The propagation of plane hydromagnetic waves in a fluid rotating with angular velocity ω and permeated by a magnetic field B = {Bx(z), By(z), 0} varying in both magnitude and direction with z is studied by techniques recently applied to the propagation of internal gravity waves in a shear flow (Bretherton 1966; Booker & Bretherton 1967). Particular attention is paid to a class of 'slow’ hydromagnetic waves of interest in connexion with the dynamics of the earth's liquid core. While, in general, rotation permits propagation across the lines of force, there is associated with each wave a ‘critical level’ z = zc, which acts as a valve by effectively permitting the wave to penetrate it from one side only. A slow hydromagnetic wave with frequency ω and wavenumber components k,l normal to the magnetic field gradient can only effectively penetrate its critical level if its propagation speed across field lines W is such that Wωz,(ωx,k+ ωy,l)ω < 0. The phenomenon of ‘critical-layer absorption’ evidently does not in general require the presence of a mean shear flow; a non-uniform magnetic field gives rise to similar effects provided that some other restoring mechanism (in this case the Coriolis force) is available to permit hydromagnetic waves to propagate across field lines.

A set of partial differential equations of the Navier-Stokes type is derived for external rarefied gas flows at all Knudsen numbers. Only the expressions of the stress tensor and heat flux vector are different from the customary Navier-Stokes relations and Fourier law. The new expressions are calculated from an approximate distribution function constructed through analysis of the BGK model of the Boltzmann equation so that it retains the qualitative property of the collision term and reproduces accurately the two extreme continuum and free-molecule regimes. Explicit forms of the stress tensor and heat flux components are given for low-speed two-dimensional flows. Solutions for vanishing Mach number and compressibility effects are then discussed. For a nearly isothermal cylinder, the present formulation leads to only one governing equation of the Navier-Stokes type for all flow regimes.

The Navier-Stokes like equation derived by Atassi & Shen for low-speed rarefied flows is analysed for a circular geometry. Asymptotic solutions for both Kn [Lt ] 1 and Kn B [Gt ] 1 are deduced using the method of matched asymptotic expansions. For Kn [Lt ] 1, fust- and second-order solutions are obtained and compared with the corresponding BGK treatment. For Kn [Gt ] 1, this equation is used only to describe the far field, the inner solution being of the free-molecule type. The matching procedure leads to a blockage correction for the drag and heat-transfer rate. Results are compared with other theories and known experimental data.

This paper describes an experimental study of the distortion of grid-generated turbulence as it approaches the stagnation region of a two-dimensional bluff body. When Lx/D [Gt ] 1, where Lx, is the scale of turbulence and D is a typical body dimension, along the mean stagnation streamline $(\overline{u^2})^{\frac{1}{2}$ attenuates like the mean flow, whereas if Lx/D [Lt ] 1 the turbulence is distorted by the mean flow field and $(\overline{u^2})^{\frac{1}{2}}$ will amplify because of vortex stretching. When Lx/D = O(1) there is found to be a combination of these effects with attenuation of energy at low wavenumbers and amplification at high wavenumbers. Measurements of the pressure fluctuations at the stagnation point show that at low wavenumbers the level of the pressure fluctuations can be predicted by a direct application of Bernoulli's equation.

It is shown that some variational methods of quantum mechanics can be re-formulated for application to the problem of the diffusion of marked particles in a fluid in random motion. As an example, a variational derivation of the Wiener-Hermite approximation procedure is given. The simplest such approxi- mation is examined and it is made plausible that it gives the exact form for both the one- and two-particle distribution functions in a certain limiting situation.

A quantitative two-dimensional theoretical model is developed to describe the movement of water and salt along the long narrow extracellular channels which appear to be a common structural feature of all epithelial membranes. This study examines the transport behaviour of both open and closed membrane systems as a function of the geometric specialization of the channel and the active transport site location under the influence of three driving forces: trans-membrane osmotic and hydrodynamic pressure differentials and active transport. The previous one-dimensional hydrodynamic model of Diamond & Bossert (1967) and Segel(1970) was confined to closed channel systems such as the gall bladder in which the only mechanism for water movement is local osmosis due to active transport.

Approximate analytical solutions are presented for long constant-area open channels in which the active transport sites have been idealized as point solute sources. A streamwise co-ordinate straining technique has been used in these solutions to describe the nonlinear effects of convection over long distances. Closed-form solutions are also presented for the pressure and solute concentration distributions within simplified models of channel constrictions with varying degrees of occlusion.

Numerical results of the model have been compared with Cole's (1961, 1962) in vivo and in vitro experiments on the rabbit ciliary body. Satisfying agreement with the measured values of the solute and water fluxes has been obtained for both the living eye and the excised ciliary body. These results strongly suggest that the formation of aqueous humour in the rabbit is a pressure-dependent process in which local osmosis due to active transport accounts for only one-third of the total aqueous flow. The model has also been applied to the gall bladder epithelium using more general boundary conditions than allowed for in the model of Diamond & Bossert. New solutions yielding a vanishing diffusional flux at the channel exit were obtained. However, the model, like that of Diamond & Bossert, does not provide a rational explanation as to how the water in the cell interior is replenished.

The flow of a stratified fluid in a channel with small and large deformations is investigated. The analogy of this flow with swirling flow in tubes with non-uniform cross-sections is studied. The flow near the wall is blocked when the Froude number takes certain critical values. The possibility of preventing the stagnation zones in the flow field is also discussed.

Measurements of mass flow rate and mean density have been made in separated laminar boundary layers with large transverse density gradients. Two-dimensional shear layers were formed by exhausting a half-jet of one gas into a reservoir of another gas with a different molecular weight. Two freons with a density ratio of 1-98 and unusual properties which permitted the measurement of the mass flow rate with a single hot wire were used. A n analysis of the mass flow rate fluctuations showed that a negative density gradient (i.e. light gas flowing into heavy) increases the amplification rate of the instability oscillations and reduces the frequency and wave number. Opposite trends were observed when the density gradient was positive. These findings are in agreement with recent theoretical predictions.

In this paper we study the generation and propagation of hydromagnetic waves during the spin-up of an electrically conducting, viscous fluid. The time-dependent motion is set up by the impulsive rotation of an infinitely conducting plane which is initially in rigid-body rotation with the fluid, the applied magnetic field being parallel to the axis of rotation. The hydromagnetic flow is permeated by two distinct wave modes and the region between their wave fronts supports harmonic waves produced by the inertial oscillations. The characteristics of these modes are discussed for some limiting cases of interest.

The eigenvalue problem for internal waves in a thermocline for which the density profile may exhibit finite discontinuities is formulated as a homogeneous Fredholm integral equation. The corresponding quadratic functional yields both upper and lower bounds for the dominant-mode eigenvalue and lower bounds for the remaining eigenvalues (which are proportional to the square of the wave speed) for any value of the wavenumber. The lower bounds based on simple trial functions appear to provide adequate approximations for typical density profiles and all wavenumbers. The upper bound (which is based on a Schwarz inequality and does not require the choice of a trial function) is sharp only for relatively long waves. A simple approximation is developed for the effect of the free surface on the dominant mode. Two algebraic formulations are given for a thermocline of homogeneous layers separated by a finite number of sheets, across each of which the density is discontinuous. The various approximations are compared with the exact results for a thermocline with a hyperbolic-tangent density profile, a three-sheeted thermocline, and a five-sheeted model of the summer thermocline in the Mediterranean.

The influence of drag-reducing polymers on the time-averaged velocity gradient $\overline{S_x}$ and on the two components of the fluctuating velocity gradient a t the wall, sx and sz, has been studied for turbulent flow in a 1 in. pipe. The time-averaged velocity gradient is related to the time-averaged wall stress by Newton's law of viscosity. The view is taken that a comparison of the turbulence structure with and without polymers should be made at the same $\overline{S_x} $. Therefore all turbulence measurements have been normalized with wall parameters. On this basis we find that the changes in the intensity of sx, and in the shape of the spectral density functions for sx and sz are not as striking as the decrease in pressure drop caused by the addition of a polymer. The root-mean-square value of sz shows a significant decrease with an increase in drag reduction but the most spectacular change in turbulence structure seems to be associated with the transverse correlation coefficient. We conclude that the increase in drag reduction is accompanied by an increase of the size of longitudinally oriented eddies in the viscous sublayer. One possible explanation for this increase in size is that the influence of the polymer additives is such as to increase the viscous resistance in the transverse direction more than in the direction of mean flow.