Torsional Couette flow between a rotating disk and a stationary wall is studied experimentally. The surface of the disk is either rigid or covered with a compliant coating. The influence of wall compliance on characteristic flow instabilities and on the laminar–turbulent flow transition is investigated. Data obtained from analysing flow visualizations are discussed. It is found that wall compliance favours two of the three characteristic wave patterns associated with the transition process and broadens the parameter regime in which these patterns are observed. The results for the effects of wall compliance on the third pattern are inconclusive. However, the experiments indicate that the third pattern is not a primary constituent of the laminar–turbulent transition process of torsional Couette flow.

This paper examines the low-Reynolds-number migration of an insulating and rigid particle that is freely suspended in a viscous liquid metal and subject to uniform ambient electric and magnetic fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ Under the same physical assumptions as Part 1, a whole boundary formulation of the problem is established. It allows the determination of the particle rigid-body motion without calculating the modified electric field and the flow induced by the Lorentz body force in the fluid domain. The advocated boundary approach, well-adapted for future numerical implementation, makes it possible to obtain an analytical expression for the translational velocity of any ellipsoidal particle (the simplest case of non-spherical orthotropic particles). The behaviour of a spheroid is carefully investigated and discussed both without and with gravity. The migration of this simple non-spherical particle is found to depend on both its nature (prolate or oblate) and the ambient uniform fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ The spheroid translates without rotation, and not necessarily parallel to $\hbox{\bf\itshape E}\wedge \hbox{\bf\itshape B}.$ For adequately selected fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B},$ the spheroid may either migrate parallel or anti-parallel to a sphere and even be motionless.

Two inequalities are proposed for the purpose of bounding from below the mean shear U′o(z) in turbulent channel flow. The first of these inequalities pertains to the energetics of the boundary layer, and the second pertains to the logarithmic form of the asymptote to Uo. These inequalities imply a maximum value of von Kármán's constant, the numerical value of which lies between the measurements of Laufer (1951) and the value obtained from bulk discharge measurements in a pipe. The formalism, which contains no adjustable parameters, is then applied to the turbulent thermal convection problem, and a lower bound for the mean temperature $\overline{T}_o(z)$ is obtained. The minimum value of the latter at large distances from the boundary is in fair agreement with Townsend's (1959) measurements. Although the proposed inequalities have not been deduced from the equations of motion, they provide facts which may be useful in the search for new variational formulations of theturbulent transport problem.

We investigate experimentally and numerically the filling of a collapsible tube, motivated by venous hemodynamics in the lower limbs. The experiments are performed by filling an initially collapsed flexible tube, applying pressure through a hydraulic circuit. The tube law and the tube tension have been previously measured. The tube shape, the flow rate and the pressure at the two ends of the tube are measured continuously. The filling occurs in three stages: a rapid equilibration of the pressure near the tube entry with atmospheric pressure, a quasi-steady filling of the tube with a linearly rising pressure, and a final stage of tube inflation. Our numerical model is the classical one-dimensional collapsible tube equations. Excellent quantitative agreement is found between computations and experimental data. We show experimentally observed shapes near the tube end that indicate possible three-dimensional effects; however these effects do not impair significantly the ability of the one-dimensional model to describe the experiment. Travelling waves of large amplitude are observed in the simulations and the experiments.

An examination of the isotropic vorticity theory was made in an adverse pressure gradient flow based on experimental data obtained in a conical diffuser having a total divergence angle of 8° and an area ratio of 4:1 with fully-developed pipe flow at entry. The results showed that the rates and the ratio of production and dissipation of the turbulent vorticity were constant in the core region of the diffuser but increase significantly in the wall layer. The overall vorticity balance was essentially the same at all axial stations. The analysis of Batchelor & Townsend (1947) for isotropic vorticity was found to be valid in the core region of the diffuser for an order-of-magnitude higher Rλ (200 [les ] Rλ 600) than in grid turbulence. The magnitude of the skewness of ∂u1/∂t was constant in the core region and comparable to that for grid turbulence. Also, this region of constant skewness extended over a larger portion of the flow cross-section compared to pipe flow. On the basis of these results, it was concluded that assumptions of isotropy in the fine structure are valid in the core region of the diffuser.

A general solution approach for the electrohydrodynamic instability of stratified immiscible fluids is presented. The problems of two and three fluid layers subject to normal electric fields are analysed. Analytical solutions are obtained by employing the transfer relations (Melcher 1981 Continuum Electromechanics. MIT Press) relating the disturbance stresses to the flow variables at the interface(s). This approach provides a convenient alternative to the direct solution of the linearized problem. The results assume a general format. Both new dispersion relations and those from various previous works are shown to be special cases when proper simplifications are considered. As a specific example, the instability behaviour of a three-layer channel flow is investigated in detail using this framework. This work provides a unifying method to treat a generic class of instability problems.

A physical theory is proposed for the skin friction on a flat plate at zero incidence in the transition flow regime, i.e. in the flow of a moderately rarefied gas. The ratio of the molecular mean free path to the characteristic size of the plate is assumed of order unity or larger. A general formula for the perturbation to the well-known friction of the free-molecule theory is given. This perturbation is attributed to the intermolecular collisions which are neglected on the basis of the free-molecule hypothesis. The expected rate of collisions are calculated for rigid spheres, using the classical kinetic theory.

Although this is intended as an approximate theory, the theoretical results check surprisingly well with the limited experimental data that are available. The present theory shows that the ratio of the Reynolds number to the Mach number squared is the governing parameter for determining the intermolecular collision effect on skin friction in the transition flow regime.

We examine the fracture of a quasi-two-dimensional surfactant-laden aqueous foam under an applied driving pressure, using a network modelling approach developed for metallic foams by Stewart & Davis (J. Rheol., vol. 56, 2012, p. 543). In agreement with experiments, we observe two distinct mechanisms of failure analogous to those observed in a crystalline solid: a slow ductile mode when the driving pressure is applied slowly, where the void propagates as bubbles interchange neighbours through the T1 process; and a rapid brittle mode for faster application of pressures, where the void advances by successive rupture of liquid films driven by Rayleigh–Taylor instability. The simulations allow detailed insight into the mechanics of the fracturing medium and the role of its microstructure. In particular, we examine the stress distribution around the crack tip and investigate how brittle fracture localizes into a single line of breakages. We also confirm that pre-existing microstructural defects can alter the course of fracture.

In a process involving electromagnetic shaping, a high-frequency electromagnetic field is used to deform a liquid conductor into a required shape. This is particularly relevant to processes such as levitation melting. In this paper the stability of such configurations are investigated. The second variation of an appropriate energy functional is derived whose minimum states correspond to stable configurations, thus providing a stability criterion. As an example, this is applied to the shaping of a levitated cylinder of circular cross-section and to an almost spherical axisymmetric shape. In both cases we find that these shapes are unstable. We then consider enclosing the entire shaping device in a metal shield, thus preventing the escape of the magnetic field. It is then shown that in general the shield has a stabilizing effect, whose exact nature depends on the topology of the liquid shape and on the field structure on its surface. This differing behaviour is discussed for two-dimensional spherical and toroidal shapes.

This paper deals with the Soret separation of a binary mixture in a cylinder subjected to an axial temperature gradient. The study is connected to an experiment designed to measure the Soret coefficient of an AgI-KI mixture corresponding to a moderate Prandtl number (Pr = 0.6) and a high Schmidt number (Sc = 60). In such an experiment the species separation is often hidden by a mixing effect due to the buoyancy-driven convection generated by a horizontal temperature gradient induced by some defect of the heating system. Here, such a defect is simulated by a slight misorientation of the cell with respect to the vertical; a small inclination (γ = 1°) of the cell has been considered, but the results can be generalized for any other small γ. For situations corresponding to a top heating and a positive Soret parameter, S, two quite different regimes have been exhibited depending on the value of S. For moderate S, the induced solutal buoyancy balances the imposed thermal buoyancy, slowing down the flow and giving a good separation rate. For small S this balance does not exist (except in the centre), leading to a remixing of the species and thus to poor separation (the separation would be still worse for negative S). The smaller the (positive) Soret parameter is, the smaller the cell misorientation γ has to be to allow a good separation rate.

Granular flows such as landslides, debris flows and avalanches are systems of particles with a large range of particle sizes that typically segregate while flowing. The physical mechanisms responsible for this process, however, are still poorly understood, and there is no predictive framework for ascertaining the segregation behaviour of a given system of particles. Here, we provide experimental evidence of individual large intruder particles being attracted to a fixed point in a dry two-dimensional flow of particles of otherwise uniform size. A continuum theory is proposed which captures this effect using only a single fitting parameter that describes the rate of segregation, given knowledge of the bulk flow field. Predictions of the continuum theory are compared with the experimental findings, both for the typical location and velocity field of a range of intruder sizes. For large intruder particle sizes, the continuum model successfully predicts that a fixed point attractor will form, where intruders are drawn to a single location.

The geometric properties are quantified for concentration iso-surfaces of a high-Schmidt-number passive scalar field produced by an iso-kinetic source with an initial finite characteristic length scale released into the inertial layer of fully developed open-channel-flow turbulent boundary layers. The coverage dimension and other measures of two-dimensional transects of the passive scalar iso-surfaces are found to be scale dependent. The coverage dimension is around 1.0 at the order of the Batchelor length scale and based on our data increases in a universal manner to reach a local maximum at a length scale around the Kolmogorov scale. We introduce a new parameter called the coverage length underestimate, which demonstrates universal behaviour in the viscous–convective regime for these data and hence is a potentially useful practical tool for many mixing applications. At larger scales (in the inertial–convective regime), the fractal geometry measures are dependent on the Reynolds number, injection length scale, and concentration threshold of the iso-surfaces. Finally, the lacunarity of the iso-surface structure shows that the instantaneous scalar field is most inhomogenous around the length scale corresponding to the Kolmogorov scale.

The elastic effects of an insoluble surfactant on the formation and evolution of two-dimensional Faraday waves is investigated numerically. We analyse the influence of the elasticity of the surface-active agent on the amplitude of the vertical vibration needed to excite two-dimensional standing waves on the free surface. The numerical solutions show that the interface is always subharmonically excited at the onset and that the presence of the surfactant requires a higher external force to induce standing waves. They also show that the magnitude of the external amplitude is related to the temporal phase shift that exists between the evolution of the surfactant concentration and the free-surface shape. A detailed description of the time-varying velocity fields and interfacial distribution of surfactants helps to provide insight into the mechanisms ruling the phenomenon.

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode ‘mild-slope’ approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

The flow field produced when a strong shock wave propagates into a steady flow expansion has been investigated numerically, experimentally and analytically. The experiments were conducted with a shock tube which was modified to allow steady flow to be established in a hypersonic nozzle prior to arrival of the shock. It has been found that the axial density distribution associated with the prior steady flow allows the unsteady flow following the nozzle primary starting shock to accelerate from supersonic to hypersonic speeds, whereas a uniform density distribution causes it to decelerate to subsonic speeds. The prior steady flow also allows the starting shock system to propagate through the nozzle a t nearly the same velocity as the incident primary shock, and therefore provides a convenient method of ensuring rapid steady flow initiation in shock tunnel nozzles. The analysis shows that the flow behaviour can be understood in terms of two approximate models. The first is applicable to a wide range of flow conditions, and allows calculation of the trajectory of the centre of mass of the starting shock system. The second is applicable to cases involving a prior steady flow, and predicts detailed features of the flow structure.

A finite-difference Navier-Stokes model has been used to study rotating baroclinic flow for Richardson number [lsim ] 1, assuming no variations except in the vertical plane wholly containing the density-gradient vector. A section of a horizontally infinite channel has been studied, assuming periodic boundary conditions at the vertical computational boundaries and no-slip conducting horizontal boundaries. Two configurations were studied, both of which have an analytic basic solution with no horizontal variations in the velocities or density gradients. Symmetric baroclinic waves developed in the flows, as long as the Richardson number was not too large and the thermal Rossby number was large enough (for fixed diffusion parameters), consistent with linear theory. The structures and energetics of the fully developed waves were found to be especially dependent upon the Prandtl number Pr. Potential energy was the ultimate wave-energy source in all cases, and the average zonal flow was never much affected by the waves. For Pr > 1 the conversion from potential energy to wave kinetic energy was direct, via temperature and vertical-motion correlation. For Pr < 1 the conversion was from potential energy, to average kinetic energy by virtue of an induced meridional flow, to wave kinetic energy. For Pr = 1 the energy conversion was by either or both of the above, depending upon the other parameters.

The way in which energy propagates away from a two-dimensional oscillatory disturbance in a thermocline is considered theoretically and experimentally. It is shown how the St. Andrew's-cross-wave is modified by reflections and how the cross-wave can develop into thermocline waves. A linear shear flow is then superimposed on the thermocline. Ray theory is used to evaluate the wave shapes and these are compared to finite-difference solutions of the full Navier–Stokes equations.

Bilger, Saetran & Krishnamoorthy (1991) give measured values of the variance, cross-correlation coefficient, autospectra, coherence and phase shift of the reactant concentration fluctuations for an irreversible second-order reaction in an incompressible turbulent scalar mixing layer. The present paper approaches the interpretation of the measured data by evaluating the above quantities in the frozen (slow) and equilibrium (fast) chemistry limits. We assume that the limiting values bracket the corresponding intermediate rate data.

The analysis leads to values that correspond with the measured variances and correlation coefficients. The paper offers simple procedures for experimenters to evaluate the fast chemistry limit of the spectral characteristics from the measured mixture fraction fluctuations. The investigation of the limiting spectra suggests that, in the frequency region considered in the Bilger et al. measurements, the shape of the autospectrum is quite insensitive to the chemistry rate. The cross-spectrum is much more sensitive to the chemistry than the autospectrum. The analysis predicts correctly that the coherence decreases with increasing frequency while the phase stays equal to π until the decrease of the coherence leads to indeterminate phase results.

Thermal convection with a continuous finite bandwidth of modes in a porous layer with horizontal walls at different mean temperatures is considered when a spatially non-uniform temperature is prescribed at the lower wall. The nonlinear problem of three-dimensional convection for values of the Rayleigh number close to the classical critical value is solved by using multiple scales and perturbation techniques. The preferred flow solutions are determined by a stability analysis. It is found that for the case of near-resonant wavelength excitation regular or non-regular solutions in the form of superposition of small-scale multi-modal solutions with large-scale multimodal (or non-modal) amplitude can become preferred, provided the wave vectors of the solutions are contained in the set of wave vectors due to the modal form of the boundary imperfections and the form of the large-scale part is the same as that due to the boundary imperfections. For the case of non-resonant wavelength excitation some three-dimensional solutions in the form of superposition of small-scale multi-modal solutions with large-scale multi-modal (or non-modal) amplitudes can be preferred, provided that the wavelength of the small-scale modulation is not too small. Large-scale flow structures are quite different from the small-scale flow structures in a number of cases and, in particular, they can exhibit kinks and can be non-modal in nature. The resulting flow patterns are affected accordingly, and they can provide quite unusual and non-regular three-dimensional preferred patterns. In particular, they are multiples of irregular rectangular patterns, and they can be non-periodic.