Disturbances in a two-dimensional jet of a viscous incompressible fluid are examined for the case where the jet has a parabolic velocity distribution at the nozzle mouth. Partial differential equations for a finite amplitude disturbance are solved by use of the finite-difference approximation, in which case the jet is analogous to that excited externally. Numerical calculations for various disturbance amplitudes clarify the nonlinearity of the solution. Moreover, the behaviour of the finite disturbance is compared with the behaviour of an infinitesimal disturbance determined from a linearized stability theory. Streaklines calculated from the finite amplitude solution indicate ‘rolling-up’. The computations are carried out over the range of jet Reynolds numbers 500-2000.

The effects of substantial temperature and hence density changes on a low-speed ‘incompressible’ flow can be modelled by adopting Charles’ Law as one of the equations of state. It is found that planar radial inflows or outflows constitute a group of solutions which are self-similar for arbitrary temperature variations. When the temperature is written as a separable function of radius and polar angle, ordinary differential equations result. Permissible solutions include some with discontinuities in the temperature gradient across a radial line (streamline); this is a rough model of a diffusion flame and it is used to illustrate some features of a variable-density flow in a channel with radial walls in the presence of such a ‘flame’.

Exact analytical solutions are given for the situation in which temperature increases linearly with radius; no boundary layers appear for either outflow or inflow. Approximate analytical solutions are presented for the case of a relatively rapid inflow with temperature independent of the radius; a velocity boundary layer exists at the walls and in the neigh bourhood of the ‘flame’, although the latter is of small velocity amplitude.

The problem of small heave and pitch motions of a slender ship in shallow water including the effect of forward speed is analysed using the method of matched asymptotic expansions. Formulae valid to first order in slenderness are given for the added-mass and damping coefficients in terms of the frequency and subcritical Froude number.

In this paper we develop a separation of variables theory for solving problems of Stokes flow in wedge-shaped trenches bounded by radial lines and concentric circles centred at the vertex of the wedge. The theory leads to a set of Stokes flow eigenfunctions which in the full wedge reduce to the corner eigenfunctions studied by Dean & Montagnon (1949) and Moffatt (1964). Asymptotic formulae for the distribution of eigenvalues are derived, an adjoint system is defined and is used to develop an algorithm for the computation of the coefficients in an eigenfunction expansion of edge data prescribed on the circular boundaries. To illustrate the algorithm we find the motion and the shape of the free surface in a wedge-shaped cavity heated from its side.

The effects of inertia on the upstream-growing boundary layer over a finite horizontal flat plate of length b moving uniformly with speed U0 in a linearly stratified \[ [(dp/dy)_{-\infty}- = - \rho_0\beta)], \] viscous, non-diffusive fluid under the Boussinesq approximation are studied. The nonlinear inertia terms are linearized by the Oseen approximation, but no boundary-layer approximation is required. The flow is governed by two parameters, namely the internal Froude number Fr[= U0/(βgb2)½] and a parameter L3[= βgb3/U0ν], where $L^{\frac{3}{4}}$ is proportional to the ratio of boundary-layer thickness to plate length for the case Fr = 0. Large values of L3 and Fr2 = 0 correspond to the case of an upstream boundary layer. By increasing the Froude number gradually, a transition occurs from an upstream boundary layer accompanied by an upstream wake to a downstream boundary layer with a downstream wake. The upstream boundary layer and wake are characterized by a balance of viscous and buoyancy forces, whereas the downstream boundary layer and wake are characterized by a balance of inertia and viscous forces. In the so-called critical-boundary-layer case, Fr4L3 = O(1), inertia, viscous and buoyancy forces are all important and this boundary layer is accompanied by both upstream and downstream wakes. Complete transition occurs when Fr4L3 increases from 10·0 to 1000·0. The drag on the plate is also calculated.

Numerical solutions are presented for the detailed characteristics of the drift current and Langmuir circulation system produced according to a theory described in part 1 of this paper. The motions that develop are traced from an initially quiescent state, and the results are compared with field observations of currents in Langmuir circulations. Qualitative features of the phenomenon appear to be reproduced by the theory and, with the appropriate choice of an empirical parameter, the solutions seem to be quantitatively consistent with field data.

This paper is concerned with a general study of the modal structure for stratified viscous plane Couette flow with a constant buoyancy frequency. When the overall Richardson number Ri is zero, the velocity and temperature modes are distinct but as Ri is increased there is an intricate interaction between them. Some simple analytical results are obtained for large and small values of the Reynolds number and more detailed results are given for $Ri = 0, \frac{1}{8}, \frac{1}{4}$ and ½. The present theory would appear to be reasonably complete for 0 [les ] Ri [les ] ¼; for Ri > ¼, however, an important open question concerns the relationship between the limiting form of the viscous modes as the Reynolds number tends to infinity and the spectrum of internal gravity waves.

A statically stable stratification with buoyancy frequency N2(z) = z2 is found to cause large changes in the modal structure for viscous plane Couette flow (as compared with the case when N2(z) = 1) and it also has a strongly destabilizing effect on the flow. On minimizing with respect to both wavenumber and Richardson number, it is found that the flow is unstable if the Reynolds number is greater than about 183. A study of the Reynolds stress and vertical buoyancy flux shows that there is a large transfer of energy from the basic flow to the velocity disturbances and this is consistent with such a surprisingly low value of the minimum critical Reynolds number.

The structure of the turbulent shear flow over a propagating wave of fixed frequency is examined. The vertical and horizontal velocities were measured in a wind-wave facility at Stanford University. The structure of the wave-perturbed turbulence was found to depend significantly on the ratio of the local mean velocity to the wave speed. The results support the idea of ‘cat's-eye’ type flow about the mean critical height.

Singular perturbation techniques are used to calculate the migration velocity of a spherical particle sedimenting, at low Reynolds numbers, in a stagnant viscous fluid bounded by one or two infinite vertical plane walls. The method is then used to study the migration of a pair of spherical particles sedimenting either in unbounded fluid or in fluid bounded by a plane vertical wall. The migration phenomenon is studied experimentally by recording the trajectory of a spherical particle settling through a viscous fluid bounded by parallel vertical plane walls. Duct- to particle-diameter ratios in the range of 27 to 48 were used with the Reynolds number based on the particle radius being between 0·03 and 0·136.

In all cases the particle is observed to migrate away from the walls until it reaches an equilibrium position at the axis of the duct. The experimentally determined migration velocities agree well with those predicted by the present theory.

A short-wave asymptotic solution is derived for the problem of the diffraction of a surface wave train, in deep water, by a two-dimensional obstacle with plane vertical sides near its intersection with the free surface. Using matched asymptótic approximations, a detailed analysis is presented for the special case of a rectangular scatterer of depth a and width 2b, and the solution is then generalized to deal with a wider class of geometries. It is found that the transmission coefficient, at small wavelengths, has an exponentially small factor that depends on the depths of the plane sides, and an algebraically small factor that depends on the corner angles.