After a brief description of the Milne generalization of the Galilean invariance group for the space–time of Newtonian kinematics, it is shown how the generalized Eulerian dynamical equations for the motion of a multiconstituent perfect (nonconducting) fluid can be expressed in terms of interior products of current 4-vectors with exterior derivatives of the appropriate 4-momentum 1-forms (whose role is central in this approach) in a fully covariant standard form whose structure is identical in the Newtonian case to that of the corresponding equation for the case of (special or general) relativistic perfect fluid mechanics. In addition to space–time covariance, this standard form exhibits chemical covariance in the sense that it is manifestly invariant under redefinition of the number densities of the independent conserved chemical constituents in terms of linear combinations of each other. It is shown how, in the strictly conservative case when no chemical reactions occur, this standard form, can be used (via the construction of suitably generalized Clebsch potentials) for setting up an Eulerian (fixed-point) variation principle in a form that is simultaneously valid for both Newtonian and relativistic cases.

When a steadily moving load is applied to a floating ice plate, the disturbance will generally approach a steady state (relative to the load) as time t → ∞. However, for certain ‘critical’ load speeds the disturbance may grow continuously with time, which represents some danger to vehicles driving on ice. To understand this phenomenon and the overall time development of the ice response, this paper analyses the problem of an impulsively applied, concentrated line load on a perfectly elastic homogeneous floating ice plate. An exact expression for the ice deflection is derived, and then interpreted by means of asymptotic expansions for large t in the vicinity of the source. The spatial development of the disturbance is analysed by considering asymptotic expansions as t → ∞ near an observer moving away from the load. Theoretical results are compared with field measurements, and some hitherto unexplained features can be understood.

The generation of instability waves in free shear layers is investigated theoretically. The model assumes an infinitesimally thin shear layer shed from a semi-infinite plate which is exposed to sound excitation. For this model the forced instability waves are calculated. The shear-layer excitation by a source farther away from the plate edge in the downstream direction is very weak while upstream from the plate edge the excitation is relatively efficient. A special solution is given for the source at the plate edge. Any type of source farther away from the plate edge produces a parabolic pressure field near the edge. For this latter, fairly general case, a reference quantity is found for the magnitude of the excited instability waves. The theory is then extended to two streams, one on each side of the shear layer, having different velocities and densities. Furthermore, the excitation of a shear layer in a channel is calculated. The limitations to the theory and some aspects related to experiments are discussed. In particular, for a comparison with measurements, numerical computations of the velocity field outside the shear layer have been carried out.

The acoustical excitation of shear layers is investigated experimentally. Acoustical excitation causes, for example, the so-called ‘orderly structures’ in shear layers and jets. Also, the deviations in the spreading rate between different turbulent-shear-layer experiments are due to the same excitation mechanism. The present investigations focus on measurements in the linear interaction region close to the edge from which the shear layer is shed. We report on two sets of experiments (Houston 1981 and Berlin 1983/84). The measurements have been carried out with laminar shear layers in air using hot-wire anemometers and microphones. The agreement between these measurements and the theory is good. Even details of the fluctuating flow field correspond to theoretical predictions, such as the local occurrence of negative phase speeds.

An investigation of wave processes in liquids with vapour bubbles with interphase heat and mass transfer is presented. A single-velocity two-pressure model is used which takes into account both the liquid radial inertia due to medium volume changes, and the temperature distribution around the bubbles. An analysis of the microscopic fields of physical parameters is aimed at closing the system of equations for averaged characteristics. The original system of differential equations of the model is modified to a form suitable for numerical integration. An elliptic equation is obtained to determine the field of the mixture average pressure at an arbitrary time through the known fields of the remaining quantities. The existence of the steady structure of shock waves, either monotonic or oscillatory, is proved. The effect of the initial conditions, shock strength, volume fraction, and dispersity of the vapour phase and of the thermophysical properties of the phases on shock-wave structure and relaxation time is studied. The influence of nonlinear, dispersion and dissipative effects on the wave evolution is also investigated. The shock adiabat for reflected waves is analysed. The results obtained have proved that the interphase heat and mass transfer determined by the thermal diffusivity of the liquid greatly influences the wave structure. The possible enhancement of disturbances in the region of their initiation is shown. The model has been tested for suitability and the results of calculations have been compared with experimental data.

Luke's (1967) variational formulation for surface waves is extended to incorporate the motion of a wavemaker and applied to the cross-wave problem. Whitham's average-Lagrangian method then is invoked to obtain the evolution equations for the slowly varying complex amplitude of the parametrically excited cross-wave that is associated with symmetric excitation of standing waves in a rectangular tank of width π/k, length l and depth d for which kl = O(1) and kd [Gt ] 1. These evolution equations are Hamiltonian and isomorphic to those for parametric excitation of surface waves in a cylinder that is subjected to a vertical oscillation, for which phase-plane trajectories, stability criteria and the effects of damping are known (Miles 1984a). The formulation and results differ from those of Garrett (1970) in consequence of his linearization of the boundary condition at the wavemaker and his neglect of self-interaction of the cross-waves in the free-surface conditions (although Garrett does incorporate self-interaction in his calculation of the equilibrium amplitude of the cross-waves). These differences have only a small effect on the criterion for the stability of plane waves, but the self-interaction is crucial for the determination of the stability of the cross-waves.

The variational formulation of the nonlinear wavemaker problem, previously applied (Miles 1988) to cross-waves in a short tank, is extended to allow for slow spatial, as well as slow temporal, variation of cross-waves in a long tank. The resulting evolution equations for the envelope of the cross-waves are equivalent to those derived by Jones (1984) and may be combined to obtain a cubic Schrödinger equation in a semi-infinite domain. The corresponding criterion for the stability of plane waves (i.e. for the temporal decay of cross-waves) agrees with Jones but differs from Mahony (1972). Weak damping is incorporated, and those stationary envelopes that are evanescent at large distances from the wavemaker are determined through analytical approximations and numerical integration and compared with the experimental observations of Barnard & Pritchard (1972) and the numerical calculations of Lichter & Chen (1987). These comparisons suggest that stationary envelopes with either no or one maximum are stable for sufficiently small amplitudes (solutions with multiple maxima may be stable but more difficult to attain) and evolve into limit cycles for somewhat larger amplitudes, but the analytical question of stability remains open.

A recent theory of Bragg scattering of surface waves by sinusoidal sandbars on a seabed is applied to three cases not examined heretofore: (1) oblique incidence on a strip of infinitely long bars, (2) oblique incidence on the corner of a bar field and (3) seabed with a mean slope. While the Bragg mechanism has been studied previously for sandbars present on many shorelines, it can be a basis for breakwaters where the soil is not strong enough to support a single massive breakwater.

We discuss our present knowledge of the flow and stability of helium II between concentric cylinders. The flow problem for helium II leads us to consider the formation of quantized vortices in the uniform rotation of helium II in an open bucket as well as quantized circulation states and vortices in a rotating annulus. We then consider how to treat the first appearance of vortices in the presence of shear, which allows us to characterize the basic flow which must be examined for stability. The results suggest an explanation for heretofore unexplained experiments. Future directions for research on the stability of helium II are suggested.

Particle dispersion in an axisymmetric jet is analysed numerically by following particle trajectories in a jet flow simulated by discrete vortex rings. Important global and local flow quantities reported in experimental measurements are successfully simulated by this method.

The particle dispersion results demonstrate that the extent of particle dispersion depends strongly on γτ, the ratio of particle aerodynamic response time to the characteristic time of the jet flow. Particles with relatively small γτ values are dispersed at approximately the fluid dispersion rate. Particles with large γτ values are dispersed less than the fluid. Particles at intermediate values of γτ may be dispersed faster than the fluid and actually be flung outside the fluid mixing region of the jet. This result is in agreement with some previous experimental observations. As a consequence of this analysis, it is suggested that there exists a specific range of intermediate γτ at which optimal dispersion of particles in the turbulent mixing layer of a free jet may be achieved.

The compressible laminar boundary-layer flows of a dilute gas-particle mixture over a semi-infinite flat plate are investigated analytically. The governing equations are presented in a general form where more reasonable relations for the two-phase interaction and the gas viscosity are included. The detailed flow structures of the gas and particle phases are given in three distinct regions: the large-slip region near the leading edge, the moderate-slip region and the small-slip region far downstream. The asymptotic solutions for the two limiting regions are obtained by using a series-expansion method. The finite-difference solutions along the whole length of the plate are obtained by using implicit four-point and six-point schemes. The results from these two methods are compared and very good agreement is achieved. The characteristic quantities of the boundary layer are calculated and the effects on the flow produced by the particles are discussed. It is found that in the case of laminar boundary-layer flows, the skin friction and wall heat-transfer are higher and the displacement thickness is lower than in the pure-gas case alone. The results indicate that the Stokes-interaction relation is reasonable qualitatively but not correct quantitatively and a relevant non-Stokes relation of the interaction between the two phases should be specified when the particle Reynolds number is higher than unity.

The results of an experimental investigation on the flow of a non-Newtonian fluid between rotating, parallel disks are described in this paper. These results are qualitatively different from those exhibited by linearly viscous fluids in that a narrow layer of exceedingly high velocity gradients appears in the non-Newtonian fluid.

A numerical and experimental study is reported of natural convection in a vertical rectangular fluid enclosure that is partially filled with a fluid-saturated porous medium. Velocities, stresses, temperatures, and heat fluxes are assumed to be continuous across the fluid/porous-medium interface, and the conservation equations for the fluid and the porous regions are combined into a single set of equations for numerical solution. Thermocouples as well as a Mach-Zehnder interferometer are used to measure temperature distributions and infer fluid flow patterns within the fluid and the porous medium. For various test cells, porous-layer configurations and fluid-solid combinations, the model predictions show excellent agreement with the experimental measurements. It is found that the intensity of natural convection is always much stronger in the fluid regions, while the amount of fluid penetrating into the porous medium increases with increasing Darcy and Rayleigh numbers. The degree of penetration of fluid into the porous medium depends strongly on the porous-layer geometry and is less for a horizontal porous layer occupying the lower half of the test cell. If penetration takes place, the flow patterns in the fluid regions are significantly altered and the streamlines show cusps at the fluid/porous-medium interfaces. For a high effective-thermal-conductivity porous medium, natural convection in the medium is suppressed, while the isotherms bend sharply at the fluid/porous-medium interface.

The motion of a flat body towards a parallel plane surface in incompressible fluid is considered both in the presence and absence of an applied force for a non-vanishing initial velocity. In the inviscid limit, a first integral of the equations is obtained and analytic solutions are presented for the cases of finite body inertia with zero applied force and finite applied force with negligible body inertia. In the former case when the ratio of body inertia to fluid inertia is large, a singular behaviour is observed in the arrest of the body before impact wherein the time-dependent pressure and radial velocity of the fluid exhibit a sharp peak and there is a large transfer of kinetic energy from the body to the thin fluid layer. For a real fluid, a general procedure is described to obtain solutions at arbitrary Reynolds number for naturally occurring initial velocity conditions. Solutions to the full Navier-Stokes equations are obtained for an arbitrary Reynolds number based on gap height which are valid provided the flow remains laminar and the gap height is small. In general, the equations of motion of the body and fluid are both dynamically and kinematically coupled. The dynamic coupling, however, is removed when the body inertia is neglected. In particular, the cases of hydrodynamic arrest with zero applied force, and draining of the fluid under a constant applied force are considered. The natural initial conditions lead to a new exact similarity solution of the Navier-Stokes equations which is valid for an instantaneous time-dependent Re based on gap height of greater than approximately 100, wherein the top and bottom boundary layers remain distinct. The longer time portions of the motion and the final arrest are described by a numerical calculation for intermediate Reynolds number and a low-Reynolds-number analysis.

An unsteady lifting-line theory is presented for a general motion of a wing of high aspect ratio. Our matched-asymptotic-expansions analysis parallels that of Van Dyke (1964) in his solution for the steady lifting line, but is complicated by the shedding of transverse vortices associated with variation of circulation with time. The principal result is an expression for the downwash due to three-dimensional effects. Numerical calculations are presented for a wing of elliptic planform following a curved path.

Some simple relations between the Lagrangian moments and cumulants in a steady finite-amplitude gravity wave on deep water are here generalized to water of finite depth. The first three Lagrangian moments are shown to be given in term of the mass-transport velocity U at the free surface, the potential and kinetic energy densities V and T, and the mean-square particle velocity $\overline{u^2_B}$ on the bottom.

A simple method of calculation is described, which exploits certain quadratic relations between the Fourier coefficients in Stokes's series. The ratio U/c and the associated Lagrangian skewness is calculated for periodic waves, as a function of the wave steepness and the mean water depth.

For limiting waves, i.e. those with sharp crests, it is found that the most symmetric orbits, in the Lagrangian sense, occur not in very deep or very shallow water, but at one intermediate value of the ratio of depth to wavelength. When the depth parameter kd equals 1.93 the vertical displacement of a marked particle at the free surface is closely sinusoidal in the time t.

The vertical oscillation of a partially submerged body produces a surface wave that carries energy away from the body. The amplitude of this wave, when surface-tension effects are not negligible, depends on the conditions applied at the line of contact between the body and the free surface of the fluid. An edge condition that includes both dynamic variation of the contact angle and contact-angle hysteresis is used in this paper; the condition implies a dissipation of energy at the contact line. The amplitude of the wave and the amount of energy dissipated are calculated for a horizontal circular cylinder and for a simple source-and-plate model. This model is shown to be an adequate representation for the qualitative description of heaving motions and to simplify the calculations considerably. The effects of varying the relative importance of surface tension and gravity, the dynamic behaviour of the contact angle, and the amount of hysteresis, are calculated for the source-and-plate model.

Dead zones tend to hold back the downstream travel, to increase the longitudinal spreading and to provide a long tail of low concentration for passive contaminant releases in natural streams. Here it is shown how the presence of a random distribution of dead zones can be accommodated into the method of moments by choosing an appropriate composite averaging. The individual roles of the cross-stream velocity shear, the dead-zones mean volume fraction and the dead-zones probability distribution are clearly revealed in the longitudinal shear-dispersion coefficient. The inevitable deviations from Gaussianity are examined by means of skewness and kurtosis. Simple examples are used to quantify the effects of the dead zones upon contaminant dispersion in Couette flow, pipe and plane Poiseuille flows.

Two mechanisms for the generation of standing edge waves over a sloping beach are described using classical linear water-wave theory. The first is an extension of the result of Yih (1984) to a class of localized bottom protrusions on a sloping beach in the presence of a longshore current. The second is a class of longshore surface-pressure distributions over a beach. In both cases it is shown that Ursell-type standing edge-wave modes can be generated in an appropriate frame of reference. Typical curves of the mode shapes are presented and it is shown how in certain circumstances the dominant mode is not the lowest.

Propulsion of a foil moving in the water close to a free surface is examined. The foil moves with a forward speed U and is subjected to heaving and pitching motions in calm water, head waves or following waves. The model is two-dimensional and all equations are linearized. The fluid is assumed to be inviscid and the motion irrotational, except for the vortex wake. The fluid layer is infinitely deep.

The problem is solved by applying a vortex distribution along the centreline of the foil and the wake. The local vortex strength is found by solving a singular integral equation of the first kind, which appropriately is transformed to a non-singular Fredholm equation of the second kind. The vortex wake, the forward thrust upon the foil and the power supplied to maintain the motion of the foil are investigated. The scattered free surface waves are computed. For moderate values of Uσ/g (U is forward speed of the foil, σ is frequency of oscillation, g is acceleration due to gravity) it is found that the free surface strongly influences the vortex wake and the forces upon the foil. When the foil is moving in incoming waves it is found that a relatively large amount of the wave energy may be extracted for propulsion. As an application of the theory the propulsion of ships by a foil propeller is examined. The theory is compared with experiments.