Among the parameters which determine the erosion damage sustained by the walls of a nozzle, in which a mixture of gas and particles is flowing is the speed attained by the particle before collision with the wall surface. This work is concerned with the determination of the particle velocity, and a number of relationships are given from which the variation in particle velocity can be obtained for a variety of gas conditions. The changes of state and velocity of the gas, occasioned by the interchange of heat and work between the gas and the particles are dependent on the ratio of the mass flow rate of particles to the mass flow rate of gas. It is shown that if this ratio is small the particle velocity may be obtained without serious error by assuming that the gas conditions are not affected by the presence of particles. Figures for the limiting value of this ratio for certain flows are given. The effects of particle size, density and initial relative velocity are investigated analytically and experimentally.

The stability of the flow of two layers of viscous liquids down an incline is investigated. The problem is governed by two competing long-wavelength modes associated with the surfaces. One mode, calling it the first mode, is faster than the second one. When the two layers have the same coefficient of dynamic viscosity, the second mode was found earlier by the author to be the governing mode for most cases. The situation is greatly altered when viscosity is not the same for the two layers. The second mode is dramatically stabilized for the range of viscosity ratio, m, less than unity, and the first mode is now generally the governing mode in that range. The overall effect is stabilizing compared with m = 1.

A relative stability index is also introduced to compare the result with that of the homogeneous case. It is found that the presence of the upper layer is generally destabilizing compared with that of a homogeneous fluid of the same total depth.

A two-dimensional, semi-infinite, stratified shear flow in which the upstream dynamic pressure and density gradient are constant (Long's model) is considered. A complete set of lee-wave functions, each of which satisfies the condition of no upstream reflexion, is determined in polar co-ordinates. These functions are used to determine the lee-wave field produced by, and the consequent drag on, a semicircular obstacle as functions of the Froude number within the range of stable flow. The Green's function (point-source solution) for the half-space also is determined in polar co-ordinates.

Chandrasekhar's (1961) solution to the eigenvalue equation arising from the Kelvin-Helmholtz stability problem for a rotating fluid is shown to be incorrect. The unstable modes are correctly enumerated with the aid of Cauchy's principle of the argument. Various previously published solutions using Chandrasekhar's analysis are corrected and extended.

The formation and transient dynamics of Stewartson layers are examined herein. The results are applied to a transient rotating fluid flow for which (i) Stewartson layers are presumed to be important, and (ii) the Ekman layers are found to be non-divergent, and hence unable to induce the crucial secondary circulation occurring in the spin-up process. The steady state is nevertheless reached after a spin-up time.

A third-order theory has been developed to study capillary instability of a liquid jet. The result shows that the asymmetrical development of an initially sinusoidal wave is a non-linear effect with generation of higher harmonics as well as feedback into the fundamental. The growth of the surface wave is found to depend explicitly on the dimensionless initial amplitude of the disturbance and the dimensionless wave-number k of the wave. For the same initial disturbance, the wave is found to have a maximum growth rate at k = 0·7 in agreement with the linearized theory. For the same wave-number, the growth is proportional to the initial amplitude of the disturbance. The cut-off wave-number and the fundamental frequency (or growth rate for the unstable case) of the wave for a given k are found to be different from the linearized theory. Furthermore, at the cut-off wave-number, the present theory shows the disturbance experiences a growth which is proportional to t2. The excellent agreement between Donnelly & Glaberson's experiment and Rayleigh's linearized theory is found to be due to their method of measurement.

The current I flowing in the steady electrocapillary discharge of saturated aqueous and alcohol solutions of sodium chloride was found to be proportional to e−α/V, where V is the applied potential and α is a constant under fixed experimental conditions.

An analysis of the structure of the wakes and waves in steady compressible magnetohydrodynamics is presented. No restriction is made on the equation of state of the gas or on the ratios of the various dissipative parameters. An asymptotic solution is obtained which furnishes directly the flow far from a body and which may be used in the construction of the entire flow field. The non-dissipative solutions are obtained as a non-uniform limit for vanishing dissipation; no matter how small the dissipation, one can go far enough from the origin that the flow is essentially dissipative. For non-aligned fields the wave pattern consists of a downstream wake and either two or four standing waves, depending on the flow regime. For aligned fields, two of these waves become wakes, so that the wake is a superposition of three structured layers, with either all downstream or two downstream and one up-stream. It is found that the non-dissipative limit of the wake is non-unique for the aligned fields case. Different limits are obtained depending on how the various dissipative parameters vanish.

An analysis of the mixing of a turbulent half-jet along a curved streamline is presented. Equations of motion referred to streamline co-ordinates are simplified by boundary-layer approximations and integrated under the assumption of similarity. A curved potential flow is dissipated by turbulence in the mixing region, resulting in a maximum value in the velocity distribution across the stream for convex flows, while the distribution is monotonic for concave flows. Pressure distributions across the stream are also presented.

A theoretical description is given for infinitesimal non-axisymmetric disturbances of a shear flow caused by differential rotation. It is assumed that the deviations from the state of rigid rotation are small corresponding to the case of small Rossby number. The shear flow becomes unstable at a finite critical value of the Rossby number, at which the inertial forces overcome the friction in the Ekman boundary layers and the constraint imposed by the variation in depth of the container. The governing equation is closely related to the Rayleigh stability equation in the theory of hydrodynamic stability of plane parallel flow. Experimental observations by R. Hide show reasonable agreement with the theoretical predictions.

A shear-acceleration wave is a propagating singular surface across which the velocity vector and the normal component of the acceleration are continuous, while the tangential component $\dot{v}$ of the acceleration suffers a jump discontinuity [$\dot{v}$]. We here discuss plane-rectilinear shearing flows of general, non-linear, incompressible simple fluids with fading memory. Working within the framework of such planar motions, we derive a general and exact formula for the time-dependence of the amplitude a = [$\dot{v}$] of a shear-acceleration wave propagating into a region undergoing a steady but not necessarily homogeneous shearing flow. When this expression is specialized to the case in which the velocity gradient is constant in space ahead of the wave, it assumes a form familiar in the theory of longitudinal acceleration waves in compressible materials with fading memory (cf., e.g., Coleman & Gurtin 1965, equation (4.12)).

In earlier work (1965) we observed that a planar shear-acceleration wave cannot grow in amplitude if it is propagating into a fluid in a state of equilibrium. It is clear from our present results that if the fluid ahead of the wave is being sheared, |a(t)| not only increases, but can approach infinity in a finite time, provided a(0) is of proper sign and |a(0)| exceeds a certain critical amplitude. We expect this critical amplitude to decrease as the rate of shear ahead of the wave is increased.

The first-order laminar shear layer between two parallel streams has been calculated exactly for compressible and incompressible two-dimensional flow. Turbulent flow has been included by using Prandtl's hypothesis for the eddy viscosity. Results are presented for a wide range of the appropriate parameters. The problem is solved by writing the momentum and energy equations as a pair of coupled integral equations in Crocco variables; these are solved by the method of successive approximation after using a simple transformation to weaken the singularity at the outer edges of the boundary layer. This approach yields uniquely the shear stress and temperature as functions of the tangential velocity u. The so-called third boundary condition, derived by Ting (1959), is then readily satisfied in evaluating the transverse component of velocity v by quadrature. Experimental results are presented in the turbulent incompressible case. Good agreement with the exact theoretical results is obtained when one stream is much faster than the other, but this falls off as the speeds of the streams tend to equalize.

The motions resulting when a linear, stable salt gradient is heated uniformly and at a steady rate from below are investigated theoretically and by laboratory experiment. A convecting, growing layer is first formed whose depth, temperature and salinity differences from the fluid above, are all increasing as t½. The way in which these quantities depend on the salinity gradient and heating rate is also predicted, and verified experimentally. A stability criterion is then developed which describes the breakdown of the diffusive boundary layer ahead of the advancing front, and leads to an expression for the thickness of the bottom layer when a second layer forms above it. The predicted form of dependence of layer thickness on the given parameters is again borne out by the experiments.

An experimental study of the effects of large longitudinal magnetic fields on the pipe flow of mercury has been made. The experiments were designed to provide large magnetic interaction parameters and thereby to have an important effect upon transition from laminar to turbulent flow and upon the structure of any turbulence. The friction factor and the transition Reynolds number were measured as functions of the magnetic field strength. The friction factor was obtained from accurate measurement of the pressure gradient along the tube. The transition Reynolds number was determined from measurements of the intermittency factor obtained using a hot-wire anemometer.

The results indicate that all relevant stability theories vastly over-estimate the stabilizing effect of the field. The results also indicate that viscosity plays a critical role in delaying transition and in reducing the friction factor of fully developed flows. This, in turn, leads to the important conclusion that the magnetic field has an important effect on the generation of new turbulence, as well as upon the damping of already generated turbulence.

The Galerkin method is applied in a new way to problems of stationary and oscillatory convective instability. By retaining the time derivatives in the equations rather than assuming an exponential time-dependence, the exact solution is approximated by the solution to a set of ordinary differential equations in time. Computations are simplified because the stability of this set of equations can be determined without finding the detailed solution. Furthermore, both stationary and oscillatory instability can be studied by means of the same trial functions. Previous studies which have treated only stationary instability by the Galerkin method can now be extended easily to include oscillatory instability. The method is illustrated for convective instability of a rotating fluid layer transferring heat.

Estimates of the eigenvalues C belonging to the manifold of solutions of the Orr-Sommerfeld equation are constructed by application of elementary isoperimetric inequalities. The inequalities also lead to a considerable improvement on the estimate of (αR) regions of linear stability given by Synge.