The paper provides a model of turbulence which effects closure through approximated transport equations for the Reynolds stress tensor $\overline{u_iu_j}$ and for the turbulence energy-dissipation rate ε. In its most general form the model thus entails the solution of seven transport equations for turbulence quantities but contains only six constants to be determined by experiment. It is demonstrated that the proposed approximation to the pressure-rate-of-strain correlations leads to satisfactory predictions of the component stress levels in plane homogeneous turbulence, including the non-equality of the lateral and transverse normal-stress components.

For boundary-layer flows a simpler version of the model is derived wherein transport equations are solved only for the shear stress $-\overline{u_1u_2}$ the turbulence energy κ and ε. This model has been incorporated in the numerical solution procedure of Patankar & Spalding (1970) and applied to the prediction of a number of boundary-layer flows including examples of flow remote from walls, those developing along one wall and those confined within ducts. Three of the flows are strongly asymmetric with respect to the surface of zero shear stress and here the turbulent shear stress does not vanish where the mean rate of strain goes to zero. In most cases the predicted profiles and other quantities accord with the data within the probable accuracy of the measurements.

The tearing of a pressure-sensitive (‘tacky’) adhesive is examined. Two flexible strips bonded by a layer of adhesive are passed between adjacent cylindrical guides and peeled apart, causing the adhesive layer to separate into two about a surface tension membrane. Treating the adhesive as a Newtonian viscous fluid, the slow-flow problem is solved by an iterative numerical scheme in which the surface tension membrane boundary in the vicinity of the region of separation is approximated by a shear-free boundary given by a sixth-degree polynomial expression. The energy dissipation rate, a measure of the ‘strength’ of the adhesive, is obtained from the flow.

The solution method is also used to determine the similar flow induced by two counter-rotating rollers partially immersed in a large bath of fluid. The results are in fairly good agreement with available experimental data. The symmetrical eddies observed under the lowest point of the surface tension membrane in the stable flow between the rollers are reproduced in the solution, proving that fluid inertia effects are not essential for their existence.

The paper presents a comprehensive examination of self-similar blast waves with respect to two parameters, one describing the front velocity and the other the variation of the ambient density immediately ahead of the front. All possible front trajectories are taken into account, including limiting cases of the exponential and logarithmic form. The structure of the waves is analysed by means of a phase plane defined in terms of two reduced co-ordinates F ≡ (t/rμ) u and Z ≡ [(t/rμ)a]2, where t and r are the independent (time and space) variables, μ ≡ d ln rn/d Intn the subscript n denoting the co-ordinates of the front, and u and a are, respectively, the particle velocity and the speed of sound. Loci of extrema of the integral curves in the phase plane are traced and loci of singularities are determined on the basis of their intersections. Boundary conditions are introduced for the case when the medium into which the waves propagate is at rest. Representative solutions, pertaining to all the possible cases of blast waves bounded by shock fronts propagating into an atmosphere of uniform density, are obtained by evaluating the integral curves and determining the corresponding profiles of the gasdynamic parameters. Particular examples of integral curves for waves bounded by detonations are given and all the degenerate solutions, corresponding to cases where the integral curve is reduced to a point, are delineated.

The effect of rotary Brownian motion on the rheology of a dilute suspension of rigid spheroids in shear flow is considered for various limiting cases of the particle aspect ratio r and dimensionless shear rate γ/D. As a preliminary the probability distribution function is calculated for the orientation of a single, axisymmetric particle in steady shear flow, assuming small particle Reynolds number. The result for the case of weak-shear flows, γ/D [Lt ] 1, has been known for many years. After briefly reviewing this limiting case, we present expressions for the case of strong shear where (r3 + r−3) [Lt ] γ/D, and for an intermediate regime relevant for extreme aspect ratios where 1 [Lt ] γ/D [Lt ] (r3 + r−3). The bulk stress is then calculated for these cases, as well as the case of nearly spherical particles r ∼ 1, which has not hitherto been discussed in detail. Various non-Newtonian features of the suspension rheology are discussed in terms of prior continuum mechanical and experimental results.

The interaction of nonlinear gravity waves and steady non-uniform currents is studied using the averaged Lagrangian method due to Whitham (1965a,b). The results are compared with the essentially linear theory of Longuet-Higgins & Stewart (1961, 1964) for three specific problems: waves on a stream (U(x), 0) with variations in the stream balanced by upwelling from below or inflow from the sides, and waves on a shear flow (0, V(x)). It appears that rates of growth of large waves are less than those predicted by linear theory and that the energy density can sometimes decrease when the wave height and steepness are still increasing. The final section discusses the form of the energy equation in terms of the Lagrangian.

We show that the rippling often observed on small progressive gravity waves can be explained in terms of a nearly resonant harmonic nonlinear interaction. The resonance condition is that the phase speeds of the two waves must be nearly identical. The in viscid analysis is generalized to any order in a small parameter proportional to the wave steepness. Wave tank measurements provide experimental evidence for most of the predicted results. The phenomenon of resonant rippling is further shown to be not just peculiar to capillary-gravity waves, but in fact possible for any weakly nonlinear dispersive wave system whose dispersion relation has discrete pairs of solutions nearly satisfying the resonance conditions.

We define a generalized vortex to have azimuthal velocity proportional to a power of radius r−n. The properties of the steady laminar boundary layer generated by such a vortex over a fixed coaxial disk of radius a are examined. Though the boundary-layer thickness is zero a t the edge of the disk, reversals of the radial component of velocity u must occur, so that an extra boundary condition is needed at any interior boundary radius rE to make the structure unique. Numerical integrations of the unsteady governing equations were carried out for n = − 1, 0, ½ and 1. When n = 0 and − 1 solutions of the self-similar equations are known for an infinite disk. Assuming terminal similarity to fix the boundary conditions at r = rE when ur > 0, a consistent solution was found which agrees with those of the self-similar equations when rE is small. However, if n = ½ and 1, no similarity solutions are known, although the terminal structure for n = 1 was deduced earlier by the present authors. From the numerical integration for n = ½, we are able to deduce the limit structure for r → 0 by using a combination of analytic and numerical techniques with the proviso of a consistent self-similar form as rE → 0. The structure is then analogous to a ladder consisting of an infinite number of regions where viscosity may be neglected, each separated by much thinner viscous transitional regions playing the role of the rungs. This structure appears to be characteristic of all generalized vortices for which 0.1217 < n < 1.

The theory of Stokes flows about slender particles is considered for the case in which the particle cross-section is arbitrary and non-uniform along the axis of the particle, subject to certain smoothness assumptions. As distinct from previous works on slender particles in which the particle is represented approximately by a line distribution of Stokeslets, the representation adopted here is the exact one of a surface distribution of Stokeslets. This representation may then be used to recover the familiar one-dimensional integral equation for the equivalent line density of Stokeslets, together with estimates for the range of particle shapes for which the equation is valid. Within this range it is found that there is a class of particles for which the established perturbation scheme, which is used to obtain a solution to the integral equation, is singular. This class of particle shapes is illustrated by the example of a uniformly twisted particle whose pitch of twist is large enough compared with the cross-sectional dimension to ensure the validity of the equation, but small enough to make the usual method of solution singular. It is shown how the equation may be transformed so that an approximate solution can be found by means of a regular perturbation scheme. The results indicate that, for each of the axial and transverse components of motion, there is an equivalent particle of circular cross-section for which the total force and couple are the same as for the original particle. However, the radii of the equivalent cylinders are different for each component, the transverse component being affected by the twist while the axial component is not.