Some properties of trapped oscillations in an inviscid rotating fluid were studied by Stewartson (1971, 1972). In this paper the modifications necessary to allow for a small viscosity in the fluid are discussed. It is found that concentrated disturbances of the kind used in ray theory are only possible if the rays are thought of as emanating from two sources outside the fluid and at opposite ends of the ray. Otherwise the ray must have a lateral tail, which vitiates the notion of a ray. The double-source character of rays is preserved at reflexions except when at glancing incidence. In this case a substantial part of the ray strength is lost and may give rise to a lateral tail if uncompensated. The focusing of the rays at the limit cycle is also discussed and shown to lead to a catastropic loss of energy, so that a free oscillation if set up rapidly decays. A simple forced oscillation is considered and shown to lead to weak resonances in the bands of trapped oscillations.

McIntyre (1972) has demonstrated that transient nonlinear self-interactions among the lee waves downstream of an obstacle in an axisymmetric, inviscid, rotating flow yield a columnar disturbance that moves upstream of the obstacle. This disturbance is calculated for a dipole, the moment of which increases slowly from 0 to a3, in a tube of radius R as a function of δ = a/R and K = 2ΩR/U (Ω and U being angular and axial velocities in basic flow). The asymptotic limit δ ↓ 0 with k ≡ K δ fixed, which is relevant for typical laboratory configurations and for the abstraction of unbounded flow, yields: (a) an upstream velocity with an axial peak of u0 = 0·011k5 δ U and a first zero at a radius of 2·8U/Ω; (b) an upstream energy flux of roughly $- \frac{3}{2}\pi\rho U^2 R^2 u_0$; (c) an upstream impulse flux of 0·20D (D = wave drag on dipole). The results (a) and (b), albeit based on the hypothesis of small disturbances, suggest that nonlinear self-interactions among the lee waves could be responsible for upstream blocking. The results (a) and (c) imply that, although upstream influence is absent from an unbounded (δ = 0) flow in the sense that the axial velocity vanishes like δ, it is present in the sense that approximately one-fifth of the impulse flux associated with the wave drag appears in the upstream flow.

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcy's law.

The energy flux in gravity-capillary wave spectra has been obtained using Hasselmann's (1962) perturbation analysis for a homogeneous Gaussian sea. As expected, resonant interactions now appear at second order, and a third-order perturbation analysis shows that energy is redistributed from waves with intermediate wavelengths (in the neighbourhood of 1·7 cm) toward gravity and capillary waves. Numerical computations are also obtained for the energy flux and the interaction time of a sharply peaked spectrum consisting of wavenumbers concentrated around a single wavenumber, superposed on a smooth background spectrum. The range of validity of the inviscid results is discussed.

The motion induced in a layer of Boussinesq fluid by moving periodic thermal waves is obtained by numerically solving the complete nonlinear two-dimensional momentum and temperature equations. Three sets of boundary conditions are treated: rigid upper and lower boundaries with symmetrical heating; free upper boundary and rigid lower boundary with heating only at the top; free upper and lower boundaries with symmetrical heating. The nonlinear streamline patterns show that, when the velocity fluctuations are larger than the phase speed of the thermal wave and the mean flow, the convection cells have shapes governed by fluctuating nonlinear interactions. Significant mean velocities can be created even without the characteristic tilt in the convection cells expected on the basis of linear theory. Nonlinear interactions can lead to a mean shear even in the absence of motion of the thermal source. When the viscous diffusion time across the fluid layer is less than or of the same order as the period of the thermal wave, the order of magnitude of the induced mean velocity does not exceed that of the phase speed of the wave, even for intense thermal forcing.

Bounds on the heat transport in a porous layer are derived using the variational method of Howard (1963) and Busse (1969b). The relatively simple structure of the variational problem in the case of porous convection allows one to formulate the theory more simply and to investigate some of the mathematical questions posed by the earlier work. A precise characterization of the solution with N wavenumbers is given. The variational problem is solved exactly among functions with a single overall wavenumber and this solution is in good agreement with a nonlinear perturbation solution of the governing equations and with experiments. An N-wavenumber solution is constructed for large Nusselt numbers by boundary-layer methods. The asymptotic solution is compared with a numerical solution of the problem for N = 2. The comparison supports the boundary-layer assumptions introduced in the asymptotic analysis.

Conjugate flows have been defined generally as flows uniform in the direction of streaming that separately satisfy the relevant hydrodynamical equations, so allowing a transition from one flow to its conjugate to be consistent with mass and energy conservation. In previous studies of various examples, certain general principles have been found to apply to conjugate flows: in particular, one in a pair of such flows is subcritical (subsonic) and the other supercritical (supersonic), the former having greater flow force (i.e. momentum flux plus pressure force). In this paper these principles are confirmed in another field of application, for which the theory of conjugate flows takes a novel course.

The theoretical model defined in § 2 consists of a straight duct of arbitrary cross-section filled with a perfect fluid whose constitutive properties vary with cross-sectional position, and whose primary, prescribed flow is axial with a velocity distribution that may be non-uniform. In § 3 the possibility of a conjugate flow in the same duct is investigated, and its principal properties relative to those of the primary flow are deduced from certain simple inequalities between integrals over the cross-section. A Lagrangian description of the conjugate flow is essential, but the properties in question are established without the necessity of determining this flow explicitly. At the end of § 3, a modification of the model is discussed accounting for dissipative, flow-force conserving transitions (shocks). The application of the theory to flows of non-uniform suspensions of gas bubbles is considered in § 4.

The present work is concerned with the flow past spheres in the Reynolds number range 5 × 104 [les ] Re [les ] 6 × 106. Results are reported for the case of a smooth surface. The total drag, the local static pressure and the local skin friction distribution were measured at a turbulence level of about 0·45%. The present results are compared with other available data as far as possible. Information is obtained from the local flow parameters on the positions of boundary-layer transition from laminar to turbulent flow and of boundary-layer separation. Finally the dependence of friction forces on Reynolds number is pointed out.