Numerical calculations show that the flow of viscous incompressible fluid in a circular pipe is stable to small axisymmetric disturbances at all Reynolds numbers. These calculations are linked with known asymptotic results.

The flow under discussion represents an idealization of the bath-tub vortex; distortions of the free surface, finite sink size, and all rigid boundaries have been eliminated from the problem in order to isolate the effect of the non-uniform stretching of vortex lines produced by the sink flow. A boundary-layer type of approximation is made about the axis, which requires that the meridional Reynolds number (N) be large, and since the problem is still intractable, an expansion is made in powers of K = R2/N (where R is the swirl Reynolds number), which measures the strength of the interaction between the swirl and meridional velocity fields. In the limit of zero K the flow is a modified Burgers vortex whose radius decreases to zero at the sink. For non-zero K, the interaction is not restricted to the vortex core, because the presence of the vortex modifies the outer irrotational flow, inducing a radial mass flux into the core, whose dependence on the axial co-ordinate is calculated to the first order in K. The structure of the core is obtained, again to the first order in K, from two co-ordinate expansions, one near the stagnation point on the axis, and the other near the sink, although only the first few terms of the latter can be determined explicitly. It is shown how the methods can be extended not only to higher orders in K, but also to any other narrow viscous vortex in which the vortex lines are stretched non-uniformly away from an internal stagnation point.

Numerical solutions to the Boussinesq equations containing a temperature-dependent viscosity are presented for the case of axisymmetric buoyancy-driven convective flow in a cylindrical cell. Two solutions, one with upflow and the other with downflow at the centre of the cell, were found for each set of boundary conditions that were considered. The existence of these two steady-state régimes was verified experimentally for the case of a cylindrical cell having rigid insulating lateral boundaries and isothermal top and bottom planes.

Using a perturbation expansion it is also shown that only one of these solutions remains stable in the subcritical régime. This, however, seems to be confined to a very narrow range of Rayleigh numbers, beyond which, according to all the evidence presently at hand, both solutions are equally stable for those values of the Rayleigh and Prandtl numbers for which axisymmetric motions occur.

Finally, certain fundamental differences between the problem considered here and that of thermal convection in a layer of infinite horizontal extent are briefly discussed.

It is shown that the principal non-linear interaction of inviscid inertial modes does not produce a resonant response in the steady geostrophic circulation. The rectified geostrophic flow manifested in a closed rotating container by a periodic excitation seems to result from a combination of viscous and non-linear effects within the boundary layers.

The uniform motion of a closed, axisymmetric body along the axis of an unbounded, rotating, inviscid, incompressible fluid is considered on Long's hypotheses that: the flow is steady; the flow is uniform far upstream of the body; the inertial waves excited by the body cannot propagate upstream. The appropriate similarity parameters are k, an inverse Rossby number based on the body length, and δ, the slenderness ratio of the body. It is conjectured that an upper bound to the parametric régime in which the solution implied by Long's hypotheses remains valid, say kδ ≡ k < kc, is determined by the first occurrence, with increasing k, of a local reversal of the flow.

A general solution for the stream function is established in terms of an assumed distribution of dipoles along the axis of the body. The disturbance upstream of the body is found to be proportional to the product of k2 and the dipole moment (total dipole strength) and to fall off only as the inverse distance, as compared with the inverse cube of the distance for a potential flow. The corresponding wave drag is found to depend on the power spectrum of the dipole distribution in the axial wave-number interval (0, k) and to be a monotonically decreasing function of the axial velocity for fixed angular velocity. Asymptotic solutions for prescribed bodies are established in the following limits: (i) K → 0 with δ fixed; (ii) δ → 0 with k fixed; (iii) k → ∞ with kδ fixed. Both the upstream disturbance and the wave drag in the limit (i) depend essentially on the dipole moment of the body with respect to a uniform, potential flow. The limit (ii) is analogous to conventional slender-body theory and yields a dipole density that is proportional to the cross-sectional area of the body. The limit (iii) leads to a singular integral equation that is solved to determine kc and the dipole moment for a slender body.

The results are applied to a sphere and a slender ellipsoid. The upstream axial velocity and the drag coefficient based on Stewartson's results for a sphere are found to differ significantly from Maxworthy's (1969) measurements, presumably in consequence of viscous separation effects. Maxworthy's measured values of upstream axial velocity are found to agree with the theoretical values for an equivalent ellipsoid, based on the sphere plus its upstream wake, for k [lsim ] kc.

Time-dependent motion of a rotating stratified fluid is analyzed within the quasigeostrophic approximation. A few examples of mechanically driven flow are analyzed. It is found that the motion is characterized by the ratio B of the stability frequency and the Coriolis parameter. Thus the ratio of the horizontal and vertical characteristic scale is in general O(B). In particular the decay process caused by a horizontal boundary will penetrate a distance B−1L into the fluid, L denoting the horizontal scale of the motion.

An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.

The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.

Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

The nature of a shear flow over a wavy boundary of small amplitude is investigated. It is found that if the viscosity is small, the nature of the flow is highly dependent on the wave amplitude. If the wave amplitude is truly infinitesimal, the flow is described by the Orr-Sommerfeld equation and in the neighbourhood of the critical layer viscous stresses are important even in the limit of vanishing viscosity. However, if the wave is sufficiently large, viscous stresses may be neglected even in the critical layer. An approximate solution of the inviscid equations of motion is obtained to describe the flow over a small but finite wave in the limit of infinite Reynolds number.

A comprehensive set of exact solutions to the first-order boundary-layer equations has been computed using the finite difference computer programme of Sells, with and without wall cooling. The effects of Prandtl number, wall cooling and Mach number on separation point location were studied. Values of displacement thickness, skin friction coefficient and Stanton number are displayed graphically for the supersonic flow over a circular concave arc, for a subsonic cooled cylinder and for the case of a linearly retarded velocity distribution. The influence of pressure gradient on recovery factor was studied. Velocity and temperature profiles are shown for four cold wall cases. The exact computer results show the errors in many of the more approximate methods available for the case where Ue = U∞(1 - X/L). The importance of second-order effects and the applicability of a first-order solution are discussed briefly.

A computational method is presented for the analysis of two-dimensional flow through a non-uniform gauze. The method, based upon the linearized theory due to Elder (1959), permits solutions for most practical cases to be obtained using relatively simple numerical techniques. Comparison with experimental data shows that the computed solutions are satisfactory provided the restrictions inherent in the linearized theory are observed.

Onset times for convection induced by buoyancy forces have been measured. Results on three aqueous systems are summarized in terms of a critical value for a time-dependent Rayleigh number. Because of the presence of minute traces of unavoidable surface-active contaminants, a ‘free’ water surface behaved as if it were inflexible and laterally rigid so far as determining first convective motion was concerned. The form of the onset motion was observed with schlieren photography for both top and side view. The pattern at onset was frequently in the form of plunging rings. Surface effects were demonstrated with an organic liquid layer in which onset times were measured for convection driven by surface tension gradients as well as by buoyancy forces. The data are compared with some recent predictions of linear stability analyses.

A linear theory is presented for the steady, axially symmetric motion of a stratified fluid in a narrow, rotating spherical annulus with a spherically symmetric gravitational field.

The fluid is driven by a combination of differential rotation of the two shells and differential heating applied at the surfaces of the spheres.

It is shown that the effect of stratification becomes increasingly important at lower latitudes with the Ekman layers on the spheres’ surfaces fading in strength as the geostrophic interior velocities themselves tend toward the shell speeds at lower latitudes.

The Singularities In The Geostrophic Solutions At The Equator Are Removed By A Boundary Layer Whose Detailed Structure Depends On The Ratio Of Horizontal To Vertical Mixing Coefficients Of Momentum And Heat.