A bifurcation study is made of laminar flow in curved ducts. The problem is formulated in a curvilinear coordinate system, and the governing equations, after orthogonal mapping is applied, are solved numerically by an iterative finite-difference method. Many computer runs were made with various duct cross-sections ranging from a circle to a square, to learn the transition of bifurcation structure with this change in cross-section and to reconcile the differences between them. In addition, a simpler technique is proposed to generate symmetric four-cell solutions in a circular pipe and a means is put forward to stabilize four-vortex structures in a complete cross-section.

The relative role of outer- and wall-layer structures in the dynamics of the near-wall region of a turbulent boundary layer was explored by examining the scaling of the spanwise correlation coefficient between the wall shear stress and the streamwise velocity fluctuation, Rτu(z). within narrow frequency bands spanning the entire turbulence spectrum. The scaling characteristics of Rτu(z) within the individual frequency bands indicate that one can separate the contribution to the streamwise velocity fluctuations, in the buffer and logarithmic regions, due to outer-layer structures from those due to wall-layer events. Results provide insight into the lack of wall scaling for the conventional correlation coefficient, Rτu(z), in the near-wall region. Moreover, the ‘negative dip’ in Rτu(z+), which has often been associated with the low-speed streaks, was found to exist within certain frequency bands for all Reynolds numbers investigated (1579 [Lt ] Reθ [Lt ] 5961). More interestingly, it is shown that, although the energy of the outer-layer structures increases with Reynolds number to overwhelm the streamwise velocity fluctuations in the near-wall region, the production of the Reynolds shear stress is dominated by wall-layer eddies. The findings of the current investigation provide strong support for Townsend's hypothesis of ‘active’ and ‘inactive’ motions.

The formation of stable cones in electrified liquid interfaces was explained by Taylor as a balance between electrical and capillary tensions, where the electrostatic potential varies as ϕ ∼ r½ with the distance r from the cone tip. Although Taylor's predictions for the dependence of the onset voltage for cone formation on the liquid surface tension γ and the cone dimensions agree with observed trends, his conclusion that the cone semiangle α can only take the value α = αT = 49.3° does not. A more general theory free from this paradox is constructed for highly conducting fluids by accounting for the space charge of the droplets emanating from the cone apex, whose potential has the remarkable property of also obeying Taylor's r½ law. In this formulation, where the apex of a conical meniscus of semiangle α emits an angularly uniform opposed coaxial conical spray of semiangle π—β, both β and the spray current I turn out to be fixed as functions of α; namely, β = β(α), and I = 2πγKqG(α), where Kq and q are the droplet's electrical mobility and total charge, respectively. In experiments with 5% H2SO4 in 1-octanol, the observed sprays are approximately conical with an apex nearly touching the meniscus tip. The measured and predicted β(α) relations are in reasonable agreement in the range 46° > α > 32°, where the liquid cone is stable and the spray is visible, though the data fall clearly below the theoretical curve. The predicted spray current I is also in rough agreement with preliminary experiments. The analysis applies neither to sprays of large droplets with significant inertia, nor to liquid cones in vacuo.

This paper concerns the asymptotic behaviour (for the limiting case of small amplitudes) of small disturbances as they evolve in time to produce the quasi-steady pattern of roll waves first discussed by Dressler in 1949. Roll waves exist if F, the undisturbed Froude number (dimensionless speed) of the flow, exceeds 2, and consist of a periodic pattern of bores separating two special continuous solutions of the governing equations in a uniformly translating frame. The mathematical problem is rather interesting as solutions of the linearized equations are unstable for F > 2. Thus, it is crucial to account for the cumulative effect of small nonlinearities to obtain a correct description of the flow over long times. We concentrate on the weakly unstable problem (0 < F − 2 [Lt ] 1) and use multiple scale expansions to derive the dominant evolution equation that governs the solution behaviour for long times. This turns out to be an integro-partial differential equation of first order that we solve numerically in conjunction with the jump condition that follows from the exact bore conditions. We present asymptotic and numerical results for periodic as well as isolated initial disturbances, and show that our theory predicts the solution accurately for both the transient and quasi-steady phases.

An experimental investigation of the fine-scale structure of turbulence was carried out. Five different shear flows were studied: three in a wind tunnel with an open working section and an elliptical nozzle and two in a wind tunnel of closed working section and square cross-section. The experiments tested two approaches to the theory of fine-scale turbulence structure: one based on the Navier-Stokes equations and the other on some similarity hypotheses. The variability of all fine-scale constants (including exponents in inertial-subrange power laws and the Kolmogorov constant) is revealed. A correlation between all fine-scale constants and the external intermittency coefficient is established.

Consistently employing the assumption of localness of wave–wave interactions in the wavenumber space, the Kolmogorov treatment of the energy cascade is applied to the case of wind-generated surface gravity waves. The effective number v of resonantly interacting wave harmonics is not limited to four but is found as a solution of a coupled system of equations expressing: (i) the dependence of the spectrum shape on the degree of the wave nonlinearity, and (ii) the continuity of the wave action flux through the spectrum given a continuous positive input from wind. The latter is specified in a Miles-type fashion, and a simple scaling relationship based on the concept of the turnover time is derived in place of the kinetic equation. The mathematical problem is reduced to an ordinary differential equation of first order. The exponent in the ‘power law’ for the spectral density of the wave potential energy and the effective number of resonantly interacting wave harmonics are found as functions of the wave frequency and of external factors of wind—wave interaction. The solution is close to the Zakharov—Filonenko spectrum at low frequencies and low wind input while approaching the Phillips spectrum at high frequencies and sufficiently high wind.

The propagation of nonlinear hydromagnetic waves in a highly conducting, self-gravitating fluid in a cylindrical geometry, subject to the convective forces produced by a radial temperature gradient, is treated in a Boussinesq approximation. Exact wave solutions of the nonlinear magnetohydrodynamic equations (in the Boussinesq approximation) in the presence of convective forces are obtained for the case when the physical quantities are independent of the axial coordinate or the azimuthal angle in the cylindrical coordinates. The solutions represent waves propagating in the azimuthal or axial direction under the influence of the helical magnetic and velocity fields and the convective forces. The solutions may be applicable to the hydromagnetic waves in the Earth's fluid core and the solar convection zone.

We study the linear stability of a vertical, perfectly concentric, core–annular flow in the limit in which the gap is much thinner than the core radius. The analysis includes the effects of viscosity and density stratification, interfacial tension, gravity and pressure-driven forcing. In the limit of small annular thicknesses, several terms of the expression for the growth rate are found in order to identify and characterize the competing effects of the various physical mechanisms present. For the sets of parameters describing physical situations they allow immediate determination of which mechanisms dominate the stability. Comparisons between the asymptotic formula and available full numerical computations show excellent agreement for non-dimensional ratio of undisturbed annular thickness to core radius as large as 0.2.

The expansion leads to new linear stability results (an expression for the growth rate in powers of the capillary number to the $\frac{1}{3}$ power) for wetting layers in low-capillary-number liquid–liquid displacements. The expression includes both capillarity and viscosity stratification and agrees well with the experimental results of Aul & Olbrich (1990).

Finally, we derive Kuramoto–Sivashinsky-type integro-differential equation for the later nonlinear stages of the interfacial dynamics, and discuss their solutions.

We have calculated the mobility and resistivity functions for two equal-sized cylinders moving in a thin viscous sheet surrounded by a lower-viscosity fluid. These functions describe the hydrodynamic interactions between proteins embedded in a lipid bilayer surrounded by an aqueous solution. For a protein of radius a embedded in a biological membrane of thickness h and viscosity μ, a key parameter λ = μh/aμ′, which characterizes the viscosity ratio between the bilayer and surounding solution, is O(100). The method of solution for the hydrodynamic interactions differs depending on the separation distance, r, between the cylinders. When r = O(a), the particles are in the near-field regime, and the solution of the Stokes equations is divided between an inner and outer domain based on the asymptotically large value of λ. The inner solution neglects the flow in the lower-viscosity fluid and is solved numerically using a bipolar expansion. The outer solution is based on the fluid flows in all phases, but the cylinders are approximated by net forces. When r [Gt ] O(a), the particles are in the far-field regime, and we use the method of reflections to solve for the hydrodynamic interactions. The uniformly valid approximations, constructed from a combination of the near-field and far-field solutions, agree with the analytic solutions obtained within the lubrication and far-field regimes. Our results show that the hydrodynamic interactions between the cylinders are long range, perarting on a lengthscale of O(λ). The range of the hydrodynamic interactions is much longer than that of non-hydrodynamic interparticle forces, suggesting that hydrodynamic interactions will be significant determinants of the structure and dynamics of biological membranes.

The results of an experimental investigation of a turbulent jet flow issuing from a circular nozzle beneath and parallel to a free surface are presented. Measurements of the mean velocity vector and all components of the Reynolds stress tensor were made using a three-component, underwater laser-Doppler velocimeter (LDV). Visualizations of the flow field using both fluorescent dye and free-surface shadowgraphs were made in support of the measurements. The jet is observed to form a shallow surface current, the lateral extent of which is significantly greater than that of the primary jet flow that produces it. Flow visualization reveals the surface current to consist largely of fluid structures ejected from the jet. These structures remain coherent within the current, apparently as a consequence of reduced turbulent mixing just beneath the surface. LDV measurements of the turbulence within the surface current reveal that near the jet centreline, where the interaction between the jet and the free surface is most intense, the velocity fluctuations normal to the free surface are diminished in approaching the surface, while the fluctuations parallel to the surface are enhanced. Away from the jet centreline, toward the edges of the surface current, the vertical and cross-stream fluctuations become approximately equal in magnitude, whereas the streamwise fluctuations become diminished. This is attributed, in part, to orbital motions beneath surface waves generated by the interaction of large-scale jet structures with the surface. When oleyl alcohol, an insoluble surface-active agent, was added to the free surface, the surface current was not observed. Comparisons between a jet issuing beneath both clean and surfactant-contaminated free surfaces, and a jet issuing beneath a solid wall are made to identify the role of streamwise vorticity and of secondary vorticity generated at boundaries on the development of these flows.