The results of Proudman & Pearson (1957) for a sphere are generalized to apply to all ellipsoids of revolution both prolate and oblate.

If the viscosity and specific weight of a fluid are variable, the equations governing its flow in a porous medium are non-linear and in general very difficult to solve. It has been found, however, that steady flows of a fluid of variable viscosity but constant specific weight can be reduced to those of a homogeneous fluid by a remarkably simple transformation, which indicates that the flow patterns of the fluid are the same as those of a homogeneous fluid with the same boundary conditions, and that only the speed need be modified. The speed of the actual flow is obtained by dividing the speed of the homogeneous-fluid flow by a factor proportional to the actual viscosity. The transformation is also used to derive the equations governing steady two-dimensional flows and steady axisymmetric flows of a fluid of variable viscosity and specific weight. In a good many cases of practical importance these equations are exactly linear, in spite of the fact that the governing equations obtained without the use of the above-mentioned transformation are non-linear. An exact solution for a steady two-dimensional flow with prescribed boundary conditions is given. Two inverse methods for generating exact solutions for two-dimensional flows are presented, together with two illustrative examples. The theory also applies to Hele-Shaw flows, so that it can be easily verified in the laboratory.

A method is described for the measurement of turbulent shearing stress and vertical heat flux by way of ‘structure functions’—the mean square velocity and temperature differences between two points a known distance apart. The method is particularly suitable for shipboard use because it does not require the measurement of velocity components relative to a fixed frame of reference. An analysis of the available observations, although they are not ideally suited to the purpose, is encouraging and points the way to further development of the method.

This paper is a study of the flow of a viscous incompressible fluid in the immediate neighbourhood of a saddle point of attachment, near which the external flow is irrotational with components {ax, by, −(a + b)z}, where a > 0 > b. It is shown that the flow is of a boundary-layer character, and that part of the boundary-layer flow is reversed when b/a < − 0·4294.

On the assumption that such flows are physically plausible, the problem may be solved for all values of b/a [ges ] −1. Even in the limiting case b/a = −1, an effect of the boundary layer is everywhere to draw fluid towards the wall, so that vorticity is still convected towards the wall.

Numerical solutions have been computed, and some of the results are presented in the tables and diagrams.

The problem described by the title is investigated when both the magnetic field and the streaming motion of the fluid at infinity are uniform and parallel to the axis of symmetry of the body. The flow pattern depends on three parameters, the Reynolds number R, the magnetic Reynolds number Rm and the Hartmann number M. In this paper it is assumed that M [Gt ] 1, M [Gt ] R, M [Gt ] Rm (no other restrictions on the parameters are imposed, so that R and Rm need not be small). The flow pattern then consists of an undisturbed uniform stream outside a cylinder circumscribing the body with generators parallel to the stream. Inside this cylinder the fluid is at rest. The leading term in the expression for the drag on the body is obtained.

By expanding the dependent variables in series of orthogonal functions on the one hand, and expanding the coefficients of these functions in power series of a parameter η on the other hand, a solution has been obtained for the system of non-linear equations of cellular convection. The expansion parameter η is chosen in such a way as to make it remain less than 1 for all finite values of the Rayleigh number. The solution so obtained is found to be valid for a large range of the imposed temperature difference, and converges rapidly. This solution provides a quantitative theory for the convective heat transport as a function of the temperature difference in the range of laminar flow.

The solution also reveals that when the actual Rayleigh number is greater than twice the critical Rayleigh number, a layer of isothermal (adiabatic lapserate in a gas medium) mean temperature develops in the middle of the fluid layer. The thickness of this layer increases as the actual Rayleigh number increases, and at the same time the temperature gradient increases in the boundary layer so that an increase in the heat transport is accomplished.

The solution reveals further that the large temperture gradients are concentrated in the region where the cold descending current approaches the lower boundary and where the warm ascending current approaches the upper boundary. It is also shown that these ascending and descending currents spread out in mushroom-like patterns, a feature characteristic of the convection of isolated hot bubbles, but one which never has been considered as the form for finite cellular convection. Recent optical observations indicate that this is the most common form of the temperature field.

The heat transport given by this solution fits a power law of exponent 1.24, which is very close to the observed power law of exponent 1.25 for laminar flow.

The flow engendered by the steady motion of a cylindrical insulator through an inviscid, incompressible fluid of small conductivity σ is not close to potential flow when the applied magnetic cross-field H0 is sufficiently strong. Here we determine the limiting form of this flow as σ → 0 with $\sigma H^2_0 \rightarrow \infty$, the latter representing the ponderomotive force.

The limit equations do not have a unique solution, but it is possible to make a selection by taking into account the inertia of the fluid during the limiting process, i.e. without recourse to considerations of how the motion was set up from rest. The forces on the cylinder are found to be asymptotically proportional to $\surd {\sigma} H_0$.

The case of an elliptic cylinder and that of a flat plate are worked out in detail.

In the previous paper (Chester 1961) it was shown that, for large values of the Hartmann number, the asymptotic solution for the flow past a body of revolution has a discontinuity on the surface of a cylinder which circumscribes the body. The flow in the region of this discontinuity is now investigated in more detail when the body is a circular disk broadside-on to the flow. It will be shown that there is actually a region of transition whose thickness is O(|x|½/M½), where x is the axial distance from the disk and M is the Hartmann number. This region is thin near the disk, but gradually thickens until it merges into the over-all flow field for x = O(M).

The leading terms in the expression for the drag are given by $\frac{D}{D_s} = \frac{M \pi }{8} \left( 1 + \frac{2}{M} \right) $, where Ds is the Stokes drag.

A one-dimensional model with no magnetic field is considered. It is supposed that the plasma starts in thermal equilibrium and then a current is forced to grow. Instability leads to the growth of waves, which are shown to stir the distribution in phase space, but only over a limited range of velocity. It is concluded that in order to restore stability the energy in the wave must become comparable to the energy of drift.