It has been argued that the no-slip boundary condition, applicable when a viscous fluid flows over a solid surface, may be an inevitable consequence of the fact that all such surfaces are, in practice, rough on a microscopic scale: the energy lost through viscous dissipation as a fluid passes over and around these irregularities is sufficient to ensure that it is effectively brought to rest. The present paper analyses the flow over a particularly simple model of such a rough wall to support these physical ideas.

Equations for the development of streamwise (or secondary) vorticity, for stationary and rotating systems, are derived directly from the vector equation for vorticity, using intrinsic co-ordinates. This approach places emphasis on the coupled equations for the vorticity components parallel and normal to the streamline. The equations derived are more general than those hitherto available, being valid for compressible, stratified and viscous flow. Some interpretation is given of the physical mechanism causing secondary flow.

Conditionally sampled hot-wire measurements were taken in a pipe at Reynolds numbers corresponding to the onset of turbulence. The pipe was smooth and carefully aligned so that turbulent slugs appeared naturally at Re > 5 × 104. Transition could be initiated at lower Re by introducing disturbances into the inlet. For smooth or only slightly disturbed inlets, transition occurs as a result of instabilities in the boundary layer long before the flow becomes fully developed in the pipe. This type of transition gives rise to turbulent slugs which occupy the entire cross-section of the pipe, and they grow in length as they proceed downstream. The leading and trailing ‘fronts’ of a turbulent slug are clearly defined. A unique relation seems to exist between the velocity of the interface and the velocity of the fluid by which relaminarization of turbulent fluid is prevented. The length of slugs is of the same order of magnitude as the length of the pipe, although the lengths of individual slugs differ at the same flow conditions. The structure of the flow in the interior of a slug is identical to that in a fully developed turbulent pipe flow. Near the interfaces, where the mean motion changes from a laminar to a turbulent state, the velocity profiles develop inflexions. The total turbulent intensity near the interfaces is very high and it may reach 15% of the velocity at the centre of the pipe. A turbulent energy balance was made for the flow near the interfaces. All of the terms contributing to the energy balance must vanish identically somewhere on the interface if that portion of the interface does not entrain non-turbulent fluid. It appears that diffusion which also includes pressure transport is the most likely mechanism by which turbulent energy can be transferred to non-turbulent fluid. The dissipation term at the interface is negligible and increases with increasing turbulent energy towards the interior of the slug.

Mixed laminar and turbulent flows were observed far downstream for \[ 2000 < Re < 2700 \] when a large disturbance was introduced into the inlet. The flow in the vicinity of the inlet, however, was turbulent at much lower Re. The turbulent regions which are convected downstream at a velocity which is slightly smaller than the average velocity in the pipe we shall henceforth call puffs. The leading front of a puff does not have a clearly defined interface and the trailing front is clearly defined only in the vicinity of the centre-line. The length and structure of the puff is independent of the character of the obstruction which created it, provided that the latter is big enough to produce turbulent flow at the inlet. The puff will be discussed in more detail later.

Numerical methods are used to investigate the separated flow over a finite flat plate when the flow at large distances is given by the stream function ψ∞ = −xy and the plate is situated on the x axis from -1 to 1. The range of nominal Reynolds number is 10–800. Reduced-mesh calculations are used for fine resolution of the flow field in the immediate vicinity of the separation point. Streamlines, equi-vorticity lines, and shear stress and pressure gradient at the plate surface illustrate the overall structure of the flow. In each case the streamwise pressure gradient is less than that for undisturbed potential flow and the position of separation is consequently downstream of that predicted by classical boundary-layer theory. The boundary-layer structure in the vicinity of the separation point shows a direct transition between the regular upstream behaviour and Dean's (1950) solution right at separation with no sign whatever of intermediate singular behaviour of the type predicted by Goldstein (1948). The implications of these results for the structure of high Reynolds number, steady, laminar flow are discussed.

Numerical solutions of finite-difference approximations to the Navier–Stokes equations have been obtained for the axisymmetric motion of a Boussinesq liquid in a rigidly bounded rotating annulus. For most of the cases studied, a temperature difference is maintained between the top and bottom surfaces such that essentially a basic stable density stratification is imposed on the fluid. The side walls are thermally insulated and the motion is driven by a differential rotation of the top surface. Approximate steady-state solutions are obtained for various values of the Rossby number ε and the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The changes in the flow field with the variation of these parameters is studied. Particular attention is given to an investigation of the meridional, or up welling, circulation and its dependence on the stratification parameter. The effects on the flow of different boundary conditions, such as an applied stress driving, specified temperature at the side walls and an applied heat flux at the top and bottom surfaces, are also investigated.

Some of the statistical characteristics of the breakdown coefficient, defined as the ratio of averages over different spatial regions of positive variables characterizing the fine structure and internal intermittency in high Reynolds number turbulence, have been investigated using experimental data for the streamwise velocity derivative ∂u/∂t measured in an atmospheric boundary layer. The assumptions and predictions of the hypothesis of scale similarity developed by Novikov and by Gurvich & Yaglom do not adequately describe or predict the statistical characteristics of the breakdown coefficient qr,l of the square of the streamwise velocity derivative. Systematic variations in the measured probability densities and consistent variations in the measured moments show that the assumption that the probability density of the breakdown coefficient is a function only of the scale ratio is not satisfied. The small positive correlation between adjoint values of qr,l and measurements of higher moments indicate that the assumption that the probability densities for adjoint values of qr,l are statistically independent is also not satisfied. The moments of qr,l do not have the simple power-law character that is a consequence of scale similarity.

As the scale ratio l/r changes, the probability density of qr,l evolves from a sharply peaked, highly negatively skewed density for large values of the scale ratio to a very symmetrical distribution when the scale ratio is equal to two, and then to a highly positively skewed density as the scale ratio approaches one. There is a considerable effect of heterogeneity on the values of the higher moments, and a small but measurable effect on the mean value. The moments are roughly symmetrical functions of the displacement of the shorter segment from the centre of the larger one, with a minimum value when the shorter segment is centrally located within the larger one.

The method of solution of the Korteweg–de Vries equation outlined by Gardner et al. (1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.

In this paper, we obtain the first-order necessary optimality conditions of an optimal control problem for a distributed parameter system with geometric control, namely, the minimum-drag problem in Stokes flow (flow at a very low Reynolds number). We find that the unit-volume body with smallest drag must be such that the magnitude of the normal derivative of the velocity of the fluid is constant on the boundary of the body. In a three-dimensional uniform flow, this condition implies that the body with minimum drag has the shape of a pointed body similar in general shape to a prolate spheroid but with some differences including conical front and rear ends of angle 120°.

The stability of a layer of Newtonian fluid confined between two horizontal disks which rotate with different angular velocities is studied. Both isothermal and adversely stratified fluids are considered for small shear rates at low to moderate Taylor numbers. The linearized formulation of the stability problem is given a finite-difference representation, and the resulting algebraic eigenvalue problem is solved using efficient numerical techniques. The critical parameters and disturbance orientations are determined as a function of the Taylor number for the isothermal flow, and for the stratified flow for Prandtl numbers of 0·025, 1·0 and 6·0.

At high Taylor numbers, the unstratified fluid flows in Ekman-like layers near the disks, and two modes of instability are noted: the viscous-type ‘class A’ travelling wave, whose existence depends on Coriolis forces, and the inflexional ‘class B’ mode, which is nearly stationary with respect to the nearer bounding disk. As the Taylor number is decreased, the Ekman layers coalesce to form a fully developed flow. In this regime there is a Taylor number below which the class A waves are always damped. The critical Reynolds number for the class B waves increases rapidly as the Taylor number approaches zero.

For Prandtl numbers of 1·0 and 6·0, the adversely stratified flow exhibits two distinct types of instability: convective and dynamical. At low Reynolds numbers, a stationary mode associated with Bénard convection in a rotating fluid is critical. It is stabilized and given orientation by the shear. At higher Reynolds numbers, the critical mode is a travelling wave of the nature of either the class A or class B waves, depending upon the Taylor number. For a Prandtl number of 0·025, the critical mode resembles oscillatory convection at small Reynolds numbers and a class A wave at larger shear rates.

An experimental investigation into flow through shaped gauze screens in two-dimensional and axisymmetric situations has shown that there is disagreement between measured velocity profiles and those computed from the method developed by Elder (1958). This disagreement has been shown to be attributable to the retention of a term of second order in the basic linearization and comparisons between experimental and theoretical profiles omitting this term are in good agreement.

A short-wave asymptotic analysis is undertaken for problems concerned with the radiation and scattering of surface waves by a cylinder whose cross-section S intersects the free surface normally. It is assumed that S is locally smooth and convex at the two intersection points with the fluid, which may be of infinite or finite depth. For both the scattering and radiation problem, a matched expansion technique is used to provide asymptotic estimates, in terms of relatively simple wave-free limit potentials, for the amplitudes of the surface wave trains that propagate from S. Explicit details are given for some particular geometries, confirming and extending earlier results. The method can, in principle, be extended to deal with other geometries.

A unique stagnation condition is derived as a guide for analysing systems of combined or branched flows through the use of the free-streamline theory and the Schwarz–Christoffel transformation. Moreover, the physical significance of such a condition is illustrated through its application to the analysis of the penetration and deflexion of jets normal to a bounded stream.

The radiation properties of partially immersed three-dimensional bodies, in time-periodic motion, are examined in the short-wave asymptotic limit ε → 0, where ε is a non-dimensional wavelength. The method of matched expansions is used to specify an outer approximation, away from the surface wave region, and an inner approximation where the potential, in the vicinity of the obstacle and free surface, depends only on the local geometry. Finally, the radially spreading surface wave field is estimated by ray-theory arguments. Explicit details are given for the heaving and rolling of a circular dock and for the heaving motion of a hemisphere. Some speculations are made regarding the scattering properties of such obstacles.

The interaction of two pure helical (circularly polarized) velocity waves according to the incompressible Navier–Stokes equation produces modulation products of mixed helicity. In general, the interaction of waves of opposite helicity is stronger than that of waves with the same helicity. The inference is that strong net helicity depresses overall turbulent energy transfer. The conservation laws strongly inhibit energy transfer from higher to lower wavenumbers, when the helicity is large. The absolute equilibrium spectra of velocity and helicity for an inviscid flow system truncated at an upper wavenumber k2 are \[ U(k) = 2\alpha/(\alpha^2-\beta^2k^2),\quad Q(k) = 2\beta k^2/(\alpha^2-\beta^2k^2), \] where the velocity variance and helicity/unit volume are ∫U(k)d3k and ∫Q(k)d3k, respectively. The temperature parameters α and β are constrained by α > 0 and |βk2| < α. There are no analogues of the negative-temperature equilibrium states known for two-dimensional inviscid flow. It is argued that the inertial-range energy cascade in isotropic turbulence driven by helical input should not differ asymptotically from that of non-helical turbulence. The absolute equilibrium distributions suggest that, in contrast to the analogous two-dimensional situation, statistically steady helical input at middle wavenumbers should not produce a significant downward cascade of energy to lower wavenumbers.

The proper corrections for non-parallel flow to the eigenvalues for small disturbances on a nearly parallel shear flow have been determined through a perturbation about the parallel flow solutions. The resulting shifts in the neutral stability curves have been calculated for the Blasius boundary layer, for the two-dimensional jet, and for the two-dimensional flat-plate wake.

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

Existing analyses of the very low Reynolds number Stokes jet, e.g. Birkhoff & Zarantonello (1957) and Förste (1963), have been confined to a single-component fluid satisfying the usual zero-slip boundary conditions at a solid surface. In contrast, the low Reynolds number biological jets which emerge from the pores and channels at the secreting surfaces of numerous human, animal and insect organs entail the movement of water and solute subject to novel boundary conditions that arise from the local osmotic driving forces at the secreting surfaces. These boundary conditions introduce a nonlinear coupling between the fluid momentum and solute conservation equations. This paper first discusses the fundamental dimensionless groups and length scales that characterize these biological jet flows and then examines in detail, using the technique of matched inner and outer expansions, one flow situation important in epithelial membrane transport, namely, the two-dimensional mixing of an inhomogeneous jet with a quiescent outer bathing solution which is bounded by a semi-permeable mem brane in the plane of the exit. The paper concludes with a discussion of other physiologically relevant problems which arise from different orderings of the length scales, and different overall geometrical configurations and flow conditions.

Small amplitude forced horizontal or vertical oscillations of a container of liquid with a free surface may give rise to motions in the liquid on a scale much greater than the forcing amplitude. Three such situations are analysed and, in those cases where the response is still small compared with the dimensions of the container, explicit asymptotic solutions for the liquid motion are obtained.

A finite-difference method recently developed to study three-dimensional viscous flow is applied here to the supersonic boundary layer on a sharp cone at moderate angles of incidence (α/θ [les ] 2, angle of attack α, cone half-angle θ). The present analysis differs from previous investigations of this region in that (i) boundary-layer similarity is not assumed, (ii) the system of governing equations incorporates lateral diffusion and centrifugal force effects, and (iii) an improved numerical scheme for three-dimensional viscous flows of the type considered here is used. Solutions are shown to be non-similar at the separation streamline with local shear-layer formation. Detailed flow structure, including surface heat transfer, boundary-layer profiles and thickness, and the formation of swirling pairwise symmetric vortices, associated with cross-flow separation, are obtained. Good agreement is obtained between the present theoretical results and the existing experimental data.

We consider a drop of liquid hanging from a horizontal support and sandwiched between two vertical plates separated by a very narrow gap. Equilibrium profiles of such ‘two-dimensional’ drops were calculated by Neumann (1894) for the case when the angle of contact between the liquid and the horizontal support is zero. This paper gives the equilibrium profiles for other contact angles and the criterion for their stability. Neumann showed that, as the drop height increases, its cross-sectional area increases until a maximum is reached. Thereafter, as the height increases, the equilibrium area decreases. This behaviour is shown to be typical of all contact angles. When the maximum area is reached, the total energy is a minimum. It is shown that the drops are stable as long as the height and the area increase together.