Conditions for the existence of similar solutions for the two-dimensional and axisymmetric boundary layers are obtained for the steady or unsteady flow for purely viscous non-Newtonian fluids, where the shear stress is proportional to the nth power of the velocity gradient and n > 0. These conditions are shown to be a generalization of the similarity conditions for Newtonian fluids. In particular it is found that the velocity at the outer edge of the boundary layer must be proportional to {R(x) + A}m or exp [AR(x)] for steady flows and to (t + A)m, (dx + A)/(t + B) or exp [At] for unsteady flows. Here x is the distance along the wall from the forward stagnation point, t is the time, R(x) = x for two-dimensional flows and for axisymmetric flows, r is the distance from the axis to the wall, and A, B, d, m are constants.

Several examples of the similar solutions are calculated analytically for both steady and unsteady pseudoplastic flows (n < 1). In these solutions the velocity in the boundary layer tends to the outside velocity in such a manner that the difference tends to zero as an inverse power of the distance from the wall, whereas such a difference tends to zero exponentially for the corresponding flow in Newtonian fluids.

This paper is a continuation of the experimental study (Elder 1965), of free convection in a vertical slot, to unsteady and turbulent motion. The flow is specified by three dimensionless parameters: σ, the Prandtl number; h = H|L, the aspect ratio and $A = \gamma g \Delta TL^3|kv$, the Rayleigh number. The present experiments are generally for σ = 7 (water), h = 10–30 and A > 106.

For A above about 8 × 108S|h3, travelling, wave-like motions grow up the hot wall of the slot and also down the cold wall. These waves grow most readily mid-way between the two ends. At higher values of the Rayleigh number when the wave amplitude is finite, the phase of successive wave fronts becomes increasingly random till near A = 1·0 × 1010|h3 an intense entrainment and mixing process commences between the wall region and the interior. The middle portion of the interior is then turbulent, the extent of the region growing further toward the ends as A increases.

Measurements of the mean temperature and the probability distribution of temperature fluctuations of the turbulent flow are reported. Except within the thin wall layers and distant form the ends, the interior has a mean temperature, uniform to within 0·1%, superimposed on which is a nearly Gaussian fluctuation-field of variance of order 0·01ΔT. A comparison is made with recent theories of turbulent convection; moderate agreement is found with the similarity ideas of Priestley (1959). The wall layer is seen as a marginally unstable sublayer.

This paper presents a discussion of some aspects of the linear stability problem for the asymptotic suction profile. An exact solution of the inviscid equation is first obtained in terms of the usual hypergeometric function and its analytical continuation. This exact solution provides both a corrected version of an earlier treatment by Freeman and an independent check on the more general method suggested for solving the inviscid equation numerically. Various approximations to the characteristic equation, and hence to the curve of neutral stability, are then considered. In particular, it is found that, in a consistent asymptotic treatment of the related adjoint problem, at least one viscous correction to the singular inviscid solution must be considered. Based on the present results for the adjoint problem, it is suggested that Tollmien's original treatment of the viscous corrections must be slightly modified.

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.

The effect of an adverse pressure gradient on the stability of a laminar boundary layer is considered in the limiting case when the skin friction at the wall vanishes, i.e. when U′(0) = 0. Such flows are not absolutely unstable as might have been expected but have a minimum critical Reynolds number of the order of 25. General results are given for the asymptotic behaviour of both the upper and lower branches of the neutral curve and a complete neutral curve is obtained for Pohlhausen's simple fourth-degree polynomial profile at separation.

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.

The equations of motion of an incompressible viscous fluid are given in a rotating helical co-ordinate system, which is non-orthogonal. Partial differential equations are derived for the boundary-layer flow on a rotating helical blade. Numerical solutions of these equations show that the radial flow in the boundary layer is strongly dependent upon the stagger and speed of rotation of the blade.

In a horizontal layer of fluid, thermal expansion or the presence of dissolved salt may cause a density gradient opposite to the direction of gravity. In such cases, when the buoyancy forces are sufficient to overcome the dissipative effects, the static state becomes unstable and convective motions arise. If the layer is infinitely large in horizontal extent, the non-linear convection problem is highly degenerate, admitting many different steady-state solutions. A general necessary criterion for stability of such non-linear steady solutions is developed here for the case in which a homogeneous vertical magnetic field acts on the fluid. The criterion is demonstrated for two rigid bounding surfaces which are perfect thermal and electrical conductors, but it is applicable to more general kinds of boundary conditions.

The static state of a horizontal layer of fluid heated from below may become unstable. If the layer is infinitely large in horizontal extent, the Boussinesq equations admit many different steady solutions. A systematic method is presented here which yields the finite-amplitude steady solutions by means of successive approximations. It turns out that not every solution of the linear problem is an approximation to the non-linear problem, yet there are still an infinite number of finite amplitude solutions. A similar procedure has been applied to the stability problem for these steady finite amplitude solutions with the result that three-dimensional solutions are unstable but there is a class of two-dimensional flows which are stable. The problem has been treated for both rigid and free boundaries.

The terminal velocity of rise of small, distorted gas bubbles in a liquid of small viscosity is calculated. Small viscosity means that the dimensionless group gμ4/ρT3 where g is the acceleration of gravity, μ the viscosity, ρ the density and T the surface tension, is less than 10−8. It is assumed—and the numerical accuracy of the assumption is discussed—that the distorted bubbles are oblate ellipsoids of revolution. The drag coefficient is found by extending the theory given recently (Moore 1963) for the boundary layer on a spherical gas bubble. The results are in reasonable quantitative agreement with the experimental data.

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.

Density waves in low-pressure are discharges have been observed travelling towards the anode with speeds near . A dispersion relation for these waves is developed, taking into account the boundary effects, namely, the continuous loss of ions to the containing walls. This theory is in fair agreement with experiment, except in its prediction of a rather high value for the wave damping. However, by allowing for the effect of electrostatic instabilities on anode-approaching waves, due to electron drift, this discrepancy is satisfactorily explained.

The phenomena which occur in a fluid contained in a rotating system are strongly influenced by the density gradients in that fluid and by certain features of the geometry. In this paper, we study several specific phenomena which involve stratification and/or irregular surface topography and some phenomena which arise because of an equatorial geometry. Although several of these studies are motivated by particular geophysical questions, we do not treat comprehensively any geophysical phenomenon. Nevertheless, the inferences we draw from these studies do much to clarify the roles of the mechanisms underlying various geophysical phenomena and these inferences should be of value when comprehensive geophysical investigations are attempted.

The development of a turbulent boundary layer in a strong adverse pressure gradient can be described by the two-layer model proposed by Stratford (1959), in which the outer part of the flow is assumed to be unmodified by the pressure-rise and the inner part described by two local parameters, the surface stress and the pressure gradient. The description suggests that the modification of the original flow is in some sense self-preserving, and it is shown here that self-preserving development of the modification is consistent with the Reynolds equations of turbulent flow in particular pressure distributions. For these distributions, the predictions of the two-layer model are confirmed without any need to make the sharp and arbitrary distinction between the two parts of the boundary layer.

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.

The Laplace equation for the pressure drop across curved liquid-gas interfaces is applied to the solution of the profile of a static liquid meniscus on the outside of a wire of circular cross-section. The resulting differential equation is integrated numerically, an operation complicated by the existence of boundary conditions at two points making a trial-and-error solution necessary. The accuracy of the solution is substantiated by comparison of computed profiles with experiments in which menisci of a blue dye in water are photographed clinging to the outside of brass wires, whose diameters lie within the range of technological interest.

A luminescent region was observed near the axis of a cylindrical duct confining a supersonic swirling air flow at ambient temperature. The luminescence was identified as a glow discharge induced by a strong internal electric field, which was produced by condensed water droplets impinging on the duct wall.

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.

Two-dimensional thermal convection of air between horizontal plates of length much greater than their separation distance is studied numerically by solution constraining motions to lie in a single vertical plane. Rayleigh numbers from 105 to 107 are employed. Steady rolls with wavelength twice the plate-separation were obtained in both cases. As the experimental two-dimensional constraint is relaxed, short-period turbulent fluctuations in temperature develop, the rolls or cells become only quasi-steady, and their wavelength increases. For the three-dimensional case, very large width-to-height ratios and averaging periods are found necessary before the temperature variance in time approaches the variance in the horizontal.

An investigation is made into the propagation of a one-dimensional combustion wave, which consists of a flame front and a precursor shock wave which pass down a tube closed at one end, in the presence of a transverse magnetic field in the undisturbed gas at rest. The shock wave is assumed to be of sufficient strength to ionize completely the initially non-electrically-conducting gas and the conditions at the flame front are taken to satisfy the Chapman–Jouguet condition. Details of the solution are compared with the corresponding results for ordinary gasdynamic deflagration.