The general solutions of the Stokes approximate equations of motion are derived for two-dimensional and axisymmetric flows in the half-space x > 0, for an arbitrarily given velocity field in the plane x = 0. There is assumed to be no solid surface in the half-space. According to whether the velocity at infinity is zero or not, the solutions can be said to describe either jet-type or wake-type flows. Only the latter category is considered; numerical examples are worked out and properties of the base flow at very low Reynolds numbers are investigated. A recirculating flow region may exist, but the flow properties are not sensitive to this feature.

The influence of inertia terms in the equations of motion on the properties of the base-flow and near-wake flow at very low Reynolds numbers is investigated by using Oseen's approximation and by comparing with the results obtained in Part 1 for Stokes flow. The general solutions of Oseen's equations of motion are derived for two-dimensional and axisymmetric flows in the half-space x > 0, for an arbitrarily given velocity field in the plane x = 0. Numerical examples are given for two-dimensional flow and compared with Stokes-flow examples.

The constraint imposed by rapid rotation stabilizes convective motions in a plane liquid layer, so that the critical Rayleigh number which marks their onset increases as the two-thirds power of the Taylor number. Here we formulate an asymptotic approximation to the equations governing an analogous problem in a sphere. The solutions we obtain for the asymptotic equations predict the same power law for the dependence of the corresponding critical Rayleigh number on the Taylor number. The analysis is limited at the outset to the case of axially symmetric motions, and the present result, taken in conjunction with earlier results for slow rotation, essentially completes the treatment of symmetric, convective modes of instability.

The effect of viscosity on the trailing-wave system well downstream of a uniformly moving surface disturbance is determined. Solutions are obtained by series expansions in inverse powers of a Reynolds number and coefficients in the viscous exponential decay-factor found. It is seen that the diverging wave pattern is much more heavily damped than the transverse waves.

The non-linear equations of the Bénard convection problem are transformed to the spectral domain. The spectral basis consists of the supercritical normal modes of the characteristic-value problem in which the exponential growth rates are characteristic values. The norm of the spectrum is the variance of an arbitrary finite-amplitude state of convection. The equations that govern the spectrum are solvable by linear methods when the spectrum is truncated by exclusion of all convective modes except those of lowest-order symmetric vertical structure. Numerical computations of heat flux for a spectrum that contains only one convective mode are in good agreement with experimental data for water in the laminar régime.

The flow between two solid boundary surfaces of revolution and also between two rotating co-axial cones, of certain orientable fluids which are unoriented at rest, has been studied. It is found that for sufficiently low velocity gradients the fluids behave like a Newtonian fluid; while at larger but still very moderate velocity gradients the predicted behaviour is similar to that observed in elasticoviscous fluids. In the latter case it is shown that the flow of this class of fluids in horizontal circles is not possible for the boundary conditions defined above, even when the inertia terms are negligible, and that the normal stresses in the plane perpendicular to the streamlines are not equal.

Experimental investigations of shear layer instability have shown that some obviously essential features of the instability properties cannot be described by the inviscid linearized stability theory of temporally growing disturbances. Therefore an attempt is made to obtain better agreement with experimental results by means of the inviscid linearized stability theory of spatially growing disturbances. Thus using the hyperbolic-tangent velocity profile the eigenvalues and eigenfunctions were computed numerically for complex wave-numbers and real frequencies. The results so obtained showed the tendency expected from the experiments. The physical properties of the disturbed flow are discussed by means of the computed vorticity distribution and the computed streaklines. It is found that the disturbed shear layer rolls up in a complicated manner. Furthermore, the validity of the linearized theory is estimated. The result is that the error due to the linearization of the disturbance equation should be larger for the vorticity distribution than for the velocity distribution, and larger for higher disturbance frequencies than for lower ones. Finally, it can be concluded from the comparison between the results of experiments and of both the spatial and temporal theory by Freymuth that the theory of spatially-growing disturbances describes the instability properties of a disturbed shear layer more precisely, at least for small frequencies.

When a basin containing a shallow liquid is rotating with angular velocity Ω and the dimensionless number $\varepsilon = 4 \Omega ^2 L^2 | gM$ is small (L and M are typical horizontal and vertical dimensions), then, to a first approximation, the second-class motions behave as if the free surface of the liquid were fixed in its equilibrium position. The lower second-class modes of such a liquid, contained in a paraboloid, are relatively easy to describe on the basis of this approximation. When the liquid is rotating within an elliptical paraboloid and the sense of rotation is opposite to that of the container itself, the motion is unstable for a range of small angular velocities. Such unstable motions always exert a couple tending to oppose the rotation of the container.

This paper is an extension of an earlier paper by Hunt (1965) on laminar motion of a conducting liquid in a rectangular duct under a uniform transverse magnetic field. The effect of the duct having conducting walls are further explored; in this case the duct considered has perfectly conducting walls parallel to the field and non-conducting walls perpendicular to the field. A solution is obtained for high Hartmann numbers by analysing the boundary layers on the walls. This solution involves an integral equation of a standard form.

It is found that in this case, unlike the cases studied in the earlier paper, the velocity profiles in the boundary layers are monotonically decreasing. The effect of an external electrical circuit is examined, although it is found that it does not influence the form of the velocity profiles.

Murray's equation (53) for the velocity of rise of a three-dimensional (spherical) bubble of diameter dB can be written $U_B = K[gd_B|6c]^{\frac{1}{2}}$ where the unassigned coefficient c enters Murray's analysis in the linearization of the convective momentum term in the momentum equation for the solids (17). (Equations (53) and (17) are as numbered in Murray 1965b.) This compares with the expression $U_B = K[\frac{1}{2}gd_B]^{\frac{1}{2}}$ used by Rowe & Partridge (1965), who determined the velocity coefficient, K, for several fluidized systems by measuring bubble diameter and velocity from X-ray ciné photographs. Values of c calculated from the measured values of K are listed in table 1, from which it is seen that the coefficients are not truly constant. The observed values of Murray's c are also appreciably less than c = 1 or c = ⅗, the two values tentatively suggested by mathematical reasoning, and Murray's figure 13, based on earlier velocity measurements over a limited range of particle sizes, is not a critical test of the value of the constancy of c.

The classical potential solution for the flow about a thin airfoil in either steady or oscillatory motion requires application of the condition, postulated by Kutta, that the fluid velocity be finite at the trailing edge of the airfoil. The Kutta condition derives from the argument that viscous stresses will not allow a flow to turn about a sharp edge. Analytic verification of the validity of this condition, of particular interest in the unsteady case, has not previously been obtained. The problem is treated here by utilizing the Oseen formulation for viscous flow. The solution thus obtained approaches small-perturbation potential flow at a large distance from the airfoil and retains a qualitatively correct representation of the rotational flow near the airfoil. By simply assuming that the resultant force on the airfoil is finite, it is shown that the Kutta condition must apply in the limit of vanishing viscosity.

The first-order corrections, for large Reynolds number, to the lift and moment on an oscillating airfoil are explicitly determined. The effect of the Oseen approximation on the applicability of the numerical results remains to be established.

Viscous, incompressible, axially symmetric flow about an impulsively started rotating sphere is studied in terms of non-steady, partially linearized Navier-Stokes equations. The non-linear centripetal acceleration is included in full, but the other non-linear terms are neglected because of the restriction in interest to the case of large (but subcritical) Reynolds or Taylor numbers, a2Ω/v. Approximate closed-form solutions for u(r,θ,t), v(r,θ,t), w(r,θ,t) are found which satisfy all relevant boundary and initial conditions. The linearization approximation is checked for consistency and a restriction on Ωt is found. The velocity profiles, in the range of validity, are shown to be approximately similar in time, so their shapes may be qualitatively correct for larger values of Ωt. Some comparison with existing steady-state theories is given and the boundary-layer displacement thickness and viscous torque on the sphere are calculated.