In the first part of the paper, a mixing layer of tanh y form is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanh y.

Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.

In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even in y, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.

It is shown that for a dynamical system admitting wave propagation modes (i.e. a wave-guide) the cross-power spectral density for stationary random fluctuations in the system will be dominated by the waves if they are lightly damped, the reason being that these can correlate over large distances of the order the inverse of the damping ratio. For a turbulent shear flow the wave propagation constant is obtained approximately from the solution of the Orr-Sommerfeld problem for the mean flow. Numerical calculations for a flat-plate boundary layer produce results for the streamwise dependence of the cross-power spectral density for the surface pressure fluctuations in good qualitative and quantitative agreement with measurements. An exception is the convection velocity for which the theory predicts a value that is somewhat too low.

The problem of the stability of a fluid with time-dependent heating has been investigated by Morton (1957), Lick (1965) and Foster (1965). Morton and Lick assumed that the rate of change of the temperature profile is small compared with the growth rate of the disturbances (quasi-static assumption). This assumption is invalid near the onset of instability (as defined by ∂/∂t = 0), and Foster has therefore used an initial-value approach.

In this paper the range of validity of the quasi-static assumption is discussed, and results of a time-scaled analysis and calculations based on this are compared with the work of Foster; the agreement is found to be good. We restrict our attention to a semi-infinite fluid initially at a constant temperature; at time t = 0 a temperature difference ΔT is applied at the (lower) horizontal boundary (case (A) of Foster).

When a solute is injected into a straight circular tube through which a solvent is in steady laminar flow it spreads out longitudinally under the combined effect of molecular diffusion and advection with the flow. Taylor (1953) showed that, provided there is no density difference between the solute and the solvent, the distribution of mean concentration satisfies a diffusion equation with a certain longitudinal diffusivity for large times and with respect to axes moving with the discharge velocity. In the present paper it is shown by a general argument that this statement remains true if the pipe is uniformly curved. An expression is given for the diffusivity, valid when the radius of curvature is sufficiently large. For all common liquids and most gases the diffusivity is reduced by the curvature. The rest of the paper deals with the effects of buoyancy forces caused by a density difference between the solute and the solvent, when the tube is horizontal. It is shown that in general a longitudinal diffusivity does not exist and an equation is derived that replaces the diffusion equation if the effects of buoyancy are small. A prediction from this equation is that buoyancy should not have a noticeable effect on the longitudinal dispersion for Péclet numbers near a certain value at which the two opposing influences of horizontal spreading due to gravity and lateral mixing due to secondary flow are in balance. This prediction is consistent with some observations made by Reejhsinghani, Gill & Barduhn (1966).

Using the grey gas approximation, the effect of radiative heat loss on axially symmetric flows is studied. Using an expansion procedure about the axis of symmetry, a numerical solution for the stagnation region is found taking the shock to be spherical. The results of this calculation are compared with the results of Lighthill's non-radiative constant density solution.

The investigation described here is an experimental study directed toward determining flow field and surface friction characteristics of a cylindrical wall jet. The flow configuration is obtained by placing a cylindrical rod along the axis of a converging nozzle. The flow field thus produced consists of a developing turbulent boundary layer co-existing with an outer fluid layer that mixes freely with quiescent surroundings. Direct measurements of fluid friction at the rod surface, performed with a hot-film element, revealed a significant effect of transverse surface curvature on the local friction factor. Velocity profiles at various axial stations exhibited similarity in the outer mixing layer, but not in the boundary layer adjacent to the surface. Representations of the velocity field in terms of law of the wall variables and defect-law variables revealed significant transverse curvature effects. The measured wall-jet velocity profiles could be satisfactorily represented in terms of a law of the wake, wherein the wake function depends on a transverse curvature parameter. In the outer mixing layer, the eddy viscosity data correspond closely to those of free jets.

Unsteady, large-amplitude motion of a viscous liquid in a long elastic tube is investigated theoretically and experimentally, in the context of physiological problems of blood flow in the larger arteries. Based on the assumptions of long wavelength and longitudinal tethering, a quasi-one-dimensional model is adopted, in which the tube wall moves only radially, and in which only longitudinal pressure gradients and fluid accelerations are important. The effects of fluid viscosity are treated for both laminar and turbulent flow. The governing non-linear equations are solved analytically in closed form by a perturbation expansion in the amplitude parameter, and, for comparison, by numerical integration of the characteristic curves. The two types of solution are compared with each other and with experimental data. Non-linear effects due to large amplitude motion are found to be not as large as those found in similar problems in gasdynamics and water waves.

The exact equations governing three-dimensional motion of an inviscid non-diffusive incompressible fluid stratified in density or of an inviscid non-diffusive gas stratified in entropy are given and briefly discussed.

The onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis. A Fourier series method is used to obtain the eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S, for a general set of boundary conditions. Numerical solutions are obtained for selected cases. Both oscillatory and monotonic instability are considered, but only the latter is treated in detail. The former can occur when a strongly stabilizing solvent gradient is opposed by a destablizing thermal gradient. When the same boundary equations are required to be satisfied by the temperature and concentration perturbations, the monotonic stability boundary curve in the (R, S)-plane is a straight line. Otherwise this curve is concave towards the origin. For certain combinations of boundary conditions the critical value of R does not depend on S (for some range of S) or vice versa. This situation pertains when the critical horizontal wave-number is zero.

A general discussion of the possibility and significance of convection at ‘zero’ wave-number (single convection cell) is presented in an appendix.

This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these waves, the density of the (incompressible) fluid varies only within a layer whose thickness h is much smaller than the total depth, and it is h rather than the total depth that must be considered as the fundamental scale against which wave amplitude and length are to be measured. Internal-wave motions supported by the oceanic thermocline appear to be the most promising field of practical application for the theory, although applications to atmospheric studies are also a possibility.

The waves in question differ in important respects from long waves of more familiar kinds, and in § 1 their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines, such as apply to internal waves in fluids of limited depth. A summary of some simple experiments is included at the end of § 1. Then, in § 2, the comparatively easy example of two-fluid systems is examined, again to illustrate principles and to prepare the way for the main analysis in § 3. This proceeds to a second stage of approximation in powers of wave amplitude, and its leading result is an equation (3·51) determining, for arbitrary specifications of the density distribution, the form of the streamlines in the layer of heterogeneous fluid. Periodic solutions of this equation are obtained, and their properties are discussed with regard to the interpretation of internal bores and wave-resistance phenomena. Solutions representing solitary waves are then obtained in the form \[ f(x) = a\lambda^2/(x^2+\lambda^2), \] where xis the horizontal co-ordinate and where aΛ = O(h2). (The latter relation between wave amplitude and length scale contrasts with the customary one, aΛ2 = O(h3)). The main analysis is developed with particular reference to systems in which the heterogeneous layer lies on a rigid horizontal bottom below an infinite expanse of homogeneous fluid; but in § 4 ways are given to apply the results to various other systems, including ones in which the heterogeneous layer is uppermost and is bounded by a free surface. Finally, in §5, three specific examples are treated: the density variation with depth is taken, respectively, to have a discontinuous, an exponential and a ‘tanh’ profile.

A new type of solitary wave motion in incompressible fluids of non-uniform density has been investigated experimentally and theoretically. If a fluid is stratified in such a manner that there are two layers of different density joined by a thin region in which the density varies continuously, this type of wave propagates along the density gradient region without change of shape. In contrast to previously known solitary waves, these disturbances can exist even if the fluid depth is infinite. The waves are described by an approximate solution of the inviscid equations of motion. The analysis, which is based on the assumption that the wavelength of the disturbance is large compared with the thickness, L, of the region in which the density is not constant, indicates that the propagation velocity, U, is characterized by the dimensionless group (gL/U2) In (ρ1/ρ2), where g is the gravitational acceleration and ρ is the density. The value of this group, which is dependent on the wave amplitude and the form of the density gradient, is of the order one. Experimentally determined propagation velocities and wave shapes serve to verify the theoretical model.

A unified picture of the linear dynamics of rotating fluids with given arbitrary stratification is presented. The range of stratification which lies outside the region of validity of both the theories of homogeneous fluids, $\sigma S < E^{\frac{2}{3}}$ and the strongly stratified fluids, σS > E½, is studied, where σS = vαgΔT/κΩ2L and E =v/ΩL2. The transition from one dynamics to the other is elucidated by a detailed study of the intermediate region E2/3 < σS < E½. It is shown that, within this intermediate stratification range, the dynamics differs from that of either extreme case, except in the neighbourhood of horizontal boundaries where Ekman layers are present. In particular the side wall boundary layer exhibits a triple structure and is made up of (i) a buoyancy sublayer of thickness (σS)−1/4E½ in which the viscous and buoyancy forces balance, (ii) an intermediate hydrostatic, baroclinic layer of thickness (σS)½ and (iii) an outer E¼-layer which is analogous to the one occurring in a homogeneous fluid. In the interior, the dynamics is mainly controlled by Ekman-layer suction, but displays hybrid features; in particular the dynamical fields can be decomposed into a ‘homogeneous component’ which satisfies the Taylor-Proudman theorem, and into a ‘stratified component’ which is baroclinic and which satisfies a thermal wind relation. In all regions the structure of the flow is displayed in detail.