A real fluid is contained between two horizontal infinite disks which rotate about a common vertical axis with the same angular velocity. On the upper disk there is an axisymmetric non-uniform temperature distribution with a minimum at the point of intersection of the disk and the axis of rotation. The lower disk is insulated. It is assumed that inertial accelerations are negligible in comparison with Coriolis accelerations and that viscous effects are confined to Ekman layers at the disks. Outside the Ekman layers, therefore, since the motion is axisymmetric, the buoyancy forces, by the geostrophic approximation, drive only an azimuthal component of the velocity field which cannot alter the temperature field. Thus heat is convected only by the secondary circulation which is driven by the viscous forces of the Ekman layers. It is possible then for the secondary flow to be so small that heat is transferred by conduction processes.

This paper analyses the conditions necessary for either conduction or convection processes to predominate and the structure of the velocity and temperature fields in these different situations. In addition the separate effects of a temperature maximum on the upper disk and of replacing the upper disk by a stress-free surface are considered.

Precise data are presented on the Rayleigh-Jeffreys instability in air, argon and carbon dioxide at pressures between 0·6 and 6·0 atm in layers of depths 1/8, 1/4, and 3/4in. A new method yields better resolution and repeatability than the methods employed in the past. Here the gas layer is brought through the state of marginal stability by increasing the pressure while both the temperature and the temperature difference are held constant. The onset of convection is detected by means of a type of heat flow meter ascribed to Prof. L. M. K. Boelter. The equipment and procedure are described and analysed in detail.

The experimentally-determined value of the critical Rayleigh number is 1793, repeatable within a probable deviation of 1% due to random errors. Considering in addition the systematic errors of the instrumentation but not the uncertainties of property values, we place the absolute value within 1793 ± 80. The corresponding theoretical value is 1708.

Finite rates of pressure rise inhibit the development of convection. The results achieved at the least rates of pressure rise are in agreement with steady-state determinations.

The approach employed in this investigation shows great promise as a method for measuring the transport properties of gaseous mixtures or of gases at combined high temperature and high pressure.

Laminar hypersonic flow past a sphere is examined on the basis of the constant-density approximation. The Navier-Stokes equations governing the flow are reduced to a nearly parabolic form so that backward influence is essentially eliminated. Two methods of solution are then used on the resulting equations. The first method is the so-called series-truncation method (local similarity), and the second method is an implicit finite-difference method. The solutions from the two methods are compared for various values of the shock Reynolds number. These solutions are also compared with Lighthill's inviscid constant-density solution for high-shock Reynolds number.

The Orr-Sommerfeld equation is solved for large Reynolds number by use of inner and outer expansion theory. The method is shown to have distinct computational advantages over the method of solution due to Sommerfeld and Lin and to be applicable to a wider class of boundary conditions. The method can be simply extended to other characteristic problems in fluids involving viscous effects concentrated in narrow regions.

Heat-transfer coefficients are calculated for forced convection from a heated flat plate, of finite breadth and infinite span, at zero incidence to a steady stream of viscous, incompressible fluid. The complete range of the Reynolds number R is considered and the results for large R are compared with the Pohlhausen boundary-layer solution for a plate of infinite breadth with the Blasius velocity field.

It is found that the heat transfer from the trailing edge of the plate is important for small Reynolds numbers but steadily diminishes as R increases. The limiting value of the heat-transfer coefficient at the leading edge agrees to good accuracy with Pohlhausen's result and the corresponding overall heat-transfer coefficient is within 3% of Pohlhausen's value for an equal length of the infinite plate measured from the leading edge. The results over the whole Reynolds-number range are probably correct to this order of accuracy.

The Reynolds analogy between skin friction and heat transfer, exactly true at large Reynolds numbers, is found to be inadequate at small values of R, as may then be expected, due to the existence of a pressure gradient parallel to the plate.

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.

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.

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.

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.

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.

The evolution of plane switch-on and switch-off shocks in the presence of small normal disturbances is examined. (Normal disturbances are those in which the perturbed quantities are functions only of time and the distance from the plane of the shock wave.) In the situation under discussion the magnetic diffusivity of the ambient gas is much greater than each of the viscous diffusivities and the thermal diffusivity.

It is shown that small normal disturbances can eventually cause appreciable changes in the flow pattern around either switch-on or switch-off shocks. Numerical computations are carried out for both a specific null switch-on and a specific null switch-off shock, in the presence of a known disturbance. The results of each of these computations, for large times, are in agreement with analytic, asymptotic solutions which are obtained.

A mechanism is suggested whereby switch-on and switch-off shocks, across which an appreciable deflexion of the direction of the magnetic field lines takes place, can adjust themselves to small normal disturbances.

The stability of a two-dimensional Couette-Poiseuille flow is investigated. The primary unidirectional flow is between two infinite parallel plates, one of which moves relative to the other. The results for the case of Poiseuille flow agree with Lin's results and all flows for which the plate velocity exceeds 70% of the maximum velocity of the Poiseuille component of flow are found to be stable for all finite Reynolds numbers. Results of intermediate cases are also obtained.