The cells observed by Bénard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Bénard's experiments are briefly discussed.

Under the assumption that the local acceleration and inertial terms are to be neglected in the equation of motion, a comparison is made between the unsteadystate movement of a suspension of particles and the same problem in reversed (or negative) time. The formalized discussion confirms Bretherton's (1962) conclusion that, if in a steady unidirectional shear flow at small Reynolds number a rotationally symmetric particle twice assumes a position with its axis of rotation in the plane perpendicular to the flow, its orbit must be periodic. The reversibility phenomena observed by S. G. Mason (1963) in dilute and concentrated suspensions are explained as well.

It is shown in general how a two-dimensional flow can be justified as a physical approximation, notwithstanding the logarithmic singularity in pressure that occurs at infinity when the cavity expands or contracts at a varying rate. The argument presented, which affords a more natural interpretation than alternatives previously suggested, refers to the approximate equivalence-to a determinable degree of accuracy-between the hypothetical plane flow and the inner region of some real three-dimensional flow with small spanwise variations. The main ideas are illustrated by the example of a long ellipsoidal body which changes in volume while also undergoing shape perturbations.

The position of the region of transition to turbulence and the manner in which turbulence develops are investigated using a hot-wire anemometer to study the character of the flow in the wake of a circular cylinder. In the range of Reynolds numbers greater than 200 in which turbulent motion is developed, the region of transition to turbulence moves towards the cylinder with increasing Reynolds number. The manner of transition to turbulence appears to undergo a basic change as the region of transition moves from the periodic wake into the region of flow immediately behind the cylinder where the separated layers have not rolled into vortices, that is, as the Reynolds number increases from Roshko's transition range to his irregular range. When transition occurs in the periodic wake it is a result of distortion due to large-scale three-dimensional effects. Turbulence, when it develops in the separated layers, is preceded by two-dimensional Tollmien-Schlichting waves which eventually degenerate to turbulence by the action of small-scale three-dimensionalities.

A body floting on the free surface of water is given a small vertical displacement from its equilibrium position and is then held fixed. When the fluid has again come to rest the body is released. The subsequent damped motion is investigated when viscosity and surface tension are neglected and the equations of motion are linearized. The method applies to bodies of arbitrary shape in two or three dimensions, and is described in detail here for the heaving motion of a horizontal half-immersed circular cylinder of radius a.

The forced periodic motion of such a cylinder has been studied in earlier papers. In particular, the hydrodynamic forces exerted by the fluid on the body can be described by a dimensionless coefficient, $\Lambda(\omega (a|g)^{\frac{1}{2}})$, where ω is the (real) angular frequency. The function ∧ can be found by convergent infinite processes, but not explicitly, and the difficulties of the problem are due to this. The free motion of the cylinder is solved in the present paper by Fourier methods. The motion is regarded as the superposition of simple harmonic motions, and the displacement y(t) is thus obtained in the form of a Fourier integral $y_0(\tau (a|g)^{\frac{1}{2}})= -{\frac {1}{8}}iy_0(0) \int ^\infty_{-\infty}{\frac {u(1+ \Lambda (u))\; e^{-iu \tau }du} {1- \frac{1}{4}\pi u^2(1+ \Lambda(u))}}.$

It is seen that the integrand involves the force coefficient ∧ and is thus not strictly an explicit expression. The asymptotic behaviour for large times can be found explicitly when the depth is infinite: $y_0(t) \sim -\frac{4}{\pi}y_0(0) {\frac {a}{gt^2}}.$

A damped harmonic behaviour had been expected. The slow monotonic decay occurs because the function $\Lambda (\omega(a|g)^{\frac {1}{2}})$, when continued into the complex ω-plane, can be shown to be many-valued near ω = 0. No physical interpretation has yet been found for this property. The free motion of a cylinder set in motion by an applied force is also treated, with similar results.

Reasons are given why there are no rapidly oscillatory terms in the asymptotic expression. For finite constant depth the function $\Lambda (\omega(a|g)^{\frac {1}{2}})$ is single-valued near ω = 0, and the asymptotic expression for this case is not yet known.

Let D0 be the Stokes drag on an axially symmetric body moving parallel to its axis with velocity U0 through an unbounded fluid. The drag D experienced by the same body oscillating with velocity U = U0eiσt along its axis in the unbounded fluid is given by the expression $\frac{D}{D_0} = \left \{{1 +{\frac {d_0}{6 \surd{(2)} \pi \mu a U_0}}(1+i)M+O(M^2)}\right \}e^{i \sigma t},$ where a is any characteristic particle dimension and $M^2 = a^2 \sigma \rho |\mu$ is a dimensionaless number. The part of this drag formula which gives the energy dissipation is calculated for bodies of various shapes.

In a previous paper on cellular thermal convection (Palm 1960) the importance of the effect caused by temperature variation of kinematic viscosity was pointed out. It was demonstrated that this effect would, owing to non-linear interactions, lead to a tendency towards hexagonal cells. For mathematical simplicity, only the interaction of two wave-components was taken into account.

Segel & Stuart (1962), working with the same equations, have examined the stability of the various equilibrium solutions. They arrive at the important conclusion that a necessary condition for the solution corresponding to hexagons to be stable is that the variation of viscosity with temperature be sufficiently great.

In the present paper the problem is discussed from a somewhat more general point of view. First it is shown that, when the variation of viscosity with temperature is sufficiently great, the solution corresponding to hexagons is the only stable one if only two wave-components are taken into account. To examine if this result is also true when the motion consists of an arbitrary number of wave-components, the case of three wave-components is studied. It turns out that in this case also the only possible mode is the pattern consisting of hexagons. The validity of this result is easily extended to a more general class of wave-components. It is shown that the solution corresponding to hexagons is stable for all small disturbances which can possibly occur. To prove this it is necessary to take into account non-linear disturbance theory.

A reasonable conclusion from the paper by Segel & Stuart and the present paper is that a hexagonal pattern is observed only when a condition of the form (6.9) is fulfilled. Experiments concerning this problem are, however, lacking.

The classical laminar boundary layer on a parabolic cylinder is calculated using the Blasius series, with modifications to improve convergence, and supplemented by an asymptotic expansion valid far downstream from the nose. The flow due to displacement thickness is thereby found with sufficient accuracy to permit evaluation of its second-order effect upon the boundary layer near the stagnation point. The skin friction and heat transfer are found to be reduced there by both displacement and curvature.