This study considers steady-state, finite amplitude thermal convection in a non-Newtonian fluid. Pseudoplastic (power-law) fluids are considered for a power-law exponent n in the range 1 ≤ n ≤ 9. Finite-difference solutions are obtained for two-dimensional periodic convective modes in a horizontally infinite fluid layer heated from below. The results show that the patterns of convective motions differ only slightly from those in a fluid of constant viscosity for n [lsim ] 3 while for n [gsim ] 3 significant differences are observed. An average viscosity is introduced which provides a good correlation of heat transfer across the layer with the Rayleigh number for the complete range of n considered.

Flow visualization and laser-Doppler anemometry have been used to provide a detailed description of the velocity characteristics of the asymmetric flows which form in symmetric, two-dimensional, plane, sudden-expansion geometries. The flow and geometry boundary conditions which give rise to asymmetric flow are indicated, and the reason for the phenomenon is shown to lie in disturbances generated at the edge of the expansion and amplified in the shear layers. The spectral distributions of the fluctuations in velocity are quantitatively related to the dimensions of the two unequal regions of flow recirculation. It is also shown that the intensity of fluctuating energy in these low Reynolds number flows can be larger than that in corresponding turbulent flows.

Hot-wire anemometer measurements in a plane jet issuing at a harmonically oscillating angle into a moving air stream have been made to aid the understanding of oscillatory jet flows in general and flow past aerofoils with oscillating jet flaps in particular. The rates of velocity decay and jet spreading are shown to be greater and less, respectively, than those for a steady jet parallel to the air stream. The shapes of instantaneous velocity profiles and limited measurements of turbulence intensity are similar to those for a steady jet in a parallel air stream if a correction is made for the small velocity difference across the curved jet. The motion of the jet centre-line indicates that flow path lines are similar to those for a steady jet flap over a significant range of frequency of jet oscillation. Finally, a quasi-steady jet flap theory is proposed for an analytical description of the major flow features.

At shock Mach numbers Ms ∼ 16 in pure argon with initial pressures p0 ∼ 5 torr and final electron number densities ne ∼ 1017 cm−3, the translational shock front in a 10 x 18 cm hypervelocity shock tube develops sinusoidal instabilities which affect the entire shock structure including the ionization relaxation region, the electron-cascade front and the final quasi-equilibrium state. By adding a small amount of hydrogen (∼ 0·5% of the initial pressure), the entire flow is stabilized. However, the relaxation length for ionization is drastically reduced to about one-third of its pure-gas value. Using the familiar two-step collisional model coupled with radiation-energy loss and the appropriate chemical reactions, it was possible to deduce from dual-wavelength interferometric measurements a precise value for the argon-argon collisional excitation cross-section SAr Ar* = 1·0 x 10−19 cm2/eV with or without the presence of a hydrogen impurity. The reason for the success of hydrogen, and not other gases, in bringing about stabilized shock waves is not clear. It was also found that the electron-cascade front approached the translational-shock front near the shock-tube wall. This effect appears to be independent of the wall material and is not affected by the evolution of adsorbed water vapour from the walls or by water vapour added deliberately to the test gas. The sinusoidal instabilities investigated here may offer some important clues to the abatement of instabilities that lead to detonation and explosions.

This paper describes how the collocation technique previously developed by the authors for treating both unbounded (Gluckman, Pfeffer & Weinbaum 1971; Leichtberg, Weinbaum, Pfeffer & Gluckman 1976) and bounded (Leichtberg, Pfeffer & Weinbaum 1976) multiparticle axisymmetric Stokes flows can be extended to handle a wide variety of non-axisymmetric creeping-motion problems with planar symmetry where the boundaries conform to more than a single orthogonal co-ordinate system. The present paper examines in detail the strong hydrodynamic interaction between two or more closely spaced identical spheres in a plane. The various two-sphere configurations provide a convenient means of carefully testing the accuracy and convergence of the numerical solution technique for three dimensional flow with known exact spherical bipolar solutions.

The important difficulty encountered in applying the collocation technique to multi-particle non-axisymmetric flows is that the selection of boundary points is rather sensitive to the flow orientation. Despite this shortcoming one is able to obtain solutions for the quasi-steady particle velocities and drag for as many as 15 spheres in less than 30 s on an IBM 370/168 computer. The method not only gives accurate global results, but is able to predict the local fluid velocity and to resolve fine features of the flow such as the presence of separated regions of closed streamlines. Time-dependent numerical solutions are also presented for various three-sphere assemblages falling in a vertical plane. These solutions, in which the motion of each sphere is traced for several hundred diameters, are found to be in very good agreement with experimental measurements. The concluding section of the paper describes how the present collocation procedure can be extended to a number of important unsolved three-dimensional problems in Stokes flow with planar symmetry such as the arbitrary off-axis motion of a sphere in a circular cylinder or between parallel walls, or the motion of a neutrally buoyant particle at the entrance to a slit or pore.

A theory is developed for turbulence in a stably stratified fluid, for example in the experiments of Rouse & Dodu and of Turner where there is no shear and the turbulence is induced by a source of energy near the lower boundary of the fluid. A growing mixed layer of thickness D appears in the lower portion of the fluid and is separated from the non-turbulent fluid above, in which the buoyancy gradient is given, by an interfacial layer of thickness h. The lower mixed layer has a very weak buoyancy gradient and the large buoyancy difference across the interfacial layer is Δb.

As indicated by the experiments of Thompson & Turner and Hopfinger & Toly, and derived by the author in a recent paper, if u is the root-mean-square horizontal velocity and l is the integral length scale, the eddy viscosity ul is a constant in a homogeneous fluid agitated by a grid. When there is stratification, the theory indicates that the fluid motion is unaffected by buoyancy forces in the mixed layer, so that ul should again be constant in the lower portions of the mixed layer. Since l is proportional to distance, we may conveniently suppose that the source of the disturbances is at a level z = 0 where u is infinite in accordance with uz = K. Thus we may take K to be a fundamental parameter characterizing the turbulent energy source. Then z is distance above the plane of the virtual energy source. If the non-turbulent fluid has uniform buoyancy, DΔb = U2 may be shown to be constant. In general, whether constant or not, U may be taken to be a fundamental parameter expressing the stability. The quantity $\hat{R}i = U^{2}D^{2}/K^2$ is the most fundamental of the several Richardson numbers that have been introduced in this problem because, with its use, ‘constants’ of proportionality do not depend on the molecular coefficients of viscosity or diffusion (for high Reynolds number turbulence) or on the geometry of the grid.

The theory contains a number of results:

1. $u_e D/K \sim \hat{R}i^{-\frac{7}{4}},$ where $u_e = dD/dt$ is the entrainment velocity. Integration yields $D \infty t^{\frac{2}{11}}$ for a homogeneous upper fluid and $D \infty t^{\frac{2}{9}}$ for a linear upper density field. This $-\frac{7}{4}$ entrainment law compares with a $-\frac{3}{2}$ law suggested by several experimenters.

2. Turbulence in the interfacial layer is intermittent with intermittency factor $I_3 \sim \hat{R}i^{-\frac{3}{4}}$. The turbulent patches have dimension $\sigma_3 \sim D\hat{R}i^{-\frac{3}{4}}$.

3. If the (equal) root-mean-square velocities in an infinite homogeneous fluid at a distance D from the grid are denoted by u1 ∼ v1 ∼ w1, we find that the r.m.s. velocities near the interface are u2 ∼ u1, v2 ∼ u1 and $w_2 \sim u_1 \hat{R}i^{-\frac{1}{4}}$.

4. The buoyancy flux q2 near the interface may be expressed as $q_2 \sim w^3_2/D$.

5. h ∼ D, as observed in several experiments.

A model for the movement of a small viscous droplet on a surface is constructed that is based on the lubrication equations and uses the dynamic contact angle to describe the forces acting on the fluid at the contact line. The problems analysed are: the spreading or retraction of a circular droplet; the advance of a thin two-dimensional layer; the creeping of a droplet or cell on a coated surface to a region of greater adhesion; the distortion of droplet shape owing to surface contamination. Relevant biological problems concerning cell movement and adhesion are described.

Flow in a partly liquid-filled, rotating, horizontal cylinder is analysed by means of finite-element numerical simulation. Of alternative methods for locating the free surface, a boundary collocation scheme with Newton-Raphson iteration converges. This method forces the residual in the normal-stress boundary condition to zero at a finite set of points on the liquid meniscus. Solutions of the steady, two-dimensional, incompressible flow problem show circumferential variation of the liquid-film thickness and corresponding pressure and velocity fields, including recirculation zones. The complications of an unknown meniscus location and a nonlinear normal-stress condition when surface tension is significant are illustrated. The finite-element method proves an effective and convenient tool for such flows, in which inertial, gravitational, pressure, viscous and capillary forces are all important.

A transformation technique is used to solve the problem of steady nonlinear surface waves where the restoring force is either gravity or surface tension. An exact nonlinear integro-differential equation is found which yields known approximate solutions. Extensions to the method to account for more complicated geometries are also illustrated. The equation is solved numerically and results in agreement with previous solutions are obtained. In the case of capillary waves, the existence of two types of wave of greatest height is clearly indicated.

An axisymmetric gravity wave, for which each of nonlinearity, dispersion and radial spreading is weak but significant, is determined as a similarity solution with slowly varying amplitude $\frac{3}{4}{\cal S}a $ and length scale l, where $a/d \propto (r/d)^{\frac{2}{3}}, l/d \propto (r/d)^{\frac{1}{3}}, r$ is the radius, d is the depth, and [Sscr ] is the family parameter of the solutions. It is shown that the free-surface displacement η(r,t) is either a wave of elevation (η ≥ 0) or a wave of depression (η ≤ 0) and that (|η|/)½ satisfies a Painlevé equation that is a nonlinear generalization of Airy's equation. Representative numerical solutions and asymptotic approximations for small and large [Sscr ] are presented. It is shown that the similarity solution conserves energy but not mass, in consequence of which (in order to obtain a complete solution to a well-posed initial-value problem) it must either be accompanied by some other component or components or be driven by a source (or sink) in some interior domain in which the implicit restriction r [Gt ] d is violated. A linear model is developed that is valid for r [lsim ] d and compensates for the mass defect of, and matches, the nonlinear similarity solution for |[Sscr ]| [Lt ] 1.

The closed equations for the velocity correlation tensor and for the mean-squared displacement of a particle suspended in a stationary homogeneous turbulent flow, with an arbitrary linear law of fluid-particle interaction, are obtained using two assumptions suggested previously for the problem of turbulent self-diffusion: the ‘independence approximation’ and the Gaussian property of the functional distribution of particle velocities. The numerical solution of the derived equations is given for an isotropic system with a model turbulence spectrum. The following characteristics of the particle motion are obtained: (a) the mean kinetic energy, (b) diffusivity, (c) rate of energy dissipation, (d) velocity correlation function, and (e) the correlation function of the relative fluid-particle velocity. The impact of various spectral modes on the characteristics of the particle motion is discussed.