The solution of the steady-state Navier–Stokes equations in three dimensions has been obtained by a numerical method for the problem of natural convection in a rectangular cavity as a result of differential side heating. In the past, this problem has generally been treated as though it were two-dimensional. The solutions explore the three-dimensional motion generated by the presence of no-slip adiabatic end walls. For Ra = 104, the three-dimensional motion is shown to be the result of the inertial interaction of the rotating flow with the stationary walls together with a contribution arising from buoyancy forces generated by longitudinal temperature gradients. The inertial effect is inversely dependent on the Prandtl number, whereas the thermal effect is nearly constant. For higher values of Ra, multiple longitudinal flows develop which are a delicate function of Ra, Pr and the cavity aspect ratios.

Observations by Cornhill ' Roach (1976) of sudanophilic lesions in the vicinity of intercostal arteries in rabbit aortas have shown that lesions develop on the downstream side of the associated ostia. There is considerable conjecture as to the role which varying levels of wall shear stress play in the development of such lesions; Cornhill ' Roach implicate high wall shear stress levels. We develop a consistent model of steady boundary-layer flow past a side slot assuming that there is Stokes flow in the side slot and that the main body of the boundary layer remains undisturbed. Our results show that increased levels of wall shear stress occur both upstream and downstream of the slot. If the withdrawal of fluid through the side slot is sufficiently great there may be a stagnation point on the downstream side of the slot. The wall shear stress level near the slot depends on both normal and transverse motions at the mouth of the slot. Indeed, very near the slot, on a length scale comparable with the slot width, the wall shear stress level depends only on the transverse motions at the mouth of the slot.

Results of an experimental investigation of the evolution of a nonlinear wave train on deep water are reported. The initial stage of evolution is found to be characterized by exponential growth of a modulational instability, as was first discovered by Benjamin ' Feir. At later stages of evolution it is found that the instability does not lead to wave-train disintegration or loss of coherence. Instead, the modulation periodically increases and decreases, and the wave train exhibits the Fermi–Pasta–Ulam recurrence phenomenon. Results of an earlier study of nonlinear wave packets by Yuen ' Lake, in which solutions of the nonlinear Schrödinger equation were shown to provide quantitatively correct descriptions of the properties of nonlinear wave packets, are applied to describe the experimentally observed wave-train phenomena. A comparison between the laboratory data and numerical solutions of the nonlinear Schrödinger equation for the long-time evolution of nonlinear wave trains is given.

The problem of the instability of a uniform, nonlinear, deep-water wave train to infinitesimal long-wave perturbations, first studied by Benjamin ' Feir (1967) and Benjamin (1967), is re-examined. It is found that the apparent discrepancy between the experimental and theoretical growth rates of the instability is associated with the experimental generation of waves which do not have the Stokes wave profiles assumed in the theory. Experimental and theoretical results relating the initial wave steepness and the most unstable long-wave perturbation are used to obtain a correction factor, which is found to account for the mismatch in wave forms and which resolves the discrepancy in growth rates. The results illustrate that, when theory is compared with experiments in which the values of certain higher-order (nonlinear) quantities must be deduced from measurements of first-order quantities, great care must be taken to ascertain that the experimental conditions and the theoretical assumptions are indeed compatible to the required order.

The two-dimensional motion of a stably stratified fluid containing two solutes with different molecular diffusivities in an inclined slot has recently been examined by Chen (1975, hereafter referred to as I). The two solutes have continuous opposing gradients with the slower-diffusing one more dense at the bottom. It is found that, in the steady state, there exists a slow upward flow along the slope driven by the slight buoyancy difference near the wall, not unlike the solution found by Wunsch (1970) and Phillips (1970) for a single solute. For the time-dependent flow resulting from switching on the diffusivities at t = 0, there may be a flow reversal near the wall depending on the relative magnitude of λ and τ (where λ is the ratio of the density gradient and τ−1 is the ratio of the diffusivity of the faster-diffusing solute T to that of the slower-diffusing one S). By examining the distributions of S and T across the slot, it becomes apparent that in cases with flow reversal double-diffusive instability is likely to occur.

In this paper, we examine the stability of time-dependent double-diffusive convection in an inclined slot both analytically and experimentally. The time-dependent perturbation equations are numerically integrated starting with an initial distribution of small random disturbances in the vorticity. The growth or decay of the kinetic energy of the perturbations serves to indicate whether the flow is unstable or stable. The results show that the flow becomes more unstable (a) with increasing λ at a given angle of inclination with respect to the vertical and (b) with increasing angle of inclination at a given value of λ. Experiments were carried out in a 2[sdot ]54 cm wide slot using sugar and salt solutions at angles of inclination of 30°, 45° and 60°. Results obtained confirm the trends predicted by the analysis. Good agreement was obtained between the predicted and the experimental values of the critical wavelength for the case λ = 0[sdot ]7.

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order ø2, where ø([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ2 is found to be \[ \mu^{*} = \mu(1+2.5\phi + 6.2\phi^2). \] The value of the coefficient of ø2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the ø2-term.

By the use of high-speed photography, instabilities occurring in high Reynolds number water jets discharging into air have been made visible. These instabilities include the axisymmetric mode accompanying the transition from laminar to turbulent flow at the nozzle exit, spray formation as a culmination of the axisymmetric disturbances, and, further downstream, helical disturbances which result in the entire jet assuming a helical form. The final disruption of the jet is due to amplification of the helical waves. It is further shown that the amplification of the helical disturbances is due in part to aerodynamic form drag, since jets discharging into surrounding air moving at the same speed as the jet remain relatively stable, compared with the case when the jet is discharged into stagnant air.

An exact Eulerian formulation of the problem of diffusion of passive scalar and vector fields by a turbulent velocity field is obtained. It is shown that, in the short autocorrelation time limit, the diffusion equation is exact for any turbulence. For non-zero autocorrelation times the form of the first few correction terms to the diffusion equation is found. As a result of these corrections the diffusion of scalar, divergence-free and curl-free vector fields will be different. The calculations use the Kubo–Van Kampen–Terwiel technique and are carried out for zero ordinary diffusivity and for homogeneous, stationary, isotropic, incompressible, helical turbulence.

A train of surface gravity waves of wavelength λ in a channel of depth H is incident on a horizontal plate of length l that is submerged to a depth c. Under the assumption that both λ and l are large compared with H, the method of matched asymptotic expansions is used to show that, to first order, the reflexion coefficient R and the transmission coefficient T are given by \[ R = \chi \left\{\frac{\sigma l}{(gH)^{\frac{1}{2}}}\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}-2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\} \] and \[ T =\chi\left\{2i\left[\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\right]\right\} \] where \begin{eqnarray*} \chi &=& 1\left/ \left\{2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)+\frac{\sigma l}{b}\bigg(\frac{H}{g}\bigg)^{\frac{1}{2}}\bigg(1+\frac{c}{H}\bigg)\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}\right.\right.\\ &&\left. +2i\bigg(\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\}, \end{eqnarray*} σ is the angular frequency and g the acceleration due to gravity.

The boundary-value problem for irrotational surface waves is derived from a variational integral I with the Lagrangian density [Lscr ] = Ξ ηt - [Hscr ] where Ξ (X, t) is the value of the velocity potential at the free surface, y = η(x, t), and [Hscr ] is the energy density in x space. [Hscr ] then is expressed as a functional of Ξ and η, qua canonical variables, by solving a reduced boundary-value problem for the potential, after which the requirement that I be stationary with respect to independent variations of Ξ and η yields a pair of evolution equations for Ξ and η. The Fourier expansions Ξ = pn(t)ϕn*(x) and η = qn(t)ϕn(x), where {ϕn} is an orthogonal set of basis functions, reduce I to Hamilton's action integral, in which the complex amplitudes pn and qn appear as canonically conjugate co-ordinates, and yield canonical equations for pn and qn that are the spectral transforms of the evolution equations for Ξ and η. The evolution equations are reduced (asymptotically) to partial differential equations for pn and qn by expanding [Hscr ] in powers of α = a/d and β = (d/l)2, where a and l are scales of amplitude and wavelength. Explicit third approximations are developed for β = O(α).

The canonical form of the equations for the free-surface elevation and potential of an irrotational fluid is more than a coincidence. The elevation is a ‘generalized coordinate’ field sufficient to define the system Lagrangian without explicit reference to the motion of the fluid interior. The Lagrangian and the associated field equations are complete and self-contained in the two-dimensional surface co-ordinates, but non-local (integro-differential) in form; the canonical equations derived by Miles are just the Hamiltonian counterparts.

A shallow three-dimensional hump disturbs the two-dimensional incompressible boundary layer developed on an otherwise flat surface. The steady laminar flow is studied by means of a three-dimensional extension of triple-deck theory, so that there is the prospect of separation in the nonlinear motion. As a first step, however, a linearized analysis valid for certain shallow obstacles gives some insight into the flow properties. The most striking features then are the reversal of the secondary vortex motions and the emergence of a ‘corridor’ in the wake of the hump. The corridor stays of constant width downstream and within it the boundary-layer displacement and skin-friction perturbation are much greater than outside. Extending outside the corridor, there is a zone where the surface fluid is accelerated, in contrast with the deceleration near the centre of the corridor. The downstream decay (e.g. of displacement) here is much slower than in two-dimensional flows.

An asymptotic description is given of Newtonian fluid flow in a channel which is suddenly heated or cooled. The viscosity is assumed to be purely a function of temperature. The asymptotic approximation is that the downstream viscosity at the channel wall differs by an order of magnitude from that in the upstream flow. Although we make the drastic assumption that viscous dissipation is negligible, we can analyse flows where the viscosity depends either algebraically or exponentially on the temperature.

This paper presents a similarity solution for plane channel flow of a very viscous fluid, whose viscosity is exponentially dependent upon temperature, when heat generation is very large. A dimensionless formulation of the problem involves two length scales (the depth h and length l, respectively, of the channel), one velocity scale (the mean velocity V of the fluid along the channel), the thermal conductivity k, thermal diffusivity k and viscosity V of the fluid, and the temperature coefficient b of the viscosity. From these, two important dimensionless groups arise, the Graetz number (Gz = Vh2/kl) and the Nahme–Griffith number (G = μ V2b/k). In the case of steady flow with G−1 [Lt ] Gz−1 [Lt ] 1 a thin thermal boundary layer of thickness proportional to Gz−½ arises at each wall with an even thinner shear layer, detached from the wall and embedded in the thermal boundary layer, of thickness proportional to Gz−½(ln G)−1, coinciding with the region of maximum temperature (ln G)/b. The similarity variable is (Pe½y/x½) where Pe is the Péclet number (Vh/k) and y and x are measured away from and along (either) boundary wall. The analogous unsteady uniform flow solution is also given.