The nonlinear stability of gas bubbles in acoustic fields is studied using a multiple-scale type of expansion. In particular the development of a subharmonic or a synchronous perturbation to the flow is investigated. It is shown when an equilibrium non-spherical shape oscillation of a bubble is stable. If the amplitude of the sound field is ε then it is shown that subharmonic perturbations of order ε½ can exist and be stable. Furthermore synchronous perturbations of order ε can exist and be stable. It is shown that synchronous perturbations, unlike the subharmonic case where the bifurcation is symmetric, bifurcate transcritically when the driving frequency is varied and also undergo secondary bifurcations. It is further shown that, in certain cases, the latter properties of the synchronous modes cause the flow to exhibit a hysteresis phenomenon when the driving frequency is varied.

When pure solvent is separated from a solution of non-zero concentration Cb by a semi-permeable membrane, permeable to solvent (water) but not to solute, water flows osmotically across the membrane towards the solution. Its velocity J is given by J = PΔC, where P is a constant and ΔC is the concentration difference across the membrane. Because the osmotic flow advects solute away from the membrane, ΔC is usually less than Cb, by a factor γ which depends on the thickness of and flow in a concentration boundary layer. In this paper the layer is analysed on the assumption that the stirring motions in the bulk solution, which counter the osmotic advection, can be represented as two-dimensional stagnation-point flow. The steady-state results are compared with those of the standard physiological model in which the layer has a given thickness δ and the osmotic advection is countered only by diffusion. It turns out that the standard theory, although mechanistically inadequate, accurately predicts the value of γ over a wide range of values of the governing parameter β = PCbδ/D (where D is the solute diffusivity) if δ is given by \[ \delta = 1.59\bigg(\frac{D}{\nu}\bigg)^{\frac{1}{3}}\bigg(\frac{\nu}{\alpha}\bigg)^{\frac{1}{2}}, \] where ν is the kinematic viscosity of the fluid and α is the stirring parameter. The final approach to the steady state is also analysed, and it is shown to be achieved in a time scale (D/ν)1/3/αk′ where k′ is a dimensionless number whose dependence on β is computed. Moreover, if β exceeds a certain critical value (≈ 10), the approach to the steady state is not monotonic but takes the form of a damped oscillation (in practice, however, β is unlikely to rise significantly above 1). The theory is extended to the case where the solute concentration is non-zero on both sides of the membrane and in that case it is shown that J is bounded as β → ∞.

The decay of a jet discharging from a circular nozzle parallel to and displaced from a solid surface is investigated under conditions where the transitional process from circular-jet flow to oblate wall-jet flow begins in the initial, transition or self-preserving regions of the original jet. The influence of displacement of the nozzle from the plane on the developed three-dimensional wall jet downstream is demonstrated and it is found that the transitional interaction with the plane is more extended when the plane interacts first in the initial zone of the circular jet. Measurements of turbulence and Reynolds stress show the transverse mixing parallel to the plane to exceed that perpendicular to the plane, and are generally consistent with the spreading rates in these two directions, the ratio of which approaches 8·5 at large distances from the nozzle. It is shown that the interaction between the plane and jet involves a relatively large-scale coherent motion in which components of velocity directed towards or away from the surface are associated with outflow or inflow along the surface. This motion is more extended in the direction parallel to the surface and provides a mechanism for the increases in mixing rate in the direction parallel to the plane.