In order to gain insight into the macroscopic behaviour of gas interacting with electric and magnetic fields, while its degree of ionization is changing, the one-dimensional unsteady flow of gas with temperature-dependent conductivity is investigated. A simple continuum model is used for the gas. It is assumed to be compressible and to obey polytropic gas laws and its conductivity is supposed to change discontinuously at a critical temperature. Particular attention is devoted to the various types of transition boundaries which can occur as the conductivity changes, and it is shown that complete flow patterns can be constructed with these as elements. A wide variety of transitions are possible and some of these have remarkable properties. For example, it is shown that discontinuous deionising fronts can exist and are expansive in nature. This contrasts sharply with the situation in conventional gas dynamics and magneto-gasdynamics where only compressive discontinuities can exist.

The results given by Brenner & Cox (1963) for the resistance of a particle of arbitrary shape in translation at small Reynolds numbers are generalized. Thus we consider here a single particle of arbitrary shape moving with both translation and rotation in an infinite fluid, the Reynolds number R of the fluid motion being assumed small. With the additional assumption that the motion is steady with respect to some inertial frame of reference, we calculate both the force and couple on the body as an expansion in the Reynolds number to O(R2 In R). This force and couple are expressed entirely in terms of various Stokes flows for the given body in rotation or translation.

A discussion is given of the form taken by the formulae for the force and couple for cases in which the body possesses symmetry properties. Quantitative results are obtained for both a spheroid and a dumb-bell-shaped body in pure translation and also for a translating rotating sphere and for a dumb-bell-shaped body in pure rotation.

The application of the general results to ‘quasi-steady’ problems is considered, with particular reference to a freely falling spheroid (of small eccentricity) which is shown to orientate itself so that it is broad-side on to its direction of motion.

Finally the general results are compared with those that would be obtained by the use of the Oseen equations. By consideration of a particular example it is shown that the Oseen equations do not in general give the correct value of the force on the body to O(R).

The normal reflexion of a plane shock wave from a plane wall is investigated experimentally using a new pressure gauge which has a rise time of 0.1 μsec and no overshoot. Wall-pressure histories of 5 μsec duration can be obtained with the present version of this gauge. The experiments were conducted on the end wall of the GALCIT 17 in. diameter shock tube.

Experimental results for three aspects of the reflexion process are presented and discussed: (1) effect of a cold wall, (2) effect of a vibrationally relaxing gas, and (3) profile for a reflecting shock front.

It is concluded that the effect of a cold wall introduces a comparatively minor perturbation to the wall-pressure history, since the ‘thickness’ of the shock-front profile exhibited by the wall-pressure history is comparable to the ‘thickness’ of the incident shock front, as measured by a stationary observer, and the pressure jump across the profile is about 90% of the theoretical value for a thermally insulated wall. Also, the wall-pressure history immediately behind the reflected shock front can be approximated by boundary-layer theory. The effect of a vibrationally relaxing gas was studied in carbon dioxide, and it is shown that the relaxation process behind the incident shock wave produces a large effect on the recorded pressure history, which provides a method for measuring the vibrational relaxation time in carbon dioxide at high temperatures.

The general solutions of the Stokes approximate equations of motion are derived for two-dimensional and axisymmetric flows in the half-space x > 0, for an arbitrarily given velocity field in the plane x = 0. There is assumed to be no solid surface in the half-space. According to whether the velocity at infinity is zero or not, the solutions can be said to describe either jet-type or wake-type flows. Only the latter category is considered; numerical examples are worked out and properties of the base flow at very low Reynolds numbers are investigated. A recirculating flow region may exist, but the flow properties are not sensitive to this feature.

The influence of inertia terms in the equations of motion on the properties of the base-flow and near-wake flow at very low Reynolds numbers is investigated by using Oseen's approximation and by comparing with the results obtained in Part 1 for Stokes flow. The general solutions of Oseen's equations of motion are derived for two-dimensional and axisymmetric flows in the half-space x > 0, for an arbitrarily given velocity field in the plane x = 0. Numerical examples are given for two-dimensional flow and compared with Stokes-flow examples.

Measurements of cavities behind disks in a water tunnel show that at high enough speeds the cavity pressure is not more than 30–40% higher than water-vapour pressure, even when at atmospheric pressure the tunnel water is nearly saturated with air. No other hard conclusions can be drawn from the investigations, which showed many puzzling features, discussed below. It is hoped, however, that the qualitative discussion may prove useful to theoreticians who may seek to improve existing mathematical models of cavitation by taking a more realistic account of the physics of the flow.

An approximate analysis is developed for the propagation of water waves past long obstacles by considering separately the effects of diffraction at each end. The motion is two-dimensional, and linearized potential flow is assumed. Reflexion and transmission coefficients are obtained for the long obstacle, and it is shown that for suitably chosen values of the obstacle length there is complete transmission due to interference between the two ends. A comparison is made with experiments for the case of a rectangular obstacle.

This study is concerned with the phenomenon of a liquid jet impinging normally on the surface of a heavier immiscible liquid. In essence, the work is a continuation of a previous investigation which involved an air jet impinging on a water surface.

An analytical model is presented which relates the velocity of the jet in the neighbourhood of the interface to the depth of cavity created by the jet. The effect of interfacial tension is taken into account. Numerous experiments were conducted utilizing both circular and plane jets. An oil jet impinging on water and a water jet impinging on carbon tetrachloride were studied. The resulting experimental data were analysed in terms of the dimensionless quantities obtained from the analytical model.

Results obtained from the circular-jet experiments were in close agreement with the results of the previously conducted air-water tests. However, there were some discrepancies in the plane-jet results; side-wall boundary-layer effects may be the explanation. Depending on the relative values of the Froude and Weber numbers, interfacial tension may be important in determining the depth of cavity created by a jet. Finally two intriguing phenomena, viz. interfacial shearing and cavity oscillation, were observed in the plane-jet experiments.

The structure of a self-preserving turbulent plane jet exhausting into a slow-moving parallel airstream is studied. The investigation includes results of turbulence measurements and the structure is compared with that of a self-preserving plane wake. The results show that self-preservation is established at a distance of about thirty jet widths downstream of the jet nozzle and that, in the self-preserving region of the jet, the distributions of the turbulent intensities and shear stress across the jet are very similar to those found in the plane wake. The distribution of the intermittency factor, however, is found to be more like that found in an axi-symmetric jet than in a plane wake. The turbulent energy balance also shows important differences to that of the wake flow. The unsteady irrotational flow outside the turbulent shear layer is investigated and it is found that the experimental results agree with the predictions of the theories of Phillips (1955) and Stewart (1956). Some comments are also made on the eddy structure and the applicability of the simple theories of turbulence.

Some of the processes relevant to the propulsion of small organisms are investigated using the simple mathematical model of two-dimensional waves passing through a sheet immersed in a viscous fluid. This model was first used by Taylor, who considered an inextensible sheet moving in an unbounded fluid of negligible inertia. Here the effects of fluid inertia, of straining of the wave-bearing surface, and of nearby walls are included in the study. The applicability of the results is restricted both by the unrealistic geometry of the model and by the method of analysis which gives results valid for small Reynolds numbers and for small wave amplitudes only. However, the following general results may have counterparts in nature.

The effect of fluid inertia is to increase the propulsive speed for a particular wave amplitude. Straining of the waving surface will probably reduce the propulsive velocity for a given amplitude, although there exist modes of surface straining that give augmented propulsion. If the wave celerity and the energy output in swimming remain constant in the presence of a solid wall, the amplitude of the wave is reduced as the wall is approached while the propulsive speed first rises slightly and then drops. It appears further that an organism swimming near a wall may induce a shear pattern which directs it away from the wall.

The constraint imposed by rapid rotation stabilizes convective motions in a plane liquid layer, so that the critical Rayleigh number which marks their onset increases as the two-thirds power of the Taylor number. Here we formulate an asymptotic approximation to the equations governing an analogous problem in a sphere. The solutions we obtain for the asymptotic equations predict the same power law for the dependence of the corresponding critical Rayleigh number on the Taylor number. The analysis is limited at the outset to the case of axially symmetric motions, and the present result, taken in conjunction with earlier results for slow rotation, essentially completes the treatment of symmetric, convective modes of instability.

The results of Proudman & Pearson (1957) and Kaplun & Lagerstrom (1957) for a sphere and a cylinder are generalized to study an ellipsoid of revolution of large aspect ratio with its axis of revolution perpendicular to the uniform flow at infinity. The limiting case, where the Reynolds number based on the minor axis of the ellipsoid is small while the other Reynolds number based on the major axis is fixed, is studied. The following points are deduced: (1) although the body is three-dimensional the expansion is in inverse power of the logarithm of the Reynolds number as the case of a two-dimensional circular cylinder; (2) the existence of the ends and the variation of the diameter along the axis of revolution have no effect on the drag to the first order; (3) a formula for drag is obtained to higher order.

Similarity solutions in which the vorticity decays algebraically towards the outer limit of the viscous layer are shown to be possible limit solutions of the full boundary-layer equations with exponential decay. The discussion is illustrated by consideration of a mainstream applicable to viscous flow in a cone.

The effect of viscosity on the trailing-wave system well downstream of a uniformly moving surface disturbance is determined. Solutions are obtained by series expansions in inverse powers of a Reynolds number and coefficients in the viscous exponential decay-factor found. It is seen that the diverging wave pattern is much more heavily damped than the transverse waves.

Three results on the flow of an infinitely conducting and inviscid fluid are presented in this paper. The first result is that all steady flows with magnetic lines coincident with streamlines are reducible to flows without a magnetic field. The second result is on the establishment of a steady irrotational and current-free flow with coincident streamlines and magnetic lines. It throws some light on the controversy between Stewartson (1960) and Sears & Resler (1959) concerning the possibility of such a flow. The third result concerns the flow of a fluid through a circular cylinder of radius R into a point sink with strength m when the fluid carries a current of density j0 at infinity. It is shown that the condition of uniform flow at infinity is impossible to maintain if a dimensionless number (kR)2 involving the current density j0 exceeds the value (3·831)2, and that the current has the effect of concentrating the flow near the centre line and of producing ring eddies which become longer and longer as (kR)2 is increased.

A solution is presented of the asymptotic flow due to a point source of momentum in a uniformly rotating unbounded environment. Under the condition that the relative swirl velocity of the jet is small compared with the ambient swirl velocity, the equations of motion reduce to a set of linear equations. These equations are expressed in terms of similarity variables and a single ordinary differential equation is derived in terms of the similarity stream function. The profiles of the flow are calculated numerically.

The jet is shown to have a narrow viscous core whose thickness increases with distance z from the virtual source of momentum as ($(vz|\Omega)^{\frac {1}{3}}$, where v is the kinematic viscosity and Ω the ambient angular velocity.

For fully-developed turbulent flow in straight channels of non-circular cross-section, there exists a transverse mean flow superimposed upon the axial mean flow. This transverse flow, commonly known as secondary flow, interacts with the axial mean flow and turbulence structure in a complex manner. In this paper several heretofore unexplored aspects of this type of secondary flow are discussed on the basis of results of an extensive experimental programme which was conducted for steady, incompressible, fully-developed turbulent air flow in both square and rectangular channels. Specifically, the following aspects are examined: (a) the Reynolds-number effect on secondary flow, (b) the directional characteristics of local wall shear stress, (c) the orientation of Reynolds-stress principal planes in a plane normal to the axial flow direction, and (d) the Reynolds equation along a secondary-flow streamline.

Within the Reynolds-number range of the investigation, the results indicate that secondary-flow velocities, when non-dimensionalized with either the bulk velocity or the axial mean-flow velocity at the channel centreline, decrease for an increase in Reynolds number. Also, the greatest skewness of local wall shear-stress vectors is shown to occur in the near vicinity of corners where secondary flow is maximum. In addition, it is shown that in planes normal to the axial flow direction, traces of Reynolds stress principal planes are not tangent and normal to lines of constant axial mean-flow velocity. This behaviour is in contrast to that for less complicated turbulent flows, for example, two-dimensional channel flow or circular-pipe flow where such traces are always tangent and normal to lines of constant axial mean-flow velocity in accordance with symmetry considerations. Finally, through experimental evaluation of terms in a momentum balance along a typical secondary-flow streamline, it is shown that secondary flow is the result of small differences in magnitude of opposing forces exerted by the Reynolds stresses and static pressure gradients in planes normal to the axial flow direction.

A method is presented in this article for deriving higher-order correction terms to the well-known asymptotic results for laminar forced-convection heat and mass transfer, and a formula is obtained for computing under fairly general conditions the first correction term to the asymptotic Nusselt number at large Péclet numbers for flows with small or moderate Reynolds numbers. This result is then applied to the problem of heat transfer from a solid, isothermal sphere in Stokes flow, to yield the asymptotic expression for the average Nusselt number,

for Pe→ ∞, Re→ 0, where and Pe are based on the radius of the sphere.

The non-linear equations of the Bénard convection problem are transformed to the spectral domain. The spectral basis consists of the supercritical normal modes of the characteristic-value problem in which the exponential growth rates are characteristic values. The norm of the spectrum is the variance of an arbitrary finite-amplitude state of convection. The equations that govern the spectrum are solvable by linear methods when the spectrum is truncated by exclusion of all convective modes except those of lowest-order symmetric vertical structure. Numerical computations of heat flux for a spectrum that contains only one convective mode are in good agreement with experimental data for water in the laminar régime.

This is largely an experimental study of the interaction of buoyancy and shear forces in the free convective flow of a liquid in a rectangular cavity across which there is a uniform temperature difference, ΔT, produced by maintaining the two vertical walls at two different temperatures. The height of the cavity, H, is made larger than the width of the cavity, L, and the cavity is sufficiently long in the third dimension for the mean flow to be nearly everywhere two-dimensional. The flow is specified by three dimensionless parameters: σ, the Prandtl number; h = H|L, the aspect ratio; A = γgΔTL3|κν, the Rayleigh number. The experiments are generally restricted to h = 1–60, σ ÷ 103 and A < 108.

For A < 103 the temperature field closely satisfies Laplace's equation but a weak stable unicellular circulation is generated. The flow is vertical throughout the slot except for regions within a distance of order L from the ends.

For 103 < A < 105, large temperature gradients grow near the walls, and in an interior region a uniform vertical temperature gradient is established. The flow is similar to that near an isolated, heated, vertical plate except that the vertical growth of the wall layers is inhibited in the central part of the slot by the presence of the other layer which prevents entrainment of fluid.

Near A = 105 the interior region of the flow generates a steady secondary flow. A regular cellular pattern becomes superimposed on the basic flow to produce a ‘cats-eye’ pattern of streamlines. Near A = 106 when the secondary cell amplitude is large, a further steady cellular motion is generated in the weak shear regions between each cell.