The recent analysis by Acrivos & Lo (1978) concerning the breakup of a long slender droplet in an axisymmetric straining motion is extended to the case of a two-dimensional hyperbolic flow. It is found that, although the cross-section of the droplet becomes significantly non-circular, the theoretical criterion for breakup is effectively the same as in the axisymmetric case. The theoretical predictions are in good agreement with the available experimental results.

The inviscid flow above an obstacle in slow transverse motion inside a rotating vessel is analysed to study the influence of the container depth on the basic steady flow structure. An asymptotic theory is presented for an arbitrarily small Rossby number Ro = U0/2Ωl under a fixed H = hRo/l (where Ω is the angular velocity of the container, U0 the obstacle velocity relative to the vessel, h the depth of the container, and l a body length measured transversely to the rotation axis). The equations when linearized for a thin obstacle or shallow topography take on the form of the inertial-wave equation; their solutions for non-vanishing H are obtained for obstacles of three-dimensional as well as ridge-like two-dimensional shapes. In all cases analysed, the solution possesses a bimodal structure, of which one part is column-like with a vorticity proportional to the body elevation (or ground topography). The other part is confined mainly to a region enclosing the body, extending a distance O(H½) upstream of the obstacle and behind a wedge-shaped caustic front at large distances; its contribution consists of lee waves similar to that discussed by Cheng (1977) for an infinite depth. The field associated with the lee waves is then biased on the downstream side, but there is little indication of any tendency to tilting in the sense of Hide, Ibbetson & Lighthill (1968).

Numerical integrations of the Navier-Stokes equations for flow past a smooth, three-dimensional, surface-mounted obstacle are presented. The variation of the flow with Reynolds number, and with geometric ratios such as the maximum slope of the obstacle, are investigated. The separated flow is investigated using visualizations of the surface-stress patterns, and also particle trajectories through the flow.

This work describes experiments in which a stably stratified salt solution is subjected to steady lateral temperature gradients. The criteria for the onset of layered convection are observed to agree with previously published stability analysis. Convective layers formed in this way are found to be statically unstable; they always merge two into one to form a new system. This process continues until the density jump at the solute interface between layers is larger than the density deficit produced by the side wall heating/cooling. The eventual stable system develops into well-mixed convective layers separated by sharp solute interfaces. The conditions for any subsequent layer intermixing and its effect on the lateral heat transfer across the slot, are also described.

A new theory is given for calculating the wave pattern and wave resistance of a ship moving at low Froude number F. It applies to ships of any width, either full-bodied or slender. In this theory, the waves travel along rays which start at source points, such as the bow and stern, on the water-line. They propagate with the speed of waves in deep water, but are also advected by the double body flow. This is the flow about the ship and its image in the undisturbed free surface. The phase of a wave at any point on a ray is the optical length of the ray from the source to that point. The amplitude is determined by an excitation coefficient, which determines its initial value, and by an integral along the ray. The total wave height at any point is the sum of the heights on all the rays through the point. The theory is incomplete because the excitation coefficients are known only for thin ships. As an illustration, the theory is applied to the thin ship case, and the results then agree with Michell's thin ship solution evaluated for F small.

A new class of ships, which we call streamlined ships, is introduced next. The usual linear free surface condition applies to the waves they produce. The ray theory is developed for these waves at low F, and it involves straight rays produced at all points on the rear half of the water-line. In addition, as an alternative to the ray theory, another method is presented for obtaining the waves at low F. It involves a Schrödinger-like equation in which distance along the ship's centre-line is the time-like co-ordinate.

For a steady plane parallel flow of an inviscid incompressible fluid of variable density under gravity, it is shown that the complex wave velocity for any unstable mode lies in a semi-ellipse whose major axis coincides with the diameter of Howard's semicircle, while its minor axis depends on the stratification.

Flow-visualization experiments for Stokes flow past thin spherical caps show separation from the rims of the caps in agreement with Dorrepaal, O'Neill & Ranger's (1976) theoretical prediction.

Changes in dune properties due to a sudden change in the water discharge are analysed. The transport of sediment is assumed to occur mainly as bed load and the square of the Froude number must be much smaller than unity. The theory is based on similarity in the bed-shear distribution close to the dune top just before and just after the change in the water discharge. The model is found to agree with laboratory experiments. Further, by use of this model, flow in a river where the water discharge oscillates weakly around a constant mean value has been treated. Hereby the variation in the phase differences between sediment transport, water depth and water discharge with the period of the oscillation is calculated. The results agree qualitatively with observations.

Two inequalities are proposed for the purpose of bounding from below the mean shear U′o(z) in turbulent channel flow. The first of these inequalities pertains to the energetics of the boundary layer, and the second pertains to the logarithmic form of the asymptote to Uo. These inequalities imply a maximum value of von Kármán's constant, the numerical value of which lies between the measurements of Laufer (1951) and the value obtained from bulk discharge measurements in a pipe. The formalism, which contains no adjustable parameters, is then applied to the turbulent thermal convection problem, and a lower bound for the mean temperature $\overline{T}_o(z)$ is obtained. The minimum value of the latter at large distances from the boundary is in fair agreement with Townsend's (1959) measurements. Although the proposed inequalities have not been deduced from the equations of motion, they provide facts which may be useful in the search for new variational formulations of theturbulent transport problem.

By means of similarity principles an analytical solution is constructed for the development of the linear flow field due to the instantaneous application of a constant point force in an infinite liquid. If the force is applied at the origin O and if ν denotes distance from O, ν denotes the coefficient of kinematic viscosity of the fluid and t the time from the application of the force, the solution constructed exhibits the following features. Initially the flow field set up has a dipole structure with centre at O and axis along the direction of the impressed force. At a station r this dipole structure persists so long as 4νt [Lt ] r2. In an axial cross-section the field lines form two sets of closed loops about two stagnation points in the equatorial plane. The stagnation points occur at r = 1.76(νt)½ and thus propagate to infinity with speed 0.88(ν/t)½. The steady state is reached algebraically.

The inviscid centrifugal instability of slowly varying flows is shown to be asymptotically associated with the lowest-order spatial dependence of the basic flow satisfying Rayleigh's criterion. This result requires special attention for basic flows which reverse their direction. At the instant of reversal the growth rate of the disturbance bifurcates and the choice of the proper branch requires that viscous effects be taken into account within a conveniently small neighbourhood of the branch point. Previous results by Rosenblat (1968) are shown to be incorrect. Such results were based on overlooking the need for viscous effects to be accounted for within a neighbourhood of the bifurcation point. This led to a wrong choice of the path to be followed at bifurcation.

This paper is a summary of the Second Bat-Sheva Seminar on magnetohydrodynamic (MHD) Flows and turbulence. It was held in the University of the Negev, Israel, on 28-31 March 1978, with 64 participants from 7 countries. Reviews and research papers were presented on the general theory of MHD, MHD duct flows (with emphasis on novel aspects such as non-uniform fields and fluid properties, bends, free-surface effects and longitudinal diffusion), two-phase flows (especially those likely to occur in a liquid-metal generator), turbulence and instabilities, and electrically driven flows (with new results presented for the theory of laminar and turbulent flows in induction furnaces, and for the theory of thermo-electrically driven flows in transverse magnetic fields). One day of the conference was devoted to turbulence, mainly without magnetic fields, with reviews and new results presented on the theory and measurements of coherent structures, intermittency at high Reynolds number, methods of calculating shear flows, and measurement techniques. The seminar was a strange mixture of people and topics, which produced some interesting papers and some useful discussion.

The abridged Langrangian-history direct-interaction (ALHDI) approximation (Kraichnan 1966) and the strain-based abridged Lagrangian-history direct-interaction (SBALHDI) approximation (Kraichnan & Herring 1978) are integrated numerically for isotropic turbulence in two and three dimensions and compared with data. At moderate Reynolds numbers in three dimensions, comparison with the computer simulations by Orszag & Patterson (1972) shows that the ALHDI gives numerically excessive energy transfer in the dissipation range while the SBALHDI approximation displays satisfactory accuracy in all ranges. In two dimensions, both approximations are in reasonable agreement with the simulations of Herring et al. (1974), the ALHDI approximation showing the better accuracy of the two at low wavenumbers. At high wavenumbers the SBALHDI approximation again transfers less energy than the ALHDI approximation but the effect is less marked than in three dimensions and the two curves straddle the data. High Reynolds number integrations of both approximations in three dimensions agree well with the tidal-channel inertial- and dissipationrange data of Grant, Stewart & Moilliet (1962), the SBALHDI approximation yielding a somewhat larger value of Kolmogorov's constant than the ALHDI approximation. The origin of the difference in straining efficiency between the two approximations at high wavenumbers and of the dependence of this difference on dimensionality is exhibited by application to the stretching of small scales of a convected passive scalar field. In three dimensions the SBALHDI approximation gives markedly larger values of the constant in Batchelor's (1959) k−1 spectrum range than the ALHDI approximation and is in better agreement with experiment. The SBALHDI values of Batchelor's constant satisfy Gibson's (1968) lower bound while the ALHDI values do not.