The small-gap equations for the stability of Couette flow with respect to non-axisymmetric disturbances are derived. The eigenvalue problem is solved by a direct numerical procedure. It is found that there is a critical value of Ω2/Ω1(Ω1, Ω2 and R1, R2 are the angular velocities and radii of the inner and outer cylinders respectively) of approximately −0·78, above which the critical disturbance is axisymmetric and below which it is non-axisymmetric. In particular for R1/R2 = 0·95, Ω2/Ω1 = −1, the wave-number in the azimuthal direction of the critical disturbance is m = 4. This result is confirmed when the full linear disturbance equations are considered, i.e. the small-gap approximation is not made.

When a potential is applied between two electrodes immersed in a flowing chemi-luminescent solution a blue glow appears at the surface of the anode. This glow may be used to observe the behaviour of liquid flows past solid surfaces. An electrochemical system suitable for generating the glow is described. Photographic studies of separated flows past cylinders, flat plates and cones are presented showing the application of the technique of electrochemiluminescence to flow visualization.

An analysis of the exact equations of the inviscid flow of a perfect gas over cusped concave bodies is described. The field is examined in the limit of infinite free-stream Mach number M∞. The slope of the shock wave in a small region adjacent to the leading edge is strongly dependent on M∞, while much further downstream the shock-wave slope is controlled primarily by the body slope. Consequently the region near the leading edge introduces into the field downstream a thin layer of gas, adjacent to the body, where the entropy is much lower than that of the gas above it. This layer is so dense that the gas velocity along it is not appreciably slowed by the pressure gradient along the body. However, it is so thin that there is little pressure change across it.

The well-known self-similar solutions to the hypersonic small-disturbance equations have previously only been used to study the flow on blunted slender convex surfaces. They are known to behave singularly at the body. It is shown that there is a region on concave power-law shapes where the self-similar solutions are the correct first approximation to the exact inviscid equations in the limit M∞ → ∞; and that, further, they predict the correct first-order surface pressure.

Numerical results for surface pressure from the similar solutions are presented, and comparisons are made with certain approximate theories available for more general shapes. Pressure measurements taken on a cubic surface in the Imperial College gun tunnel are presented and compared with the theoretical distributions.

Steady circulatory motions at the surface of tap-water flowing in an open channel and past a hemispherical air bubble on a channel wall are described and explained by the presence of a film of surface-active material. Boundary conditions for the liquid phase are derived by assuming that the surface film is insoluble and incompressible. Good agreement is obtained between the theoretical and measured velocity fields in the open-channel flow.

Anisotropic grid turbulence $(\overline{w^2} = \overline{v^2} \simeq \overline{u^2}/1.4)$ is passed through an axisymmetric nozzle of small contraction ratio followed by a straight section. Inside the contraction $\overline{v^2}(= \overline{w^2})$ and $\overline{u^2}$ tend to equalize, but their ratio approaches the original pre-contraction value in the straight section behind the contraction. Experiments were repeated with three grids of different mesh sizes and gave similar results.

The case of a dielectric sphere executing small-amplitude oscillations in an incompressible, conducting fluid with a uniform magnetic field aligned along the axis of oscillation is studied. For the case of large Hartmann and Reynolds numbers, the MHD equations are linearized. Solutions correct to O(Rm) are obtained for the velocity, pressure, current, and electric and magnetic fields in the fluid. The MHD drag on an oscillating dielectric sphere in mercury was experimentally determined in aligned fields to 0·8 Wb/m2. The experimental results agree well with the theory. The theory and experiment were extended to include the régime where the perturbed velocity is not small compared with the velocity present in the absence of a magnetic field. The validity of using the inviscid velocity to compute the current is verified by measuring the angular dependence of the induced electric field.

The viscous structure of a separated eddy is investigated for two cases of simplified geometry. In § 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier–Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in § 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.

Fluid impacts on the base of a stably stratified region of fluid cause internal-wave ripples whose spread is predominantly horizontal if the duration of the impacts is long compared with the natural period of the stratified fluid. The development of a single ripple in a slightly viscous fluid is calculated, first with a constant vertical gradient of potential density and then with a gradient varying linearly with height. The single-ripple results are used to find the intensity of the statistically steady wave motion generated by impacts which are randomly distributed in space and time. Above a critical height, dependent on the viscosity and stability of the fluid and on the time and length scales of the impacts, wave energy falls off as the −5/3 power of the height with a constant density gradient and as the −25/6 power with a linearly varying gradient. The predictions are compared with observations of temperature fluctuations in the stable region of an ice-water convection system and with observations of ‘clear-air turbulence’ over strato-cumulus cloud. Reasonable numerical agreement can be obtained with plausible values for the scales of the convective motion which provides the impacts.

In this exploratory study, schlieren photography has been used to reveal a variety of convection patterns in pools of evaporating liquids. Pure materials as well as binary solutions were investigated, both in the presence and absence of surface contamination, which was shown to exert a considerable influence on the convective flow. It was found that, with the exception of water, the observed flow structures were largely dependent on the depth of the evaporating pool and to a much lesser extent on the properties of the liquid. Also, it was possible to identify certain flow patterns as being induced by a surface-tension-driven instability and others as being due to buoyancy-driven convection; still other patterns seemed to be uniquely associated with the presence of surface contamination.

Consideration is given to the possibility of overstability in the Couette flow of viscous and elastico-viscous liquids. The relevant linear perturbation equations are solved numerically using an initial-value technique. It is shown that over-stability is not possible in the case of Newtonian liquids for the cases considered. In contrast, overstability is to be expected in the case of moderately-elastic Maxwell liquids. The Taylor number associated with the overstable mode decreases steadily as the amount of elasticity in the liquid increases, and it is concluded that highly elastic Maxwell liquids can be very unstable indeed.

A simple analysis is presented for the finite-amplitude, steady motions in a rotating layer of fluid which is heated uniformly from below and cooled from above. The boundaries are considered to be ‘free’ and a solution is obtained for the two-dimensional problem using the eigenfunctions of the stability problem plus the smallest number of higher modes required to represent non-linear interactions. In his analysis of the stability problem Chandrasekhar (1953) concluded that in mercury overstable motions can occur for a value of the Rayleigh number which is as little as 1/67 of the value required for instability to steady motions. In the present paper it is shown that, for a restricted range of Taylor number, steady finite-amplitude motions can exist for values of the Rayleigh number smaller than the critical value required for overstability. The horizontal scale of these finite-amplitude steady motions is larger than that of the overstable motions. A more exact solution to the finite-amplitude problem confirms the above results. The latter solution together with additional physical results will be presented in a later paper.

Large spanwise variations of boundary-layer thickness and surface shear have been found recently in wind tunnels designed to maintain two-dimensional flow. Bradshaw (1965) argues that these variations are caused by minute deflexions in the free-stream flow rather than by any intrinsic instability of the boundary layers. This paper is a study of the effect of a small, periodic transverse flow on a flat-plate boundary layer. The perturbation flow Reynolds number is assumed to be O(1) as it is in the experiments.

Interactions between waves in a stably stratified fluid with a free or fixed upper surface may occur at second order when the horizontal wave-numbers, ki, and frequencies, σi, of the three interacting waves satisfy the relations \[ {\bf k}_1 \pm{\bf k}_2 \pm {\bf k}_3 = 0,\quad \sigma_1 \pm \sigma_2 \pm \sigma_3 = 0. \] These relations may be satisfied in the case when two free surface waves interact with an internal wave, or in the case when all three waves are internal, provided that they do not all belong to the same mode. The theory is applied to situations which might be realized in the laboratory.

The stability of forward and rearward slug flows due to the motion of a body through a perfectly conducting liquid with an embedded magnetic field aligned to the direction of motion is studied. It is found that, for sub-Alfvénic flows, the forward and rearward slug flows predicted by Stewartson, Leibovich and Ludford are stable. For super-Alfvénic flows, if at all meaningful, both the forward and the rearward slug flows are stable also.

In this paper the viscous, slightly stratified flow towards a sink is investigated. The fluid is assumed incompressible, linearly stratified in density, and the flow steady. The theoretical portion of this paper includes both the two-dimensional and the axisymmetric cases, whereas the experimental portion includes only the two-dimensional case. The solutions obtained indicate that there exists a with-drawal layer symmetrically situated about the horizontal plane of the sink. The flow occurs in this layer while outside this layer there is essentially no motion. This withdrawal layer grows in thickness with the distance x (or r) from the sink at a rate proportional to $x^{\frac{1}{3}}$ (or $r^{\frac{1}{3}}$). The velocity distributions u(y) (or u(z)) are similar from one station x (or r) to another.

The correspondence between the eigenvalues for the problem of the onset of convection in a fluid confined between two horizontal plates and for the stability of viscous flow between two cylinders rotating at almost the same angular velocity has been known for some time. The recent work of Chandrasekhar (1961) has prompted the extension of the analogy to a larger group of rotating cylinder problems and their associated convection cases in which the primary temperature distribution is parabolic. This paper shows the analogy between these two problems and presents data which give the corresponding temperature distribution for a given ratio of angular velocities between the two cylinders. The equivalent Rayleigh numbers are listed for the Taylor numbers given by Chandrasekhar (1954). The eigenfunctions for several of the parabolic temperature profiles are determined. These results show that the single vortex convection pattern becomes a double vortex for certain initial temperature distributions. The critical Rayleigh numbers for the stability of a layer of water which is near 4 °C is also found by analogy.

A viscous vortex ring is considered as an initial disturbance to a uniformly rotating fluid. Inertial waves are generated by the disturbance and propagate through the fluid affecting the diffusion of the vortex ring with time t. The rate of diffusion is found to be proportional to $t^{\frac{3}{2}}$. This result is compared with the diffusion rates for non-rotating fluids and for conducting fluids under a uniform magnetic field.

An exothermal hydromagnetic discontinuity in a perfectly conducting gas is studied for the case of an arbitrarily orientated magnetic field. The evolutionary modes and the manner of variation of the physical properties for such phenomena are determined. The Hugoniot adiabatic is obtained and analysed.

This is the second of two articles by the authors dealing with asymptotic expansions for forced-convection heat or mass transfer to laminar flows. It is shown here how the method of the first paper (Acrivos & Goddard 1965), which was used to derive a higher-order term in the large Péclet number expansion for heat or mass transfer to small Reynolds number flows, can yield equally well higher-order terms in both the large and the small Prandtl number expansions for heat transfer to laminar boundary-layer flows. By means of this method an exact expression for the first-order correction to Lighthill's (1950) asymptotic formula for heat transfer at large Prandtl numbers, as well as an additional higher-order term for the small Prandtl number expansion of Morgan, Pipkin & Warner (1958), are derived. The results thus obtained are applicable to systems with non-isothermal surfaces and arbitrary planar or axisymmetric flow geometries. For the latter geometries a derivation is given of a higher-order term in the Péclet number expansion which arises from the curvature of the thermal layer for small Prandtl numbers. Finally, some applications of the results to ‘similarity’ flows are also presented.

Numerical methods are used to investigate the steady two-dimensional motion of a viscous incompressible fluid past a flat plate of finite breadth at zero incidence to a uniform stream. Before application of numerical techniques, the governing partial differential equations for the stream function and vorticity are reduced to ordinary differential equations by an adaptation of methods normally used to solve Oseen's linearized equations. The complete range of the Reynolds number R is considered, from indefinitely small to indefinitely large. All the results are intended to represent solutions of the full Navier-Stokes equations of motion, although in practice approximations are inevitable. These are mainly brought about by the necessity of limiting the size of the calculations.

At the lower end of the Reynolds-number range, the calculated frictional drag coefficient agrees well with the results of Tomotika & Aoi (1953) based on Oseen's equations. At intermediate and higher Reynolds numbers there is good agreement with the experimental results of Janour (1951) and with the improvement of the Blasius solution given by Kuo (1953). Finally a limiting solution is obtained as R → ∞. This shows that the drag coefficient is proportional to R−½, in accordance with boundary-layer theory. The actual calculated value of the coefficient is about 4% higher than the Blasius value.

Although the present results tend generally to confirm the trend of the recently published results at R = 0·1, 1 and 10 of Janssen (1957), there are substantial discrepancies in the detailed results in a number of instances. In particular, the drag values obtained at R = 1 and 10 are some 20% higher than Janssen's although there is reasonable agreement at R = 0·1. It seems possible that Janssen's analogue is a little crude at the higher Reynolds numbers.