The hydrodynamic forces and couples that act on two spherical particles in slow motion through a quiescent fluid are determined as functions of the relative configuration of the particles from the solution of the Stokes equation for the motion of the fluid in the vicinity of the particles. General formulae that relate the translational and rotational velocities of the particles to the ratios of their radii a = a2/a1 and net densities I are obtained, as are asymptotic forms for the velocities in the limiting cases of very large and very small interparticle separation. Relative trajectories of the particles when they move solely under gravity and their own interaction are calculated for several values of I and a. A particularly interesting feature of the results is that, for certain ranges of values of I and a, trajectories of finite length and trajectories having the form of closed periodic orbits may occur.

Hot-film anemometer measurements have been carried out in a fully developed turbulent channel flow. An oil channel with a thick viscous sublayer was used, which permitted measurements very close to the wall. In the viscous sublayer between y+ ≃ 0·1 and y+ = 5, the streamwise velocity fluctuations decreased at a higher rate than the mean velocity; in the region y+ [lsim ] 0·1, these fluctuations vanished at the same rate as the mean velocity.

The streamwise velocity fluctuations u observed in the viscous sublayer and the fluctuations (∂u/∂y)0 of the gradient at the wall were almost identical in form, but the fluctuations of the gradient at the wall were found to lag behind the velocity fluctuations with a lag time proportional to the distance from the wall. Probability density distributions of the streamwise velocity fluctuations were measured. Furthermore, measurements of the skewness and flatness factors made by Kreplin (1973) in the same flow channel are discussed. Measurements of the normal velocity fluctuations v at the wall and of the instantaneous Reynolds stress −ρuv were also made. Periods of quiescence in the − ρuv signal were observed in the viscous sublayer as well as very active periods where ratios of peak to mean values as high as 30:1 occurred.

The flow normal to an infinite circular cylinder which is uniformly accelerated from rest in a viscous fluid is considered. The flow is assumed to remain symmetrical about the direction of motion of the cylinder. Two types of solution are presented. In the first an expansion in powers of the time from the start of the motion is given which extends the results of boundary-layer theory by taking into account corrections for finite Reynolds numbers. Physical properties of the flow for small times and finite but large Reynolds numbers are calculated from this expansion. In the second method of solution the Navier-Stokes equations are integrated by an accurate procedure which is a logical extension of the solution in powers of the time. Results are obtained for R2 = 97·5, 5850, 122 × 103 and ∞, where R is the Reynolds number. This is defined as R = 2a(ab)½/v, where a is the radius of the cylinder, b the uniform acceleration and v the kinematic viscosity of the fluid. The methods are in good agreement for small times.

The numerical method of integration has been carried to moderate times and various flow properties have been calculated. The growth of the length of the separated wake behind the cylinder for R2 = 97·5, 5850 and 122 × 103 is compared with the results of recent experimental measurements. The agreement is only moderate for R2 = 97·5 but it improves greatly as R increases. The numerical integrations were continued in each case until the implicit method of integration failed to converge, which terminated the procedure. A secondary vortex appeared on the surface of the cylinder for the case R2 = 122 × 103.

The laminar boundary layer on a rotating thin blade of an axial turbomachine is discussed. A method of deriving the basic equations of the flow for a generalized blade configuration is described, treating a typical blade in practical use as an example. A perturbation technique is applied to obtain a series solution composed of two groups of similar functions. The results give insight into the properties of the boundary layer on helical non-loaded blades in a uniform stream, indicating the influence purely of three-dimensionality of the blade configuration.

The sublayer of a turbulent boundary-layer flow with and without polymer additives was measured by a simple optical technique having a high degree of spatial resolution. The reduction of skin friction with polymer additives was found to be due to a decrease in the effective viscosity of water at the interface between the mucus (glycoprotein) or polymer (polyethylene oxide) and the water and a thickening of the laminar sublayer. The rate at which mucus diffuses away from a medium-sized fish is almost constant for all swimming speeds.

The body of given volume with the smallest drag in Stokes flow is obtained by making use of theoretical results due to Pironneau. A suitable family of solutions of the Stokes equations is used and the no-slip condition is expressed numerically by a technique of quadratic minimization. We find that the drag on this optimal body is 0·95425 times the drag on the sphere of equal volume.

This paper deals with the development of the flow in a curved tube near the inlet. The solution is obtained by the method of matched asymptotic expansions. Two inlet conditions are considered: (i) the condition of constant dynamic pressure at the entrance, which may be of practical interest in applications to blood flow in the aorta; and (ii) a uniform entry condition. It is shown that the geometry and the nature of the entry condition appreciably influence the initial development of the flow. The effect of the secondary flow due to the curvature on the wall shear is discussed and it is shown that the cross-over between shear maxima on the inside and the outside of the tube occurs at a downstream distance which is 1·9 times the radius of the tube for entry condition (i) while in the case of entry condition (ii) it is 0·95 times the radius, which is half the distance required in case (i). It is found that the pressure distribution is not significantly influenced by the secondary flow during the initial development of the motion. The analysis, which is developed for steady motion, can be extended to pulsatile flows, which are of greater physiological interest.

The growth of infinitesimal disturbances on an axisymmetric jet column is investigated theoretically and experimentally. The theoretical analysis is based upon inviscid stability theory, wherein axisymmetric, helical and double helical disturbances are considered from the spatial reference frame. In the jet flow field near the source, the mean velocity profile is observed to have a potential core and a thin, but finite, shear layer between the potential core and the quiescent ambient fluid. With downstream distance, the potential core diameter decreases and the shear-layer thickness increases. To incorporate these variations into the theory, a quasi-uniform assumption is adopted, whereby successive velocity profiles are analysed individually throughout the region in the jet flow where disturbances are observed to be small. The results of the theory indicate that initially, in the jet flow where the shear layer is thin and the potential core is larger, all disturbances considered are unstable. The dominant disturbance in the jet is an axisymmetric one. However, further downstream in the jet, where the half-breadth thickness of the shear layer is 55% of the potential core radius, a helical disturbance is found to dominate the axisymmetric and double helical modes. Nowhere in the jet flow field examined was the double helical disturbance found to be dominant. The cross-stream distributions of velocity and vorticity for the dominant disturbance modes are presented according to the spatial stability theory.

The downstream development of the jet column and the characteristics of the disturbances amplifying on it were also studied in a water tank. No artificial stimulation of any particular disturbance was used. The experimental results show good agreement with the results of the theory in the region where the disturbances are small. However, conclusive confirmation of the switch in the hierarchy of dominant disturbances was not found. Half of the time the disturbance observed experimentally exhibits an axisymmetric character and the other half a helical one. This apparently is due to the similar spatial amplification rates experienced by both of these disturbance modes. It is concluded that this switching of dominant modes is, in large part, responsible for (i) the well-known natural drifting of disturbance characteristics in jet flows, and (ii) the wide variety of observations made in previous jet experiments.

Kolmogorov's second hypothesis has been examined for the case of turbulence generated behind a very large grid. The turbulent Reynolds number RΛ = 280 was sufficient to obtain a short inertial subrange. The one-dimensional subrange constant a1 = 0·48 ± 0·06 is in agreement with recent determinations made in geophysical flows. Isotropy was tested by comparing the transverse velocity spectrum with the transverse spectrum predicted from the longitudinal spectrum using the isotropic relations. The comparison showed the flow to be isotropic everywhere except at the largest scales.

It was observed that at high wavenumbers the spectra were attenuated by effects of finite wire length. The wire-length corrections suggested by Wyngaard (1968) were found to be inadequate. New corrections based on experimentally determined universal spectra are proposed.

The interactions of weak nonlinear disturbances in a compressible fluid including shocks, expansion waves and contact surfaces are investigated by making use of the reductive perturbation method. It is found that the nonlinear waves belonging to different families of characteristics behave almost independently of each other, while those belonging to the same family are governed by either the Burgers equation or the equation of heat conduction. Thus the statistical properties of one-dimensional shock turbulence in a compressible fluid are reduced to those of the solutions of the Burgers equation. In particular, the law of energy decay of shock turbulence is shown to be identical to that of Burgers turbulence.

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocity u, rather than the spatial co-ordinate x, is used as the independent variable, and an equation for ∂f/∂u at a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.