Reynolds number, depends only on the value of a parameter A = (ρp/ρ0−1)(D/U2r)(dUr/dt). Here ρp and ρ0 are the densities of the droplets and of the surrounding gas, respectively, and D is the droplet diameter. In fact, in the Reynolds number range 3·2 < Re < 77, where multiple data are available having the same value of Re but having different values of A, the drag data can be expressed as CD = CDS(Re) – KA, where CDS(Re) is the steady drag at the instantaneous Reynolds number Re, and K is a constant of order 1.

Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found.

Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

A random superposition of waves in a rotating, stratified, electrically conducting fluid leads to dynamo action in the sense that it yields a mean electric field having a component parallel to the mean magnetic field (‘α-effect’). Using Fourier analysis methods, we derive an explicit expression for the mean electric field. The α-effect has tensor form. We obtain a finite α-tensor even in a case of vanishing mean helicity. The result is discussed in the context of the solar turbulent dynamo.