Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

It is shown that the surface wind drift in the ocean substantially reduces the maximum wave height ξx and wave orbital velocity that can be attained before breaking. If q is the magnitude of the surface drift at the point where the wave profile crosses the mean water level and c is the wave speed, then \[ \zeta_{\max} = \frac{c^2}{2g}\bigg(1-\frac{q}{c}\bigg)^2. \] Incipient breaking in a steady wave train is characterized by the occurrence of stagnation points at wave crests, but not necessarily by discontinuities in slope. After breaking, there is in the mean flow a stagnation point relative to the wave profile near the crest of the broken wave, on one side of which the water tumbles forward and behind which it recedes more smoothly to the rear. Some simple flow visualization studies indicate the general extent of the wake behind the breaking region.

An investigation is made of selective withdrawal of a linear stratified fluid from a line sink in a channel of depth d. The flow is characterized by a densimetric Froude number F = Q/Nd2, where Q is the discharge per unit width and N is the Väisälä frequency. The dynamics of establishment of flow are investigated theoretically. Analytic results are obtained from a linearized theory based on a systematic perturbation scheme for small values of F. These results lead to a proper identification of the successive arrival of ‘columnar disturbance modes’ as the mechanism responsible for the development of flow concentration in the withdrawal region. Viscous and diffusive effects are also examined. For larger times and higher discharges (higher values of F), nonlinear effects become important, and the full Navier-Stokes equations are now solved numerically by a finite-difference procedure in a ‘stretched’ co-ordinate system. The solutions indicate that the establishment of the steady flow field is due to the successive arrival and interaction of ‘columnar disturbance modes’. Steady-state solutions are also presented, and a similarity profile is obtained. Comparison of the theoretical findings with experimental results are presented in part 2. Agreement with experimental measurements is found to be excellent.

Experiments were conducted to examine the development and establishment of the withdrawal current in the selective withdrawal of a stratified fluid from a line sink. The experiments were performed in a Plexiglas channel 30·5 ft long 14 in. wide, and filled with a linearly stratified salt solution to a depth of 18 in. The line sink was located at mid-depth. The flow was symmetric about the middepth. Velocity measurements and flow visualization were obtained by neutrally buoyant liquid droplets and vertical dye lines. Density measurements were made by a salinity probe. The development of the upstream velocity field was found by measurement to be brought about by the successive arrival of ‘columnar disturbance modes’, discussed in part 1. Agreement with the theoretical predictions is excellent. A similarity profile of the steady-state horizontal velocity is obtained, and the Froude number based on the thickness of the flowing zone is found to approach a constant value in the essentially inviscid region. The results are compared with field data from the Tennessee Valley Authority (TVA) reservoirs.

Numerical solutions for laminar incompressible fluid flows past an abruptly started elliptic cylinder at 45° incidence are presented. Various finite-difference schemes for the stream-function/vorticity formulation are used and their merits briefly discussed. Almost steady-state solutions are obtained for Re = 15 and 30, whereas for Re = 200 a Kármán vortex street develops. The transient period from the start to the steady or quasi-steady state is investigated in terms of patterns of streamlines and lines of constant vorticity and drag, lift and moment coefficients.

Periodic dissolution patterns that result from the interaction of a soluble surface and an adjacent turbulent flow have been investigated experimentally and theoretically. They occur at a Reynolds number based on a characteristic wavelength and friction velocity of about 2200. Experimental results for the stable geometry, propagation, mass-transfer distribution, average mass-transfer correlation and friction factor are presented. An interpretation based upon the repetitive transitional nature of the flow structure is advanced to explain aspects of the origin and behaviour of this type of roughness.

A variational method involving minimization of the energy dissipation rate which was previously developed for transport in polymer systems is applied here to flow of a continuum solvent through a thin membrane. The membrane is represented by an array of spherical particles undergoing Brownian motion, subject to various interactions with one another and with the motion of the solvent. General upper bounds on the solvent permeability of the membrane are obtained in terms of equilibrium distribution functions, and applications of the method are illustrated for the case where membrane elements are confined to a plane. Calculations which treat all beads equivalently give permeability estimates whose dependence on the number n of beads per unit area of membrane has the form, at low n, \[ \kappa = (6\pi\eta n)^{-1}(1-\alpha n+\ldots), \] where η is the solvent viscosity and α is a constant. More-elaborate trials which allow the drag on a bead to be influenced by the distribution of other beads in the vicinity give the stronger bounding estimate \[ \kappa = (6\pi\eta n)^{-1}(1+\alpha^{\prime}n\ln n+\ldots). \] Comparison with a self-consistent field approach suggests that this logarithmic behaviour is the true first-order correction.

In a previous paper, the inviscid stability of a swirling far wake was investigated, and the superposition of a swirling flow on the axisymmetric wake was shown to be initially destabilizing, although all modes investigated eventually become more stable at sufficiently large swirl. The most unstable disturbances were non-axisymmetric modes with negative azimuthal wavenumber n representing helical wave paths opposite in sense to the wake rotation. The disturbance growth rate appeared to increase continuously with |n|, while all modes with |n| > 1 represented disturbances which are completely stable for the non-swirling wake. In the present analysis, both timewise and spacewise growth rates are calculated for the lowest three negative non-axisymmetric modes (n = −1, −2 and −3). Vortex intensity is characterized by a swirl parameter q proportional to the ratio of the maximum swirling velocity to the maximum axial velocity defect. The large wavenumbers associated with the disturbances at large |n| allow the n = −1 mode to have the minimum critical Reynolds number of 16 (q ≃ 0·40). The other two modes investigated have minimum Reynolds numbers on the neutral curve of 31 (n = −2, q = 0·60) and 57 (n = −3, q = 0·80). For each mode, the neutralstability curve is shown to shift rapidly towards infinite Reynolds numbers once the swirl becomes sufficiently large. Some of the most unstable swirling flows are shown to possess spacewise amplification factors almost ten times that for the most unstable wavenumber for the non-swirling wake at moderate Reynolds numbers.

The linear stability of a basic flow of two homogeneous inviscid incompressible fluids under the action of gravity is treated mathematically. In the basic state, one fluid is at rest below a horizontal plane z = 0; and the other flows above in the x direction, its speed varying slowly with the lateral co-ordinate y. The eigenvalue problem for normal modes is derived; its equation is a partial differential one, the co-ordinates y and z not being separable. The problem is solved approximately by taking the modes locally as if the basic velocity were independent of y, though the lateral wavenumber is allowed to vary slowly with y. This leads to an ordinary differential equation in y which is solved by the JWKB method. Detailed calculations are made for a parabolic profile, representing the blowing of air over water in a wide channel, and for other profiles.

This paper analyses the locomotion of a finite body propelling itself through a viscous fluid by means of travelling harmonic motions of its surface. The methods are developed with application to the propulsion of ciliated micro-organisms in mind. Provided that the metachronal wavelength (of the surface motions) is much smaller than the overall dimensions of the body, the flow can be divided into an oscillating-boundary-layer flow to which is matched an external complementary Stokes flow. The present paper employs the envelope model of fluid/cili a interaction to construct equations of motion for the oscillating boundary layer. The final solution for the propulsive velocity is obtained by application of the condition of zero total force on the self-propelling body; alternatively, if the organism is held at rest, the thrust it generates can be computed. Various optimum propulsive velocities for self-propelling bodies and optimum thrusts for restrained bodies are analysed in some simple examples. The results are compared with the relatively sparse observations for a number of microorganisms.