The steady and quasi-steady motion achieved in a rotating stratified sphere of fluid is studied in the context of a linearized Boussinesq model. In certain parameter ranges an explicit expression is obtained for the flow field as a functional of the surface stress. The non-singular interior solution is used to examine the behaviour of the boundary layer close to the equator. The results agree with previous conclusions about the behaviour of a rotating stratified fluid in simpler geometries. Viewing the problem as a simple model for the interior core of the sun, this work indicates a solar spin-down time that is within the lifetime of the sun.

A numerical calculation is presented for the slow flow of a viscous fluid in a region bounded by a right-angled isosceles triangle. The particular flow considered is the secondary flow generated in the plane of a cross-section by the primary axial flow, under a constant pressure gradient, through a slightly curved tube of triangular cross-section. The flow is assumed to be at small Dean number so that the stream function of the secondary flow satisfies the biharmonic equation. The sequence of eddies of decreasing size and intensity first identified by Moffatt (1964a) is observed in each of the corners. Numerical details of a number of these eddies are given and their properties are found to be in excellent agreement with the theory.

A study is made of the form taken by a slender jet of water whose only boundary is a free surface. The only forces acting are inertial and gravitational. Attention is paid to the cross-flow velocity components and to the development of the shape of the cross-section of the jet as it progresses. It is established that a jet with initially elliptic cross-sections can remain elliptical, and the variation in the aspect ratio along the jet is determined.

At the body-diameter Reynolds number 2000, the axisymmetric wake behind a slender streamlined body of revolution remained laminar without any indication of breakup far beyond the five-body-lengths test section. The mean-velocity profiles were measured with close spacings in order to obtain experimental information on the transient process from boundary layer to fully developed wake. A semi-empirical theory was developed to describe the flow-field characteristics.

The growth rates and phase speeds of gravity-capillary wind waves are investigated through numerical solution of a linear, viscous, coupled, shear-flow perturbation model. Numerical results are obtained by transforming the boundary-value problem of a perturbed mean laminar shear flow into a matrix-eigenvalue problem using standard finite-difference methods.

Detailed calculations are performed for a basic state composed of a logarithmic-linear mean flow profile in the air and a linear-logarithmic mean flow profile in the water. We exclude turbulent Reynolds stresses. Calculated growth rates show excellent agreement with corresponding experimental growth rates. This implies that the initial growth of gravity-capillary wind waves is almost certainly due to the instability of the coupled laminar shear flow in the air and water.

The investigation also demonstrates that the shear flow in the water cannot be ignored in wave growth studies, since the usual 3-4%, highly sheared, wind-induced surface drift produces a significant increase in the growth of wind-generated gravity-capillary waves.

A half-immersed circular cylinder is making vertical oscillations on water of finite constant depth. The virtual-mass and damping coefficients are studied in the limit as the wavelength tends to infinity. It is found that the virtual-mass coefficient tends to a finite limit and that the amplitude ratio ultimately varies as the frequency. This behaviour differs from the behaviour for infinite depth, where the virtual-mass coefficient tends to infinity and the amplitude ratio ultimately varies as (frequency).

The method developed previously (Youngren & Acrivos 1975) for obtaining numerical solutions to the Stokes equations for flows past solid particles is extended to problems with free boundaries. This technique is applied to the determination of steady shapes for an inviscid gas bubble symmetrically placed in an extensional flow. For large surface tension the computed bubble shape is found to be in excellent agreement with that obtained analytically by Barthès-Biesel & Acrivos (1973), while for small surface tension it agrees with an expression derived by Buckmaster (1972) using slender-body theory.

A circular cylinder is held vertically and withdrawn from a large bath of liquid. When the horizontal end face of the cylinder rises above the mean undisturbed liquid level, an axially symmetrical meniscus is formed, which joins the cylinder at the circumference of the end. If the cylinder is further withdrawn, at a certain height the meniscus breaks. By means of the calculus of variations the condition for stability is derived in terms of the increase in height and the increase in the angle between the tangent to the meniscus and the face of the cylinder at the point where the meniscus joins it. The meniscus is unstable as long as these changes are of the same sign.

In typical jet-noise measurements one is almost always interested in the pressure in the far field as kR → ∞. The purpose of this paper is to show that this limit is singular in the sense of matched asymptotic expansions. The inner solution (kfixed, R→ ∞) is given by geometric acoustics, whereas the outer is given by the so-called acoustic-shielding solution (k fixed, R→ ∞). A suitable composite solution is also constructed.

Now jet-noise measurements are always made at fixed and finite values of R (typically 50 jet diameters) and from these measurements one would like to infer the value of pR at R = ∞, where p is the acoustic pressure. One interesting and somewhat unexpected result of this paper is that the value of pR at infinity cannot be inferred from these measurements above a certain frequency. In other words, in order to obtain accurate estimates of pR at infinity for higher and higher frequencies, one has to be further and further away from the jet!

Boundary-layer damping of one-dimensional gravity waves of slowly varying amplitude a(t), characteristic wavenumber k, and characteristic frequency ω in water of depth d and kinematic viscosity v is calculated for a [Lt ] d, d [Lt ] 1/k and (2ν/ω)½ [Lt ] d. General results are given for the temporal evolution of the power spectral density determined by either a Fourier-integral (spatially aperiodic) or Fourier-series (spatially periodic) representation of the wave. Solitary and cnoidal waves are considered as examples. Keulegan's (1948) inverse-fourth-power decay for the solitary wave is recovered, and the numerical parameter therein is evaluated by reduction to a Riemann zeta function. A universal decay curve is obtained for the Stokes-scaled amplitude S = a/k2d3 of the cnoidal wave as a function of the boundary-layer-scaled time (νω)½t/d; the result is both more flexible and more compact than that obtained by Isaacson (1976). The decay is within 5% of that for a solitary wave (inverse fourth power) for S > 2 or that for an infinitesimal wave (exponential) for S < 2. An analytical approximation with a maximum error of less than 1% is obtained by joining an asymptotic approximation for S > 1 to the exponential approximation for S < 1.

The rolling motion of a sphere on a smooth plane boundary in a simple-harmonic water motion has been analytically and experimentally investigated. For spheres having specific gravities ranging from 0·09 to 15·18 the sphere motion was found to be sinusoidal for both low and high values of the period parameter defined by Keulegan & Carpenter. The knowledge of the sphere motion, and hence the resultant force, allowed the determination of inertia and drag coefficients from Fourier-averaging techniques. Experiments in the inertial range yielded an added-mass coefficient of 1·2, compared with 0·67 from inviscid theory for translating spheres. For values of the period parameter greater than 30 the drag coefficient is reported to be approximately 0·74.

This note provides numerical values for the long-wave limit of the virtual-mass coefficient relating to the heaving motion of a half-immersed circular cylinder on water of finite depth, found analytically by Ursell in the preceding paper; some preliminary analysis is needed, however.

A stability analysis of meandering and braiding perturbations in a model alluvial river is described. A perturbation technique, involving a small parameter representing the ratio of sediment transport to water transport, is used to obtain the following results.

Under appropriate conditions, the existence of sediment transport and friction are necessary conditions for the occurrence of instability in the flow and on the bed; thus instability is not inherent in the flow alone. An Anderson-type scale relation for longitudinal instability is obtained for meandering. A relation estimating the number of braids and differentiating between meandering and braided regimes is derived. These relations are independent of sediment transport.

The reflexion of an internal gravity wave with stream function \[ \psi = \exp\{i(kx + lz - \omega t)\} \] from a corrugated surface of the form z = a cos px is investigated using an extension of Rayleigh's method for an inviscid non-diffusive fluid. It is found that the method converges for many wall slopes ap and incoming-wave phase propagation incidence angles θ = tan−1l/k but that the appropriate series solution is only asymptotic in other cases. The accuracy of the calculations is assured by requiring that the solution satisfy the boundary condition at the wall using a least-squares error minimization technique. The accuracy is then verified through the conservation of energy flux. It is found that, as the surface slope is increased for constant θ, less energy appears in specular reflexions and more is either back-scattered or redistributed into other forward-scattered modes. As the horizontal internal wavelength is decreased to become comparable to the corrugation wavelength of the wall, substantially less energy appears in specular reflexion, but of the order of 95% of the incoming energy is specularly reflected for 30° < θ < 60° when the two wavelengths are equal. In contrast to this, it is found that the general level of the ratio of back-scattered to forward-scattered energy is reduced by O(10−3) to O(10−2) as the incident horizontal internal wavelength becomes smaller than the corrugation wavelength. The results are compared with the linear theory of Baines (1971); agreement is good for forward-scattered energy and excellent for the back-scattered flux when p ≥ k.

Experimental measurements of the mean velocity profiles produced by axially symmetric turbulent boundary layers on cylinders of various diameters are described. The profile measurements were made with very small hot wires developed for this investigation. Measurements of the wall shear stress on cylinders ranging from 0.02 to 2.0 in. in diameter are also reported. In the boundary layer on cylinders, well-defined regions exist in which the two-dimensional law of the wall and a three-dimensional wake law are valid. There was no evidence that the boundary layer was not fully turbulent even on the cylinders of smallest diameter. Measurements of wall pressure fluctuations beneath the boundary layer on a 1 in. diameter cylinder are also described. The results were much the same as those previously reported by Willmarth & Yang (1970) for a 3 in. diameter cylinder. The only difference was the discovery that the wall pressure was correlated in the transverse direction approximately half-way around the cylinder. This was not true on the 3 in. diameter cylinder.

Attenuation of a plane shock due to the interaction with a single slit is examined via spark schlieren photography for shock Mach numbers between 1·17 and 2·44 and slit widths between 0·173 and 3·175 cm. Wave speed measurements by piezoelectric transducers indicate that the attenuation is weak and adequately predicted by Whitham's ray-shock theory. The slit width is observed to produce only a secondary effect on the shock attenuation rate. Stability of the attenuating shock is demonstrated from a wave diagram constructed by the ray-shock technique.

The flow of an incompressible fluid through a curved wire-gauze screen of arbitrary shape is reconsidered. Some inconsistencies in previously published papers are indicated and the various approximations and linearizations (some of which are necessary for a complete analytic solution) are discussed and their inadequacies demonstrated. Attention is concentrated on the common practical problem of calculating the screen shape required to produce a linear shear flow and experimental work is presented which supports the contention that the theoretical solutions proposed by Elder (1959)–subsequently discussed by Turner (1969) and Livesey & Laws (1973)-and Lau & Baines (1968) are inadequate, although, for the case of small shear, Elder's theory appears to be satisfactory. Since, in addition, there are inevitable difficulties concerning both the value of the deflexion coefficient appropriate to any particular screen and inhomogeneities in the screen itself, it is concluded that the preparation of a curved screen to produce the commonly required moderate to large linear shear flow is bound to be somewhat empirical and should be attempted with caution.

Recently Lindzen (1974) has proposed a model of a shear-layer instability which allows unstable modes to co-exist with radiating internal gravity waves. The model is an infinite, continuously stratified, Boussinesq fluid, with a simple jump discontinuity in the velocity profile. Linear stability theory shows that the model is stable for wavenumbers k such that k2 < N2/2U2, where N is the Brunt—Väisälä frequency and 2U is the change in velocity across the discontinuity. For N2/2U2 < k2 < N2/U2 an unstable mode may co-exist with an internal gravity wave. This paper examines the weakly nonlinear aspects of this model for wavenumbers k close to the critical wavenumber N/2½U. An equation governing the evolution of the amplitude of the interfacial displacement is derived. It is shown that the interface may support a stable finite amplitude internal gravity wave.

This paper is concerned with convection generated by uniformly distributed internal heat sources. By a numerical method it is found that the planform is down-hexagons for infinite Prandtl numbers and Rayleigh numbers up to at least 15 times the critical value. The motion is also studied for finite Prandtl numbers and small supercritical Rayleigh numbers by using an amplitude expansion. It turns out that a small subcritical regime exists. Moreover, it also emerges that for Prandtl numbers less than 0.25 the stable planform is up-hexagons. In §3 a necessary condition in order to obtain a hexagonal planform is derived when the coefficients in the differential equations are a function of the vertical co-ordinate z.

We consider a kinetic theory model of a gas, whose molecular velocities are restricted to a set of fourteen given vectors. For this model we study the Couette flow problem, the boundary conditions on the walls being the conditions of pure diffuse reflexion. The kinetic equations can be integrated by quadrature under the assumption that the walls have opposite velocities and equal temperatures. The presence on the walls of tangential velocities leads to the consequence that the velocity slip coefficient does not in general vanish when the Knudsen number goes to zero.

Considering the same problem again after the suppression of tangential velocities, we obtain formulae for the velocity and temperature slip coefficients which generalize results of Broadwell (1964b), and which agree qualitatively with experiments.