This paper is concerned with the dispersion of a dynamically neutral material quantity in a fluid flowing throuh a porous medium. The medium is regarded as an assemblage of randomly orientated straight pores, and it is assumed that the path of a marked element of the material quantity consists of a sequence of statistically independent steps whose direction and duration vary in some random manner. The probability density function for the displacement of a single marked element is calculated and values for the dispersion of a cloud of marked elements then follow.

The case is examined in which the flow satisfies Darcy's law (i.e. the mean velocity is linearly proportional to the mean pressure gradient), and the molecular diffusivity is sufficiently small for the dispersion to be primarily due to the randomness of the streamlines, but it is not assumed that effects of molecular diffusion can be altogether neglected. It is shown that the longitudinal dispersion in the direction of the mean flow may be described asymptotically by an effective diffusivity which is a function of U, l, a, K and T. (U denotes the average velocity, l the pore length, a the pore radius which is shown to be related to the permeability, k the molecular diffusivity, and T the time from the initial instant.) Expressions for the longitudinal diffusivity kl are obtained according to the relative values of l/U, T, t0 = l2/2k and t1 = a2/8k. These are given in §4, equations (4.3), (4.4) and (4.5). Speaking roughly, when t0 [Gt ] T [Gt ] l/U,kl/Ul is alogarithmic function of UT/l and increases with T; when T [Gt ] t0 [Gt ] l/U, which must eventually be the case however small k,kl/Ul is alogarithmic function of Ul/k and independent of T. The theoretical results are compared with experimental data reported in the literature and approximate agreement is obtained when E is put equal to the average diameter of the particles composing the porous medium.

The lateral dispersion in the direction perpendicular to that of the mean velocity is found to be governed asymptotically by an effective diffusivity $k_t = \frac {3}{16} Ul$. However, it is pointed out that some of the assumptions, namely that successive steps are statistically independent and that the dispersion of a cloud follows immediately from the statistical properties of the displacement of a single marked element, may not be valid for the lateral dispersion and this result is therefore suspect.

Remarks are made in §5 on the dispersion for high values of the Reynolds number Ul/v (v = kinematic viscosity) when Darcy's law is not obeyed, and it is argued that kl/Ul should decrease as Ul/v increases.

It has been observed experimentally that small bubbles in pure liquids can be held stationary or driven downwards by means of a sufficiently strong negative temperature gradient in the vertical direction. This effect is demonstrated to be due to the stresses resulting from the thermal variation of surface tension at the bubble surface. The flow field within and around the bubble is derived, and an expression for the magnitude of the temperature gradient required to hold the bubble stationary is obtained. This expression is verified experimentally.

Measurements of heat transfer from circular wires placed normal to a horizontal airstream have been made in the Reynolds number range 0·01 to 140. The Nusselt number can be related to the Reynolds number and temperature loading by an expression of the form $N \left(\frac {T_m}{T_\infty} \right)^{-0\cdot 17} = A + BR^n,$

where the values of n, A and B (see table 3) depend on whether the Reynolds number is above or below the value for which a vortex street exists in the wake of the wire. This value of the Reynolds number (R [eDot ] 44) is independent of the intensity and scale of the stream turbulence. The theoretical heat transfer relation based on the Oseen approximation is approached asymptotically as R → 0, provided free convection is negligible.

Free convection effects diminish rapidly with increasing Reynolds number so that the orientation of the wire with respect to the vertical has a negligible influence on heat transfer except at very low velocities. For horizontal wires at very low Reynolds numbers, free convection is significant, when, roughly speaking, the Reynolds number is less than the cube root of the Grashof number.

The transfer of heat by forced convection from a hot wire in a low Mach number rarefied gas stream is investigated for small Reynolds numbers. A simple expression is found for the Nusselt number, and it is suggested that this has reasonable validity over a wide range of Knudsen numbers, even into the free-molecule flow régime.

The interactions of discontinuities, in which detonation waves participate, are analysed. For this purpose detonation polars are introduced and their properties are examined. A full list of possible interactions is then given, and the use of the polars for the quantitative solution of intersection problems is illustrated.

If a series of air trajectories is drawn from a fixed point at a certain height above the earth's surface, the end-points of the superimposed trajectories, after each has been followed for a time t, will have a standard vector deviation S. The purpose of this paper is to discuss the form of S. One expression is obtained by relating S empirically to the magnitude of the mean vector position [x] of the ends of the trajectories, so that $S = CL |[X|L]|^a,$

where L is a certain length and C and a are dimensionless constants; a is found to be about 0·870 and C is nearly proportional to $\surd T$, where T is the total period over which the series of trajectories is initiated.

A more satisfying expression for S connects it with the variability of the wind during the priod of initiation of the trajectories, i.e. with the standard vector deviation of wind (ω) which is itself dependent on T. The relationship involves the Lagrangian correlation coefficient r(t) of wind with time along the trajectories. It is found that the variation of S with time t agrees well with that deduced from the exponential formula r(t) = e-αt. Moreover, α, the reciprocal of the characteristic time of the ‘eddies’, is identified with ω2/K where K is the coefficient of eddy viscosity for lateral mixing.

When a fluid which is lighter than its surroundings is emitted by a source under a sloping roof (or a heavier fluid from a source on a sloping floor), it may flow as a relatively thin turbulent layer. The motion of this layer is governed by the rate at which it entrains the ambient fluid. A theory is presented in which it is assumed that the entrainment is proportional to the velocity of the layer multiplied by an empirical function, E(Ri), of the overall Richardson number for the layer defined by Ri = g(ρa - ρ) h/ρaV2. This theory predicts that in most practical cases the layer will rapidly attain an equilibrium state in which Ri does not vary with distance downstream, and the gravitational force on the layer is just balanced by the drag due to entrainment together with friction on the floor or roof.

Two series of laboratory experiments are described from which E(Ri) can be determined. In the first, the spread of a surface jet of fluid lighter than that over which it is flowing is measured; in the second, a study is made of the flow of a heavy liquid down the sloping floor of a channel. These experiments show that E falls off rapidly as Ri increases and is probably negligible when Ri is more than about 0·8.

The theoretical and experimental results allow predictions to be made of flow velocities once the rate of supply of density difference is known. An estimate is also given of the uniform velocity which the ambient fluid must possess in order to cause the motion of the layer to be reversed.

A method, based on the measurement of attenuation of microwaves, which allows the electron density and the ionization rate of shock-heated gases to be obtained, is described. Results obtained for air in the shock Mach range 8·2-10·4, and for nitrogen containing 0·25% oxygen in the range 7·4-8·8, show that the electron density is in agreement with theoretical calculations based on thermodynamic equilibrium. Ionization time measurements in air are presented in this range and these results extend the measurements of Niblett & Blackman (1958) to a lower Mach range.

The stability of a viscous fluid between concentric cylinders is analysed, for the case in which the basic velocity distribution is the sum of a velocity distribution due to the rotation of the cylinders (Taylor 1923), and a ‘pumping’ velocity distribution due to a pressure gradient acting round the cylinders (Dean 1928). The critical Taylor number is computed for a wide range of values of the ratio of average velocity of pumping to average velocity of rotation for the case in which the outer cylinder is stationary. It is assumed that the spacing between the cylinders is small.

The stability of plane shock waves was measured in a shock tube by perturbing the primary shock wave, formed on rupturing the diaphragm, by means of thin wedges. A time-record of the shape of the shock wave after it passed the wedges and travelled along a channel of constant cross-section was obtained by Schlieren photography. Analysis of the photographs enabled the rate at which the shock wave recovered its plane shape to be determined and this, together with the detailed shape of the wave at various instants, was compared with the first-order theory of Freeman (although all the conditions assumed in the theory could not be faithfully reproduced in the experiments).

For shock-wave Mach numbers of 1·165, 1·41 and 1·60, the time-rate of decay of the perturbations was found to agree quite well with the theoretical value, but the amplitudes of the perturbations were much larger than those given by the theory.

The experiments failed to give reliable information about the decay of the perturbations after a large time, owing, it is believed, to flow separation from the sharp corners of the wedges which constituted an additional source of disturbance to the shock waves.