We analyse the steady, fully developed, laminar flows of uniform, electrically conducting, incompressible fluids along arbitrarily shaped ducts of constant cross-section under the action of transverse magnetic fields to show that these flows are uniquely specified by certain parameters, namely: either the pressure drop along the duct per unit length or the volume flow rate and either the electric current leaving the walls of the duct at each point where the duct is connected to an external electrical circuit or the electric potential at these points. The most important, and novel, feature of the analysis is the proof that no secondary flow can exist in these fully developed flows. The analysis includes the special cases where the walls of the duct are non-conducting, and where the fluid is non-conducting (e.g. Poiseuille flow); in the latter case the uniqueness of the flow has not previously been demonstrated, at least in print.

The study of the slow motion of a solid body through a viscous fluid is of importance in many fields. We may instance viscometry and chemical engineering as two subjects in which a quantitative understanding of such motions is essential. Thus, in recent years systematic attempts have been made to calculate the drag force and couple on slowly sedimenting particles in a bounded viscous fluid. Using the ‘method of reflexions’ Brenner (1) has obtained formulae which give, for a wide class of sedimenting particles, an estimate of the effect of the container walls on the drag force, the error being O(c/l)3, where c is a characteristic particle dimension and l the distance from the centre of the particle to the nearest boundary wall. The reader is referred to the recent book by Happel and Brenner (2) for a complete account of these and other pertinent investigations; also to Williams (3) for an alternative integral equation approach to these results.

We investigate the steady rotation of an insulating body of revolution in an unbounded electrically conducting fluid permeated by a uniform axial applied magnetic field. The assumptions of a small magnetic Reynolds number (Rm ≪ 1, i.e. the weakly conducting situation) and negligible inertia forces compared with the magnetic forces (R/M2 ≪ 1) permit us to suppress the inflow at the poles and outflow at the equator, which normally occurs for a non-conducting viscous fluid ((12), pp. 436–439). Thus in the case of the sphere, we find an exact solution of the reduced equations in terms of an infinite series of Legendre polynomials of order 1 with coefficients which are the ratios of modified spherical Bessel functions. This is the canonical problem by which results for arbitrary bodies of revolution are obtained.

Maximum and minimum principles are developed for boundary-value problems with field equations T*Tφ + f(φ) = 0 in some region V, subject to φ = h(Tφ) on the boundary of V.

In recent years new materials of industrial importance have become known whose rheological properties cannot be adequately characterized by the classical Newtonian working model, many of these materials exhibiting both viscous and elastic properties. The large-scale flow behaviour of some of these new materials can be strikingly different from that of purely viscous liquids. For example, when sheared between coaxial cylinders, some liquids are observed to ‘climb up’ the inner cylinder, i.e. the free surface rises spectacularly near the inner boundary. This phenomenon has been observed by many authors (see, for example (1–4)), and is now widely known as the Weissenberg effect. Again, Merrington (5) observed that a stream of rubber solution ‘swells’ radially immediately on emerging from a capillary tube. In order to attempt to explain behaviour such as this, we must examine the consequences of a working model which is more complex than that for Newtonian liquids.

The problem stated in the title is studied for small values of the diffusivity ratio ε and the magnetic force coefficient N with the aim of finding the effect of rotation and the disc conductivity on the magnetic and velocity fields, currents, shear stresses, etc.

We have recently observed a curious phenomenon, undoubtedly well known, while mixing instant coffee in a mug. If the bottom of the mug was tapped repeatedly with the spoon as the powder was stirred into the water, the note emitted could be heard to rise in pitch by over an octave. This rise occurred in a matter of seconds, about the same time as it took the mug to heat up. Further experimentation showed that almost any powder could produce the effect in cold or hot water, but water alone would not. Vigorous stirring did not seem essential. A full mug showed as pronounced an effect as a half-filled one.

The elastic fields in an elastic circular inclusion and its surrounding infinite dissimilar elastic matrix, are determined when either the matrix or inclusion is subject to a concentrated force or edge dislocation.

Limiting solutions are derived for the flexure of simply supported many-sided regular polygons, as the number of sides is increased indefinitely. It is shown that these solutions are different from those for simply supported circular plates. For axisymmetric loading, circular plate solutions overestimate the deflexions and the moments by significant factors.

Mr K. A. Jukes has kindly pointed out that on page 398 line 21 and page 400 line 12 of my paper ‘Summability by Dirichlet Convolutions’, this journal 63 (1967), 393–400, Theorem 56 of Hardy and Riesz, The general theory of Dirichlet series, is misquoted.