The paper presents an analysis of laminar motion of a conducting liquid in a rectangular duct under a uniform transverse magnetic field. The effects of the duct having conducting walls are investigated. Exact solutions are obtained for two cases, (i) perfectly conducting walls perpendicular to the field and thin walls of arbitrary conductivity parallel to the field, and (ii) non-conducting walls parallel to the field and thin walls of arbitrary conductivity perpendicular to the field.

The boundary layers on the walls parallel to the field are studied in case (i) and it is found that at high Hartmann number (M), large positive and negative velocities of order MVc are induced, where Vc is the velocity of the core. It is suggested that contrary to previous assumptions the magnetic field may in some cases have a destabilizing effect on flow in ducts.

The structure of plane steady shock waves in a gas with several internal energy modes which relax in parallel is investigated. Transport effects are neglected. Conditions for continuity and monotonicity of the velocity profile are discussed; when all modes have constant specific heats and relaxation times it is established that velocity must decrease monotonically. Internal mode energy contents may overshoot their local equilibrium values.

Numerical results for waves in a hypothetical gas with two relaxing modes are presented for purposes of illustration.

A digital computer method for solving certain problems involving two-dimensional incompressible viscous flow is described. The time-dependent case is treated; the mathematical problem is thus that of solving a non-linear fourth-order partial differential equation in three variables. The choice of difference equations, of relaxation procedure, the kind of approximation to boundary conditions, and the resulting computational stability, speed, and accuracy are considered. Most experience so far has been for a rectangular region for which boundary velocities are prescribed as certain functions of time; an example of one such problem showing vortex formation and break-up is given.

The nature of the steady-state viscous flow between two large rotating disks has often been discussed, usually qualitatively, in the literature. Using a version of the numerical method described in the preceding paper (Pearson 1965), digital computer solutions for the time-dependent case are obtained (steady-state solutions are then obtainable as limiting cases for large times). Solutions are given for impulsively started disks, and for counter-rotating disks. Of interest is the fact that, at high Reynolds numbers, the solution for the latter problem is unsymmetrical; moreover, the main body of the fluid rotates at a higher angular velocity than that of either disk.

The stability of a viscous liquid between two concentric rotating cylinders with an axial flow has been investigated. Attention has been confined to the case when the cylinders are rotating in the same direction, the gap between the cylinders is small and the axial flow is small. A perturbation theory valid in the limit when the axial Reynolds number R → 0 has been developed and corrections have been obtained for Chandrasekhar's earlier results.

Theoretical studies of aerodynamic noise suggest that the sound field of supersonic flows will be dominated by eddy Mach waves. Recent experimental evidence supports this view. In supersonic turbulent boundary layers, and rocket exhaust flows, turbulence occurs in regions of high mean velocity gradient. At low speed, such gradients are known to amplify the sound emitted by turbulence. This paper deals with the corresponding Mach wave problem. The exact equations of sound radiation by turbulence are rearranged in a form where the equivalent sources, derivatives of the turbulence stress tensor, are shown to be dominated by one term. That term is formed from the product of the mean velocity gradient and the rate of change of density. It seems that its resemblance to the dominant source of sound in low speed shear flows is largely fortuitous. In the Mach wave case, the theory is designed to include effects of both temperature gradients and density perturbations, and the approximations of the estimate are of a type that would not be expected to be valid away from the Mach wave condition. The basic theory is used to make an estimate of the sound radiated from supersonic boundary layers, and an approximate equation relating the radiated pressure to the surface pressure is derived. Experimental evidence is then examined to show that the equation is in excellent agreement with observation. The theory is then applied to annular shear flows of the rocket exhaust type. Again an approximate equation relating near and far field pressures is established, and the paper concludes with suggestions for experiments that could check the result.

This paper deals with the calculation of the convective heat transfer rate to the end-wall of a shock tube from a monatomic gas heated by a reflected shock. We consider a range of shock strengths for which the equilibrium thermodynamic state is one of appreciable ionization. The resulting boundary-layer problem involves the thermal conductivity and ambipolar diffusion coefficient for a partially ionized monatomic gas. The formulation here is restricted to the case of a catalytic wall and equal temperatures for all species. We ignore the effect of the plasma sheath at the wall. Consideration is given to three limiting cases for which similarity-type solutions of the boundary-layer equations may be found: (1) complete thermodynamic equilibrium behind the reflected shock and within the boundary layer; (2) equilibrium behind the reflected shock, but no gas-phase recombination in the boundary layer; (3) no ionization in either region. Numerical calculations are carried out for argon using estimated values of thermal conductivity and ambipolar diffusion, and compared with shock-tube experiments of Camac & Feinberg (1965). For no ionization, calculations were made with thermal conductivity varying as the ¾ power of the temperature, which fits the estimates of Amdur & Mason (1958) up to 15,000°K. Excellent agreement with experiment is obtained confirming an extrapolation of this power law up to 75,000°K. For ionized cases, based on estimates of Fay (1964), the theory predicts heating rates 20–40% lower than measured values. Some possible reasons for this discrepancy are discussed.

An infra-red heat-transfer gauge was used in a shock tube for end-wall measurements of the convective heat transfer from argon behind the reflected shock. The thermal conductivity of neutral (un-ionized) argon was measured before the ionization-relaxation time, and was fitted with the power-law temperature dependence 4·2 × 10−5(T/300)0·76±0·03 cal/sec cm°K, where T is measured in °K, and ±0·03 refers to the probable error The free-stream temperature ranged from 20,000 to 75,000°K, corresponding to incident-shock velocities from 3 to 6mm/μsec. At later times, after the free stream established equilibrium ionization, the convective-heat-transfer rate remained the same as the initial rate with neutral argon. Theoretical predictions of Fay & Kemp (1965), assuming equilibrium-boundary-layer conditions, are 20–30% below the experimental values. Also reported in this paper are measurements of the ionization times behind the reflected shock, and these are in agreement with an extrapolation of the Petschek & Byron (1957) measurements behind the incident shock. There is a discussion of the large changes in the gas conditions behind the reflected shock due to the ionization process. The final equilibrium conditions are reached abruptly, as indicated by the continuum-radiation emission which becomes constant immediately after ionization relaxation.

The concept of a law of the wall and a velocity defect law which are related to each other through a common velocity scale and a semilogarithmic velocity profile in the region where they overlap, can be applied successfully to turbulent boundary layers with suction or injection. For turbulent boundary layers at moderate suction rates the velocity scale is proportional to $u_{r}^{2}/{{\nu }_{0}}$. For layers at arbitrary suction or blowing rates the velocity scale, which has been determined empirically, is proportional to (ur + 9v0).

A mono-energetic electron plasma is trapped in a one-dimensional parabolic potential well. In the undisturbed state a suitable background charge provides space-charge neutrality. It is shown that stationary disturbances of infinitesimal amplitude can exist in the plasma for certain critical values of the parameter $\Lambda \,=\,{4\bar{n}{{e}^{2}}}/{m{{\omega }^{2}}}\;\$, where $\bar{n}\$ is the mean electron density and ω/2π is the frequency of oscillation of the electrons in the well. The first few critical values are Λ = 0, 4·12, 8·2.

The boundary conditions at the end of the plasma are non-linear. As a result stationary disturbances of finite amplitude in a given mode, say the rth, require that Λ shall exceed Λr. Further it can be shown that disturbances of small amplitude in the rth mode are unstable when Λ exceeds Λr. This applies even for Λr = 0; in this case there exist nearby an unstable even and an unstable odd mode.

It seems likely that these results can be extended to all cases in which the potential well is symmetrical.

The flow round an elliptic cylinder immersed in a finite volume of fluid and influenced by an electric field is determined, under the boundary conditions arising from electrosmosis at the surface of the cylinder. This is a problem in slow viscous flow with difficult boundary conditions, and is solved by numerical analysis, using the relaxation method. After the flow pattern has been determined, the mechanical action on the cylinder is calculated. An experimental application based on the theoretical results is suggested as a method for the determination of electrokinetic potentials.

This paper is concerned with deducing the most important features of the steady separated flow past a circular cylinder in the limit of vanishing viscosity. First of all, it is shown that the experimental results reported in an earlier article cannot be reconciled with the notion that, as the Reynolds number Re is increased, the flow becomes inviscid everywhere and that viscous effects remain confined within infinitesimally thin shear layers. In contrast, the limiting solution is visualized as exhibiting three essential features: a viscous, closed ‘wake bubble’ of finite width but with a length increasing linearly with Re in which inertial and viscous effects are everywhere of equal order of magnitude; an outer inviscid flow; and, separating the two regions, a diffuse viscous layer covering large sections of the external field. Further properties of this asymptotic solution include: a finite form drag, a negative rear pressure coefficient at the rear stagnation point of the cylinder, and a Nusselt number for heat transfer which becomes independent of Re along the non-wetted portion of the cylinder surface. This model is shown to be consistent with all the experimental data presently available, including some new heat transfer results that are presented in this paper.

An approximate technique is also proposed which takes into account the asymptotic character of the flow in the vicinity of the cylinder and which predicts the pressure distribution around the cylinder in good agreement with the experiments.