An experimental study is made of nonlinear interactions in a laminar free shear layer. Two disturbances (f1 and f2), excited by sound, amplify and grow independently for small amplitudes. At larger amplitudes the disturbances interact to generate fluctuations of sum and difference frequencies (f2 ± f1). Harmonics and subharmonics of f1 and f2 are also generated and all fluctuations interact to generate additional fluctuations of the form (nf2/m) ± (pf1/q); n, p = 1,2,3,…, m, q = 1,2. Nonlinear mode competition suppresses the growth of f1 or f2, depending on their relative amplitudes, and contributes to finite amplitude equilibration. An upper bound on the modal integral of total u′r.m.s.2 fluctuation energy is found. Fluctuation energy tends to be distributed among all possible frequency components, and its upper bound does not increase as the number of components increases.

The problem of forced acoustic oscillations in a pipe is studied theoretically. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed to a completely open mouth at the other end are considered. All these boundary conditions are modelled by two parameters: a length correction and a reflexion coefficient equivalent to the acoustic impedance.

The linear theory predicts large amplitudes near resonance and nonlinear effects become crucially important. By expanding the equations of motion in a series in the Mach number, both the amplitude and wave form of the oscillation are predicted there.

In both the open- and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed-end case and to the cube root for the open end.

We consider the flow in a differentially heated vertical slot filled with a stably stratified solution. The stability of the flow driven by the differential heating is investigated in the limits of small but finite amplitude disturbances and very large solute Rayleigh number RS = gβ(∂Sa/∂z)D4/KSv. If the Schmidt number H = KT/KS is of order 1, the growth of an initial perturbation at the neutral point is balanced by horizontal advection of solute and heat, and a steady equilibration amplitude is attained. The Nusselt number is independent of all fluid properties and is directly proportional to the Rayleigh number excess ε = (Ra – Rac)/Rac. If H is much greater than |RS|⅙, or if the disturbance wave-number is slightly less than the critical wavenumber, subcritical instabilities are possible. In particular a resonant instability is possible. These theoretical predictions are consistent with previous experimental results and with the laboratory results described in this paper. In the experiments we find that the mixing of the initial sugar gradient is accomplished by convection cells which undergo transitions to larger wavelengths. The breakdown of the interfaces between convection cells is described.

An analysis is made of the stability of an unsteady basic flow of a conducting fluid in the presence of a parallel magnetic field. The particular profile investigated is the classical Kelvin–Helmholtz profile modified by the addition of an oscillatory component. Two cases are considered in detail: that of a perfectly conducting fluid and that of a poorly conducting fluid. The investigation leads, in both cases, to an equation of the Hill type. It is concluded that the magnetic field has a stabilizing influence but is nevertheless unable to suppress the Kelvin–Helmholtz instability in an unsteady (basic) flow.

The flow induced by impulsively starting the inner cylinder in a Couette flow apparatus is investigated by using a nonlinear analysis. Explicit finite-difference approximations are used to solve the Navier–Stokes equations for axisymmetric flows. Random small perturbations are distributed initially and periodic boundary conditions are applied in the axial direction over a length which, in general, is chosen to be the critical wavelength observed experimentally. Simultaneous occurrence of Taylor vortices is obtained at supercritical Reynolds numbers. The development of streamlines, perturbation velocity components and the kinetic energy of the perturbations is examined in detail. Many salient features of the physical flow are observed in the numerical experiments.

Equations for the development of streamwise (or secondary) vorticity, for stationary and rotating systems, are derived directly from the vector equation for vorticity, using intrinsic co-ordinates. This approach places emphasis on the coupled equations for the vorticity components parallel and normal to the streamline. The equations derived are more general than those hitherto available, being valid for compressible, stratified and viscous flow. Some interpretation is given of the physical mechanism causing secondary flow.

In this paper, we obtain the first-order necessary optimality conditions of an optimal control problem for a distributed parameter system with geometric control, namely, the minimum-drag problem in Stokes flow (flow at a very low Reynolds number). We find that the unit-volume body with smallest drag must be such that the magnitude of the normal derivative of the velocity of the fluid is constant on the boundary of the body. In a three-dimensional uniform flow, this condition implies that the body with minimum drag has the shape of a pointed body similar in general shape to a prolate spheroid but with some differences including conical front and rear ends of angle 120°.

A short-wave asymptotic analysis is undertaken for problems concerned with the radiation and scattering of surface waves by a cylinder whose cross-section S intersects the free surface normally. It is assumed that S is locally smooth and convex at the two intersection points with the fluid, which may be of infinite or finite depth. For both the scattering and radiation problem, a matched expansion technique is used to provide asymptotic estimates, in terms of relatively simple wave-free limit potentials, for the amplitudes of the surface wave trains that propagate from S. Explicit details are given for some particular geometries, confirming and extending earlier results. The method can, in principle, be extended to deal with other geometries.

The radiation properties of partially immersed three-dimensional bodies, in time-periodic motion, are examined in the short-wave asymptotic limit ε → 0, where ε is a non-dimensional wavelength. The method of matched expansions is used to specify an outer approximation, away from the surface wave region, and an inner approximation where the potential, in the vicinity of the obstacle and free surface, depends only on the local geometry. Finally, the radially spreading surface wave field is estimated by ray-theory arguments. Explicit details are given for the heaving and rolling of a circular dock and for the heaving motion of a hemisphere. Some speculations are made regarding the scattering properties of such obstacles.

Existing analyses of the very low Reynolds number Stokes jet, e.g. Birkhoff & Zarantonello (1957) and Förste (1963), have been confined to a single-component fluid satisfying the usual zero-slip boundary conditions at a solid surface. In contrast, the low Reynolds number biological jets which emerge from the pores and channels at the secreting surfaces of numerous human, animal and insect organs entail the movement of water and solute subject to novel boundary conditions that arise from the local osmotic driving forces at the secreting surfaces. These boundary conditions introduce a nonlinear coupling between the fluid momentum and solute conservation equations. This paper first discusses the fundamental dimensionless groups and length scales that characterize these biological jet flows and then examines in detail, using the technique of matched inner and outer expansions, one flow situation important in epithelial membrane transport, namely, the two-dimensional mixing of an inhomogeneous jet with a quiescent outer bathing solution which is bounded by a semi-permeable mem brane in the plane of the exit. The paper concludes with a discussion of other physiologically relevant problems which arise from different orderings of the length scales, and different overall geometrical configurations and flow conditions.

The forces acting on an elastic particle suspended in a shear field, and moving relative to it, are found for the case in which there are small deformations from an initially spherical shape. The deformation is the result of the viscous stresses acting on the particle. Of principal interest is that there is a component of the force perpendicular to the free-stream direction, so that the particle will migrate across the undisturbed streamlines.

The formalism of transport theory is adapted to a general description of bubble populations in a moving fluid. The bubble distribution, as a function of position, velocity, radius and time, satisfies a Boltzmann-type transport equation that is derived and then formally solved by the method of characteristics. In order to apply this new analytical tool to the specific problem of gas bubble transport in the upper ocean, an ocean model and a bubble dynamics model must be chosen. For the purpose of illustration, explicit solutions are written, for distributed sources in a stationary ocean with simple expressions for bubble gas diffusion and drag. Calculated results clarify the relations between observed bubble distributions at sea, proposed bubble source mechanisms and known models of single-bubble dynamics.