The general nature of the flow at large distances from a two-dimensional body moving uniformly through an unbounded, linearly stratified, non-diffusive viscous fluid is considered. The governing equations are linearized using the Oseen and Boussinesq approximations, and the boundary conditions at the body are replaced by a linearized momentum-integral equation. The solution of this linear problem shows a system of jets upstream and a pattern of waves downstream of the body. The effects of viscosity on these lee waves are considered in detail.

Conjugate solutions of the equation of swirling cylindrical flow are considered in the case where the ‘primary’ flow has constant axial velocity. Numerical solutions are exhibited, which suggest a division of primary flows into four classes, and some conditions governing membership of these classes are derived. Finally, the case when the governing equation is slightly non-linear is considered.

The lift tensor for any three-dimensional body moving in a linear shear flow at low Reynolds numbers has been calculated by asympototic methods. The tensor is applied to the problem of the motion of a dumb-bell shaped particle. The particle is shown to have a preferred periodic orbit which corresponds to maximum dissipation. The dissipation is calculated and the intrinsic viscosity of a dilute suspension of such particles is predicted. Experiments conducted with a single particle tend to confirm the stability of the predicted orientation.

A number of shear-flow phenomena can be explained qualitatively if turbulence is regarded as a continuous viscoelastic medium with respect to its action on a mean field. Conditions are sought under which the analogy is quantitative, and it is found that the turbulence must be fine-grained and the mean field weak. For geometrical convenience the turbulence is assumed to be nearly homogeneous and isotropic so that body forces are required to maintain it. The turbulence is found to respond initially to an arbitrary deformation as an elastic medium, in which Reynolds stress is linearly proportional to strain. Three processes that cause the resulting Reynolds stress to relax are distinguished: viscous diffusion, body-force agitation and non-linear scrambling. It is argued that, regardless of which process dominates, Reynolds stress evolves in a continuously changing mean field according to a viscoelastic constitutive law, relating stress to deformation history by means of a scalar memory function. The argument is carried through analytically for weak turbulence, in which non-linear scrambling is negligible, and the memory function is computed in terms of the wave-number-frequency spectrum of the background turbulence. In the course of the analysis, a new type of Reynolds stress arises related to the passage of the turbulence through its sustaining environment of body forces. It is found that the mean field must be surprisingly weak for this ‘translation stress’ to be negligible. Applications of the viscoelasticity theory of turbulent shear flow are discussed in which body forces and therefore translation stress are absent.

In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Bénard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

The rate of mixing across a density interface between two layers of liquid has been measured in a laboratory experiment which allows a direct comparison between heat and salinity transports over the same range of density differences. Low Reynolds number turbulence was produced by stirring mechanically at a fixed distance from the interface, either in one or in both layers, and the results for these two sets of experiments are also compared. The measurements cover a factor of two in stirring rate and twenty in density. Over this range of conditions the ratio of entrainment velocity to stirring velocity can be expressed as functions of an overall Richardson number Ri, and in this form the results of the one and two stirred layer experiments are indistinguishable from one another. For density differences produced by heat alone, the functional dependence is close to Ri−1 except at small values of Ri where it approaches a finite limit. For experiments with a salinity difference across the interface, the mixing rate is the same as in the heat experiments at low values of Ri, but falls progressively below this as Ri is increased, with the approximate form $Ri^{\frac{3}{2}} $.

An interpretation of these results has been attempted, using a dimensional analysis and qualitative mechanistic arguments about the nature of the motion. The Ri−1 dependence implies a rate of change of potential energy proportional to the rate of working by the stirrer. The decreased mixing rates for salt have been attributed to a slower rate of incorporation of an entrained element into its surroundings by diffusion, which increases the tendency for it to return to the interface and dissipate energy in wave-like motions.

A study is made of the stability of a viscous, incompressible fluid with a finite conductivity flowing between parallel planes in a parallel magnetic field. The general form of the magnetohydrodynamic stability equation is a sixth-order differential equation. The complete sixth-order differential equation is solved numerically as an eigenvalue problem. Stability curves are obtained for a range of values of the magnetic Reynolds number Rm and the Alfvé n number A based on two-dimensional disturbances. It is found that the minimum critical Reynolds number is raised as Rm increases for a given A2 and as A2 increases for a given Rm, respectively. The stability curve closes and finally degenerates to a point which gives the critical value for Rm or A2. Results obtained for two-dimensional disturbances are modified to take into account three-dimensional disturbances. Then the minimum critical Reynolds number where three-dimensional disturbances become apparent is obtained, below which two-dimensional disturbances are the most unstable.

The unstrained free turbulent flow generated by a composite grid has been found to possess a distinct, deeply indented boundary separating non-turbulent fluid from turbulent fluid of virtually homogeneous intensity. At this boundary, contraction, in the sense of a reduction in the volume of turbulent fluid, takes place at a rate which is independent not only of the intensity and degree of anisotropy of the turbulence, but also of the relative depth of the surface indentations. The apparent outward spreading of the turbulence is solely due to the increasing amplitude of the boundary bulges.

The subsequent plane straining of the flow in a constant area distortion shows that a reasonably high degree of anisotropy, combined with straining by the mean flow, is required before entrainment of non-turbulent fluid can occur at the boundary. Under these simple straining conditions, the rate of growth of the volume of turbulent fluid appears to be independent of the relative depth of indentations in the turbulence front.

It is considered that in free turbulent shear flows, the potential-turbulent flow boundary will advance into the non-turbulent fluid at a rate which depends on the local mean rate of strain and the local degree of anisotropy adjacent to the front. Contraction is expected to occur on those occasions when the boundary lies in a region of zero mean rate of strain, or when the anisotropy of the turbulence is low. The increased rate of spread due to large eddy activity is thought to be mainly caused by an increase in the probability of the potential-turbulent flow boundary being in a position where the mean rate of strain and local anisotropy are both high, rather than being entirely due to an increase in the surface area of the front.

The unsteady laminar boundary-layer flow induced by the impulsive motion of a semi-infinite flat plate along its length is investigated. It is found that, unlike Stewartson's (1951, 1960) conclusion, a power-series solution is possible using the ‘correctly stretched’ variables in the analysis. The small-time solution, which is developed in powers of the time, shows a smooth transition from the initial Rayleigh flow to the final Blasius flow without an essential singularity and, furthermore, its validity extends to the whole time domain. The series solution developed for large times, however, seems divergent and merely asymptotic. No evidence is found for the existence of an essential singularity in the solution as described by Stewartson.

A second-order theory for equatorial, baroclinic Rossby waves in a bounded ocean yields, among other components, terms of zero frequency. We inquire whether the associated rectified flow (or ‘streaming’, or ‘mass transport’) can be related to the submerged equatorial jet observed by Cromwell, Knauss and others. For quite reasonable models of baroclinic Rossby waves, the resultant streaming can be sharply concentrated at the equator, and varies in depth as N(z), thus giving maximum flow at the thermocline (where the Väisälä frequency N(z) is largest). In this way the rectification hypothesis might account for the observed submerged equatorial steering of the Cromwell current. The observed direction of streaming calls for a predominance of equatorially symmetric Rossby waves. The observed magnitude implies that the r.m.s. fluctuations be of the same order as the mean current; that is to say, we have applied weakinteraction theory to a strong-interaction problem. There are other difficulties.

When a horizontal layer of fluid is heated from below and cooled from above with the mean temperature and physical parameters of the fluid constant, the two-dimensional roll is known to be the stable solution near the critical Rayleigh number. In this study, with the mean temperature changing steadily at a rate η, the Rayleigh number and the velocity and temperature fields governed by the Boussinesq equations are expanded in two parameters: η, and the amplitude ε. Hexagons are shown to be the stable solution near the critical Rayleigh number. The direction of the motion depends upon the sign of η. A finite amplitude instability is possible with an associated hysteresis in the heat flux as the critical Rayleigh number is approached from below or from above.

A recent analysis by Gupta (1967) suggests that a layer of elastico-viscous fluid at rest between parallel plane boundaries may be in unstable equilibrium. This surprising result is attributable to the inadequacy of the constitutive equation adopted by Gupta as the basis for his analysis. An alternative constitutive relation, which takes account of the entire strain-history of the motion, leads to the more reasonable result that the equilibrium is stable whenever the fluid has a ‘fading memory’.

It is shown that useful information concerning the flow in the neighbourhood of the various separation and stagnation points in the laminar near wake of a blunt-based two-dimensional wedge can be learned from the locally valid Stokes type series solutions to the incompressible Navier-Stokes vorticity equation derived previously by Dean & Montagnon (1949) and Moffatt (1964). This theory, which is in qualitative agreement with the experiments of Hama (1967) and Donaldson (1967), shows that the flow separates from the base of a blunt-based body and not from its trailing edge. The series solution for the two-dimensional stagnation point is treated in detail and compared with Howarth's (1934) numerical solution in order to study the convergence and conditions for completeness of the Stokes type series solution. Finally, the wake rear stagnation point is examined to provide insight into the problem of wake closure.

A non-linear potential problem which describes the injection of a high speed jet into a uniform flow of lower total head is linearized when the jet injection angle is small. The resulting linear problem is solved using the Wiener–Hopf technique. Numerical results for the position of the streamline separating the jet from the free stream and the pressure coefficient along the upstream wall are obtained for various injection angles.

An experimental investigation of the instability of a laminar air flow over water shows two distinct modes of unstable oscillations as predicted by theory. The Tollmien–Schlichting waves instability could be excited by a ribbon vibrating in the air, and the neutral stability curve determined. The water wave instability mode could be excited by a ribbon vibrating in the water. The growth rates of these waves show only fair agreement with theoretical predictions.

It has been found in part 1 (Krishnamurti 1968) that when the mean temperature of a fluid layer is changing at a constant rate η, hexagonal flows are stable in a range of Rayleigh numbers near the critical. The direction of flow depends upon the sign of η. The static state is unstable to finite amplitude disturbances at Rayleigh numbers below the critical point predicted by linear theory.

The validity of this theory is tested in an experiment in which the heat flux is measured as a function of η and Rayleigh number. The horizontal plan form is determined from the side by continuously exposing a photographic film moving in a vertical direction as tracers in different regions of the fluid are illuminated. Finite amplitude instability and hexagonal cells are indeed observed.

New factors illustrating the effect of convection on acoustic radiation are deduced from Lighthill's theory of aerodynamic noise. For sound dependent on mean velocity variations for its generation, that is shear noise, the new factors are (1 − M cos θ)−3 and (1 − M cos θ)−1 in contrast with (1 − M cos θ)−5 for self noise.

Comparison with the measured far-field directionality patterns reported by Howes (1960) shows that the factor (1 − M cos θ)−3 provides considerably better agreement with the experimental observations than the self noise factor.

The source terms discussed by Lighthill (1954) when considering shear amplification are examined and shown to contain the terms produced by the analysis of Csanady (1966) as well as a new term dependent on the second derivative of the mean velocity.

The description of a laminar boundary layer with a constant pressure gradient parameter β but with an initial velocity profile close to that given by the solution of the Falkner-Skan equation for that β, is shown to lead to an eigenvalue problem in much the same manner as prevails for the Blasius solution. However, it is found that only for β > β0, where β0 corresponds to the separation value, and for the upper branch solutions are the eigenvalues all positive and the flow spatially stable. The lower branch solutions involve negative as well as positive eigenvalues and are spatially unstable.

This paper is an analysis of incompressible three-dimensional flows of electrically conducting fluids under the action of transverse magnetic fields which are assumed to be sufficiently strong that the interaction parameter N (= M2/R) [Gt ] 1, where M is the Hartmann number and R is the Reynolds number. We also assume that R [Gt ] 1 and Rm (magnetic Reynolds number) [Lt ] 1, so that experimental verification of the theory may be possible.

The main results are: (i) when a thick body is placed in a parallel-sided channel with non-conducting walls the flow over it is highly dependent on the conductivity of the body, in a surprising way. If the body is non-conducting, there is no flow within that cylinder which circumscribes the body and is parallel to the magnetic field; outside the cylinder the flow is plane and potential and enters or leaves the surface shear layer of this cylinder at right angles. If the body is conducting, flow over it is possible and is of a different nature outside and inside the cylinder. (ii) When a non-conducting flat plate is placed in such a channel no blocking of the flow occurs. If the plate is elongated in the flow direction, the flow over it becomes identical to that calculated by Hasimoto (1960) and, if elongated at right angles to the flow, becomes identical to that calculated by Dix (1963).

Of particular interest in our analysis are the two types of layer which occur in these flows, the first being the Hartmann boundary layer, which is shown to have a controlling influence on the vorticity of the core flow in three-dimensional situations analogous to that of the Eckman layer in rotating-fluid flows. The second type, the free shear layer at the circumscribing cylinder, is of interest because of its internal structure and effect on the external flow.

On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation.

It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.