A detailed equation is proposed for the force exerted on a sphere that accelerates rectilinearly in an otherwise still fluid. In addition to the buoyant force, the fluid exerts forces that depend on (a) the velocity of the sphere, (b) the acceleration of the sphere and (c) the history of the motion. The equation reduces to the known theoretical solution for low velocity and large acceleration.

The proposed equation was tested and found most satisfactory for a particular case in which the velocity was not small, viz. the case of simple harmonic motion along a straight line. The acceleration (added mass) and history coefficients in the equation were evaluated experimentally. They were found to depend on the ratio of the convective acceleration to the local acceleration as measured by the parameter V2/aD, in which V, a and D are the velocity, acceleration and diameter of the sphere, respectively. The Reynolds numbers varied from 0 to 62 during the tests.

This paper considers the question of how large the magnetic energy can be in stationary homogeneous turbulence at large Reynolds number of an incompressible conducting fluid for which the magnetic diffusivity λ is much less than the kinematic viscosity ν. An approximate equation, that takes account of the effect of the Lorentz forces on the turbulence, is proposed for the spectrum function of the magnetic energy for wave-numbers lying in the equilibrium range. This equation is used to determine the magnetic spectrum function and the level of magnetic energy for the case when a statistically steady magnetic field is maintained by a relatively small input of magnetic energy, by a weak applied field say, on the scale of the energy-containing eddies; it being supposed as suggested by earlier work that the magnetic energy would eventually die away in the absence of external electromotive forces. The results are complicated, there being essentially four different régimes depending in a fairly involved way on the relative values of λ/ν, the turbulent Reynolds number, and the ratio of the energy of the applied field to the energy of the turbulence. The main conclusions about the amplification factor are shown diagrammatically in figure 4. The wavenumber at which the magnetic spectrum has its maximum tends to decrease, as the applied field is increased, from the conduction cut-off wave-number $(\epsilon |v \lambda ^2)^{\frac{1}{2}}$ to values lying in the inertial subrange, much less than $(\epsilon |v ^3)^{\frac{1}{4}}$.

The case of a turbulent dynamo is also examined, and it is concluded that, if it exists, the equilibrium magnetic energy would be given by Batchelor's criterion of equipartition of energy between the magnetic field and the small, energy-dissipating eddies. The magnetic energy in the dynamo is found to lie mainly in Fourier components of wave-number about $(\epsilon |v \lambda ^2)^{\frac{1}{2}}$.

The motion at low Reynolds number of a drop in a quiescent unbounded fluid is investigated theoretically by means of a singular-perturbation solution of the axisymmetric equations of motion. Special attention is paid to the deformation of the drop. It is shown that for small values of the Weber number We the drop will first deform exactly into an oblate spheroid and then, with a further increase in We, into a geometry approaching that of a spherical cap. These results are quite insensitive to the ratio of the viscosities of the two fluid phases. The first-order effect of the deformation on the drag of the drop is also included in the analysis.

The vertical constant-speed motion of a constant-strength point source towards a horizontal free boundary is analysed. A procedure based on expansions in even powers of the Froude number is employed. The asymptotic expansion of the potential is found to satisfy a simple differential equation, which, when integrated, yields an image-type solution valid for all Froude numbers. Froude-number effects are contained in a distribution of sources along the vertical line from the image of the submerged source with respect to the undisturbed free surface upward to infinity. The solution is valid for arbitrary values of the density ratio across the free surface.