A technique is developed to model the multiple scattering of surface waves in an array of axisymmetric wave-energy devices. The matrix equation which results is inverted to yield the exciting forces, the added-damping and added-mass matrices, the optimal power absorption and the optimal device responses. The matrix method is also used on the previously unstudied problem of an unconstrained array. Finite-size effects of devices are shown to be important in producing phase-shifts, which shift the uncon- strained frequency response, but leave the opt,imal energy absorption virtually unchanged.

A theoretical model is developed for the motion of a human red blood cell in a shear field. The model consists of a tank-treading ellipsoidal membrane encapsulating an incompressible Newtonian liquid immersed in a plane shear flow of another incom- pressible Newtonian liquid. Equilibrium and energy considerations lead to a solution for the motion of the particle that depends on the ellipsoidal-axis ratios and the ratio of the inner- to outer-liquid viscosities. The effect of variation in these parameters is explored and it is shown that, depending on their values, one of two types of overall motion is exhibited: a steady stationary-orientation motion or an unsteady flipping motion. A qualitative agreement of the predicted behaviour of the model with experi- mental observations on red blood cells is found.

The steady oblique interaction of two solitary waves on the surface of water of constant depth is considered. One wave is taken to be of arbitrary (large) amplitude and the other is small with a (non-dimensional) amplitude measured by the parameter ε. A solution is sought as an asymptotic expansion, based on ε, that assumes that in some region the solution is the sum of the two waves plus interaction terms. It is shown that this expansion is not uniformly valid close to a critical angle. This angle varies from zero (parallel waves) up to about 63°, as the amplitude of the larger wave increases from infinitesimally small to the largest-possible solitary wave. I n the limit of two small waves, the details agree precisely with the results obtained by Miles (1977a).

When the angle between the two waves is dose to the critical angle, for a given large wave, an alternative asymptotic expansion is required. In this strong-interaction case, the dominant term is just the large wave but with a phase shift that is an arbitrary function of the characteristic variable associated with the small wave. This function is determined by matching to an appropriate far field, and it turns out to be proportional to the logarithm of a hypergeometric function (which itself can be expressed in terms of the associated Legendre function Pv−μ). The phase shift is then well-defined (finite, real) provided the angle is not very close to the critical value. When this occurs the phase shift can be infinite at a specific angle (which corresponds to the case |μ| = v > 0), and even closer to the critical angle (|μ| < v) the phase shift is undefined (no longer real). A real solution for the wave profile is still possible if negative amplitudes are allowed, but the resulting solution is unacceptable since the surface is then not undis- turbed at infinity. It is shown that the criterion |μ| < v matches exactly (for two small waves) with Miles’ criterion for the non-existence of a regular reflection.

For the strong interaction (and |μ| ≥ v i t is argued that the small wave cannot penetrate the large wave unchanged. The large wave suffers significant distortion (bending) in the interaction, but the small wave, if it penetrates at all, must have an amplitude O(ε). This is the main aspect of the problem which cannot be completely determined using the present methods. The difficulty can be traced to the large solitary wave which is not known in closed form: only the exponential behaviour in the ‘tail’ is used explicitly in this work.

For a non-uniform discharge of contaminant in a shear flow the initial advection velocity and the amount of shear across the contaminant cloud depend upon the discharge shape across the flow. Here i t is shown how the continuing influence of the discharge non-uniformity can be incorporated into a delay-diffusion description of the dispersion process (Smith 198 1). An important improvement over the variable coefficient diffusion equation derived by Gill 85 Sankarasubramanian (1971) is that the solutions have the physically correct superposition property.

We consider liquid in a circular cylinder that undergoes nonlinear Marangoni insta- bility. The upper free surface of the liquid is taken to have large-enough surface tension that surface deflections are neglected. The side walls are adiabatic and impenetrable, and for mathematical simplicity the liquid is allowed to slip on the side walls. The linearized stability theory for heating from below gives the critical Marangoni number Mc as a function of cylinder dimensions, surface-cooling condition and Rayleigh number. The steady nonlinear convective states near Mc are calculated using an asymptotic theory, and the stability of these states is examined. At simple eigenvalues Mc the finite-amplitude states are determined. We find th at the Prandtl number of the liquid influences the stability of axisymmetric states, distinguishing upflow at the centre from downflow. Near those aspect ratios corresponding to double eigenvalues Me, where two convective states of linear theory are equally likely, the nonlinear theory predicts sequences of transitions from one steady convective state to another as the Marangoni number is increased. These transitions are determined and discussed in detail. Time-periodic convection is possible in certain cases.

Attention is confined to roll-cell development and roll-cell interaction appropriate to one horizontal dimension larger than either the other horizontal dimension or the depth. At simple eigenvalues Mc the roll-cell amplitude and transport fields can be obtained. Near those aspect ratios corresponding to double eigenvalues Mc, where two roll-cell states of linear theory areequallylikely, thenonlinear theory predicts sequences of transitions from one steady convective state to another as the Marangoni number is increased. Direct comparisons are made of the results here with those of the previous paper for Marangoni convection in circular cylinders. Time-periodic convection is possible in certain cases.

This paper analyses the problem of a flow past an oscillating body moving with constant velocity, below and parallel to a free surface. Special attention is given to frequencies of oscillation in the neighbourhood of the critical frequency ωc= 0.25 g/U, where the classical linearized solution yields infinitely large wave amplitude. As a result both the lift and drag forces acting on the oscillating body at the resonant frequency are singular. It is demonstrated in the paper how this resonance is elimi- nated by considering higher-order free-surface effects, in particular the interaction between the first- and third-order terms. The resulting generalized solution yields finite wave amplitudes at the resonant frequency which are O(ε½) and O(εlogε) for 2 and 3 dimensions respectively. Here 6 is a measure of the singularity strength. It is also shown that inclusion of third-order terms causes a shift in the wavenumber and group velocity which eliminates the singularity in the lift and drag expressions at the resonant frequency. These results are illustrated by computing the lift and drag experienced by a submerged oscillating horizontal doublet in a uniform flow.

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

An experimental investigation has been carried out into the nature of the transport of airborne material in the near aerodynamic wakes of bluff bodies with simple shapes. The main attention was focused on the essential differences existing between axi- symmetric flows (as about disks) and two-dimensional flows (as about rectangular long thin flat plates). Measurements were made for such bodies of the near-wake residence time of injected small particles, along with other and more familiar near- wake properties such as the vortex-shedding frequency and base pressure. It was concluded for disks that the transport of material into and out of the near-wake region is dominated by turbulent diffusion, and is strongly influenced by free-stream turbulence, especially for free-stream turbulence whose length scale is substantially smaller than the disk diameter. For rectangular flat plates, transport is dominated by the periodic shedding of vortices, and to only a secondary extent by turbulent motions, and is not strongly influenced by free-stream turbulence.

Basec on somewhat simplified profiles of turbulent eddy diffusivity and mean vcLacity in turbulent flow, an expression is derived for the convection current in a pipe where electrification occurs at the wall. The expression is in explicit analytic form, and applies for all turbulent Reynolds numbers and all fluid conductivities, from conditions where the Debye length is small compared with the diffusion sublayer (typical aqueous solutions) to conditions where the Debye length is large compared with the sublayer (typical liquid hydrocarbons).

The discrete-vortex model is applied to simulate the separation bubble over a two- dimensional blunt flat plate with finite thickness and right-angled corners, which is aligned parallel to a uniform approaching stream. This flow situation is chosen because, unlike most previous applications of the model, the separation bubble is supposed to be strongly affected by a nearby solid surface. The major objective of this paper is to examine to what extent the discrete-vortex model is effective for such a flow. A simple procedure is employed to represent the effect of viscosity near the solid surface; in particular, the no-slip condition on the solid surface. A reduction in the circulation of elemental vortices is introduced as a function of their ages in order to represent the three-dimensional deformation of vortex filaments, An experiment was also performed for comparison purposes.

The calculation yielded reasonable predictions of the time-mean and r.m.s. values of the velocity and the surface-pressure fluctuations, together with correlations between their fluctuating components, over most of the separation bubble. The interrelation between instantaneous spatial variations of the surface-pressure and velocity fluctuations were also obtained. A comparison between the calculated and measured results suggests that, in the real flow, the three-dimensional deformation of vortex filaments will become more and more dominant as the reattachment point is approached.

A linearized stability analysis is carried out for the breakup of small-diameter liquid filaments of dilute polymer solutions into droplets. Oldroyd's 8-constant model expressed in a corotational reference frame is used as the rheological equation of state. The crucial idea in this theory is the recognition that the liquid may be subject to an unrelaxed axial tension due to its prior history. If the tension is zero, the present analysis predicts that jets of shear-thinning liquids are less stable than comparable jets of Newtonian liquids; this is in agreement with previous analyses. However, when the axial tension is not zero, and provided the stress relaxation time constant is sufficiently large, the new theory predicts that the axial elastic tension can be a significant stabilizing influence. With reasonable values for the tension and stress relaxation time the theory explains the great stability observed for jets of some shear- thinning, dilute polymer solutions. The theory explains why drops produced from jets of such liquids are larger than drops from corresponding Newtonian liquids. The theory also appears capable of explaining the sudden appearance of irregularly spaced bulges on jets after long distances of t,ravel with little amplification of disturbances.

A numerical method for calculating the interaction of steep (nonlinear) ocean waves with large fixed or floating structures of arbitrary shape is described. The interaction is treated as a transient problem with known initial conditions corresponding to still water in the vicinity of the structure and a prescribed incident waveform approaching it. The development of the flow, together with the associated fluid forces and structural motions, are obtained by a time-stepping procedure in which the flow at each time step is calculated by an integral-equation method based on Green's theorem. A few results are presented for two reference situations and these serve to illustrate the effects of nonlinearities in the incident waves.

This paper is an attempt at a mathematical synopsis of the theory of wave motions on glaciers. These comprise surface waves (analogous to water waves) and seasonal waves (more like compression waves). Surface waves have been often treated and are well understood, but seasonal waves, while observed, do not seem to have attracted any theoretical explanation. Additionally, the spectacular phenomenon of glacier surges, while apparently a dynamic phenomenon, has not been satisfactorily explained.

The present thesis is that the two wave motions (and probably also surging, though a discussion of this is not developed here) can both be derived from a rational theory based on conservation laws of mass and momentum, provided that the basal kinematic boundary condition involving boundary slip is taken to have a certain reasonable form. It is the opinion of this author that the form of this ‘sliding law’ is the crux of the difference between seasonal and surface waves, and that a further understanding of these motions must be based on a more satisfactory analysis of basal sliding.

Since ice is here treated in the context of a slow, shallow, non-Newtonian fluid flow, the theory that emerges is that of non-Newtonian viscous shallow-water theory; rather than balance inertia terms with gravity in the momentum equation, we balance the shear-stress gradient. The resulting set of equationsis, in essence, a first-order nonlinear hyperbolic (kinematic) wave equation, and susceptible to various kinds of analysis. We show how both surface and seasonal waves are naturally described by such a model when the basal boundary condition is appropriately specified. Shocks can naturally occur, and we identify the (small) diffusive parameters that are present, and give the shock structure: in so doing, we gain a useful understanding of the effects of surface slope and longitudinal stress in these waves.

The two-phase flow in settling vessels with walls that are inclined to the vertical is investigated. By neglecting inertial effects and the viscosity of the suspension i t is shown that the particle concentration remains constant on kinematic-wave fronts. The wave fronts are horizontal and propagate in a quasi-one-dimensional manner, but are imbedded in a two-dimensional or three-dimensional basic flow which, in turn, depends on the waves via the boundary conditions. Concentration discontinuities (interfaces) are described by kinematic-shock theory. The kinematic shocks are shown to be horizontal, with the possible exception of discontinuities that separate the suspension from the sediment.

At downward-facing inclined walls conservation of mass enforces the existence of a boundary-layer flow with relatively large velocity. As G/R2→∞ and G/R4→ 0, where G and R are respectively a sedimentation Grashof number and a sedimentation Reynolds number, the entrainment of suspended particles into the boundary-layer flow of clear liquid is negligibly small. This provides an appropriate boundary condi- tion for the basic flow of the suspension. Thus, in the double limit considered, a kine- matic theory suffices to determine the convective flow of the suspension due to the presence of inclined walls.

As an example batch sedimentation in vessels with inclined plane or conical walls is investigated. The settling process is terminated after a time that can be considerably smaller than the time required in a vertical vessel under the same conditions.Depending on the initial particle concentration, there are centred kinematic waves that are linked to a continuous increase of the particle concentration in the suspension. In an appendix, the flow in the boundary layer at a downward facing, inclined wall is investigated. With G/R2→∞ and G/R4→ 0, the boundary layer consists of an inviscid particle-free main part, a viscous sublayer at the wall, and a free shear sublayer at the liquid/particle interface.

In a recent paper Doshi, Daiya & Gill (1978) showed that the value of Taylor's longitudinal diffusivity D for laminar flow in a channel of rectangular cross-section of breadth u and height b is about 8D0, for large values of the aspect ratio a/b, where Do is the value of the longitudinal diffusivity obtained by ignoring all variation across the channel. This superficially surprising result is confirmed by an independent method, and is shown to be caused by the boundary layers on the side walls of the channel. The primary purpose of the paper, however, is to consider the value of D in turbulent flow in a flat-bottomed channel of large aspect ratio, for which arguments based on physics are adduced in support of the formula D≈[1 + B][1 - λ(b/u)], where B and λ are positive constants independent of b. It is shown that this result is consistent with laboratory experiments by Fischer (1966). The paper concludes with a discussion of the practical effects of aspect ratio on longitudinal dispersion in channels whose cross-section is approximately rectangular.

The limiting process introduced by Stewartson & Cheng (1979) is used to obtain solutions in the limit of vanishing Rossby number for deep rotating flows at arbitrary Reynolds number over cross-stream ridges of finite slope. Examination of inviscid solutions for infinite-depth flow shows strong dependence on obstacle shape of not only the magnitudes but also the positions of disturbances in the far field. In finite-depth flow there is present the Stewartson & Cheng inertial wave wake, which may be expressed as a sum of vertical modes whose amplitudes depend on the obstacle shape but are independent of distance downstream; the smoother the topography and the shallower the flow, the fewer the number of modes required to describe the motion. For abrupt topography the strength of the wake does not, however, decrease monoton- ically with decreasing container depth (or Rossby number). In very deep flows viscosity causes the wake to decay on a length scale of order the Reynolds number times the ridge width. In shallower flows, where only a few modes are present, the decay of the wake is more rapid. For Reynolds numbers and depths of the order of those in the experiments of Hide, Ibbetson & Lighthill (1968)) viscosity causes the disturbance to take on the appearance of a leaning column.

Measurements of air and solid velocities were made in an air-solid two-phase flow in a horizontal pipe by the use of a laser-Doppler velocimeter (LDV). The pipe was 30 mm inner diameter, and two kinds of plastic particles, 0.2 and 3.4 mm in diameter, were conveyed in addition to fine particles (ammonium chloride) for air-flow detection. The air velocities averaged over the pipe cross section ranged from 6 to 20m/s and the solid-to-air mass-flow ratio was up to 6. Simultaneous measurements of both air and 0.2 mm particle velocities were found possible by setting threshold values against the pedestal and Doppler components of the photomultiplier signal.

As the loading ratio increased and the air velocity decreased, mean-velocity distributions of both phases increased asymmetrical tendency. I n the presence of 0.2mm particles, a flattening of the velocity profile was remarkable. The effects of the solid particles on air-flow turbulence varied greatly with particle size. That is, 3.4 mm particles increased the turbulence markedly, while 0-2 mm ones reduced it. The probability-density function of the air flow deviated from the normal distribution (Gaussian) in the presence of particles. Finally, the frequency spectra of air-flow turbulence were obtained in the presence of 0.2 mm particles by using a fast Fourier transform (FFT). As a result, it was found that t,he higher-frequency components increased with increasing loading ratio.

The Rayleigh number R, in a horizontal layer with temperature-dependent viscosity can be based on the viscosity at T0, the mean of the boundary temperatures. The critical Rayleigh number Roc for fluids with exponential and super-exponential viscosity variation is nearly constant at low values of the ratio of the viscosities at the top and bottom boundaries; increases at moderate values of the viscosity ratio, reaching a maximum at a ratio of about 3000, and then decreases. This behaviour is explained by a simple physical argument based on the idea that convection begins first in the sublayer with maximum Rayleigh number. The prediction of Palm (1960) that certain types of temperature-dependent viscosity always decrease Roc is confirmed by numerical results but is not relevant to the viscosity variations typical of real liquids. The infinitesimal-amplitude state assumed by linear theory in calculating Roc does not exist because the convection jumps immediately to a finite amplitude at R0c. We observe a heat-flux jump at R0c exceeding 10% when the viscosity ratio exceeds 150. However, experimental measurements of R0c for glycerol up to a viscosity ratio of 3400 are in good agreement with the numerical predictions when the effects of a temperature-dependent expansion coefficient and thermal diffusivity are included.

Axisymmetric flows between counter-rotating cylinders of varying radius ratio are examined. The stability of these flows to non-axisymmetric disturbances is considered, and the results of these calculations are compared with experiments.