The propagation of a two-dimensional weakly nonlinear wavefront into a polytropic gas in a uniform state and at rest has been studied. Successive positions of the wavefront and the distribution of amplitude on it are obtained by solving a system of conservation forms of the equations of weakly nonlinear ray theory (WNLRT) using a TVB scheme based on the Lax–Friedrichs flux. The predictions of the WNLRT are found to be qualitatively quite different from the predictions of the linear theory. The linear wavefronts leading to the formation of caustics are replaced by nonlinear wavefronts with kinks. By varying the initial shape of the wavefront and the amplitude distribution on it, the formation and separation of kinks on the wavefront has been studied.

General expressions are derived for the load distribution acting on an arbitrary curved and twisted rigid or deformable slender cylindrical structure moving in an ambient non-uniform potential flow field. Further simplifications are presented for flexible shapes in the limit of a small cross-section. The general analysis is illustrated for straight, toroidal and helical shapes. These shapes are frequently encountered in nature and are good examples of typical fluid–structure interaction problems.

A lattice-Boltzmann method has been developed to simulate suspensions of both spherical and non-spherical particles in finite-Reynolds-number flows. The results for sedimentation of a single elliptical particle are shown to be in excellent agreement with the results of Huang, Hu & Joseph (1998) who used a finite-element method. Sedimentation of two-dimensional circular and rectangular particles in a two-dimensional channel and three-dimensional spherical particles in a tube with square cross-section is simulated. Computational results are consistent with experimentally observed phenomena, such as drafting, kissing and tumbling.

Rayleigh's (1926, Appendix A) method for the approximate calculation of potential flow from the open end of a semi-infinite flanged cylinder is applied to obtain analytical representations of Green's function describing the generation of sound waves within a flanged cylinder by sources located in the neighbourhood of the open end. Detailed results are given for the circular cylinder considered by Rayleigh and extension made to a flanged cylinder of rectangular cross-section. The validity of various approximations is assessed by comparison with the exact solution available (by the method of conformal transformation) for potential flow from a two-dimensional, flanged duct. The results are used to compare the profiles and the pressure gradients of compression waves generated when a high-speed train enters tunnels of circular and rectangular cross-sections.

A drop in an axisymmetric extensional ow is studied using boundary integral methods to understand the effects of a monolayer-forming surfactant on a strongly deforming interface. Surfactants occupy area, so there is an upper bound to the surface concentration that can be adsorbed in a monolayer, Γ∞. The surface tension is a highly nonlinear function of the surface concentration Γ because of this upper bound. As a result, the mechanical response of the system varies strongly with Γ for realistic material parameters. In this work, an insoluble surfactant is considered in the limit where the drop and external fluid viscosities are equal.

For Γ<Γ∞, surface convection sweeps surfactant toward the drop poles. When surface diffusion is negligible, once the stable drop shapes are attained, the interface can be divided into stagnant caps near the drop poles, where Γ is non-zero, and tangentially mobile regions near the drop equator, where the surface concentration is zero. This result is general for any axisymmetric fluid particle. For Γ near Γ∞, the stresses resisting accumulation are large in order to prevent the local concentration from reaching the upper bound. As a result, the surface is highly stressed tangentially while Γ departs only slightly from a uniform distribution. For this case, Γ is never zero, so the tangential surface velocity is zero for the steady drop shape.

This observation that Γ dilutes nearly uniformly for high surface concentrations is used to derive a simplified form for the surface mass balance that applies in the limit of high surface concentration. The balance requires that the tangential flux should balance the local dilatation in order that the surface concentration profile will remain spatially uniform. Throughout the drop evolution, this equation yields results in agreement with the full solution for moderate deformations, and underscores the dominant mechanism at high deformation. The simplified balance reduces to the stagnant interface condition at steady state.

Drop deformations vary non-monotonically with concentration; for Γ<Γ∞, the reduction of the surface tension near the poles leads to higher deformations than the clean interface case. For Γ near Γ∞, however, Γ dilutes nearly uniformly, resulting in higher mean surface tensions and smaller deformations. The drop contribution to the volume averaged stress tensor is also calculated and shown to vary non-monotonically with surface concentration.

We apply a new kind of analytic technique, namely the homotopy analysis method (HAM), to give an explicit, totally analytic, uniformly valid solution of the two-dimensional laminar viscous flow over a semi-infinite flat plate governed by f‴(η)+αf(η)f″(η)+β[1−f′2(η)]=0 under the boundary conditions f(0)=f′(0)=0, f′(+∞)=1. This analytic solution is uniformly valid in the whole region 0[les ]η<+∞. For Blasius' (1908) flow (α=1/2, β=0), this solution converges to Howarth's (1938) numerical result and gives a purely analytic value f″(0)=0.332057. For the Falkner–Skan (1931) flow (α=1), it gives the same family of solutions as Hartree's (1937) numerical results and a related analytic formula for f″(0) when 2[ges ]β[ges ]0. Also, this analytic solution proves that when −0.1988[les ]β0 Hartree's (1937) family of solutions indeed possess the property that f′→1 exponentially as η→+∞. This verifies the validity of the homotopy analysis method and shows the potential possibility of applying it to some unsolved viscous flow problems in fluid mechanics.

This paper describes the flow-induced retention of charge stabilized colloidal particles during flow through cylindrical pores. Current models describing the low-Reynolds-number flow behaviour of particulate suspensions through porous media do not predict retention of stable colloidal particles if the particles are smaller in size than the pores, and the particles and the pores have like surface charges. Retention is not expected under these conditions because the small particle size relative to the pore constriction size precludes straining (physical capture of particles larger than the pore constriction) while particle–pore surface electrostatic repulsion prevents deposition. However, the experiments show that substantial particle retention can occur under these conditions. The mechanism causing particle retention under these conditions, hydrodynamic bridging, is flow-induced. In this mechanism, hydrodynamic forces acting on particles arriving at a pore entrance do not allow their simultaneous passage through the pore, resulting in the formation of a particle bridge across the pore constriction. This paper reports experiments elucidating the effects of velocity, particle concentration, and the ratio of pore size to particle size on retention by hydrodynamic bridging. For flow through cylindrical pores, the effect of velocity on retention by bridging is opposite to that of retention by deposition. Furthermore, observations indicate the existence of a critical flow velocity necessary for particle bridging to occur. This critical velocity is a measure of the net colloidal interparticle and particle–porous medium repulsion that must be overcome by the hydrodynamic forces for bridging to occur. Approximate theoretical calculations of the trajectories of two particles approaching an isolated cylindrical pore are also presented. These calculations show that bridging is indeed possible in the Stokes flow regime for the experimental conditions considered.

This paper is concerned with the coupling mechanisms leading to the spontaneous generation of sound during flame propagation in a tube open at one end. We consider the cases of premixed gaseous combustion and of premixed spray combustion of decane droplets in air. The flame front propagates from the open to the closed end of a tube and, for a particular position, starts to amplify a longitudinal acoustic mode of the tube. We call this mode the primary acoustic instability and focus our study on the physical mechanisms responsible for sound amplification. Measured amplification rates are compared to calculated values. In the gaseous case, it is shown that the instability results from a coupling between the acoustic acceleration field and the geometry of the flame front separating the burnt gases from the denser unburnt mixture. The situation is quite different in the spray case. The primary acoustic instability is much stronger and results from a modification of the inner structure of the flame. This modification arises from the velocity lag of the droplets in the acoustic velocity field, leading to a modulation of the fuel flux at the flame.

Direct numerical solutions of the incompressible Navier–Stokes equations have been obtained under the Boussinesq approximation for the temporal evolution of a turbulent jet-like flow subjected to off-source volumetric heating, of the kind that occurs in a cloud due to latent heat release on condensation of water vapour. The results show good qualitative agreement with available experimental data on spatially growing jets. Thus, heating accelerates the flow and arrests jet growth; and turbulence velocities increase with heating but not as rapidly as mean velocities, so normalized intensities drop. It is shown that the baroclinic torque resulting from the heating enhances the vorticity dramatically in all three directions, with a preferential amplification at the higher wavenumbers that results in a rich fine structure at later times in the evolution of the jet. Streamwise vortex pairs, rendered stronger by mean flow acceleration, appear to be responsible for large expulsive motions at certain transverse cross-sections in the ambient fluid near the heated flow; together with the disruption of the toroidal component of the coherent vorticity achieved by heating, this results in an entraining velocity field that is qualitatively different from that around unheated turbulent jets. This mechanism may provide a plausible explanation for the experimentally observed drop in entrainment with off-source heating.

Single drop impact onto liquid films is simulated numerically. Surface tension and gravity are taken into account, whereas viscosity and compressibility are neglected. This permits recourse to a boundary-integral method, based on an integral equation for a scalar velocity potential. Calculations are performed for normal impacts resulting in axisymmetric flows.

For times that are small compared to the characteristic time of impact 2R/w0 (R being the drop radius, w0 its initial velocity towards the liquid film), it is found that a disk-like jet forms at the neck between the drop and the pre-existing liquid film, if the impact Weber number is high enough. This jet can pinch off a torus-shaped liquid volume at its tip or reconnect with the pre-existing liquid film, thus entraining a torus- shaped bubble. In reality, both the torus-shaped bubble and liquid torus will decay according to Rayleigh's capillary instability, thus breaking the cylindrical symmetry. This mechanism of bubble entrainment differs from those described in literature.

For times that are comparable to or larger than the characteristic time of impact, capillary waves on the film, or the well-known crowns, are obtained again according to whether the impact Weber number is low or high enough.

In this paper we describe a detailed study of the wake structures and flow dynamics associated with simulated two-dimensional flows past a circular cylinder that is either stationary or in simple harmonic cross-flow oscillation. Results are examined for Re=500 and a fixed motion amplitude of ymax/D=0.25. The study concentrates on a domain of oscillation frequencies near the natural shedding frequency of the fixed cylinder. In addition to the change in phase of vortex shedding with respect to cylinder motion observed in previous experimental studies, we note a central band of frequencies for which the wake exhibits long-time-scale relaxation oscillator behaviour. Time-periodic states with asymmetric wake structures and non-zero mean lift were also observed for oscillation frequencies near the lower edge of the relaxation oscillator band. In this regime we compute a number of bifurcations between different wake configurations and show that the flow state is not a unique function of the oscillation frequency. Results are interpreted using an analysis of vorticity generation and transport in the base region of the cylinder. We suggest that the dynamics of the change in phase of shedding arise from a competition between two different mechanisms of vorticity production.

Confined vortex breakdown generated by a rotating cone within a closed cylindrical container has been studied both by numerical simulation and by experimental techniques. A comprehensive investigation of the various flow regimes has been carried out by flow visualization. From laser–Doppler measurements of the entire flow field (three velocity components) detailed maps of the time-averaged flow structures for single and double breakdown have been constructed. Three-dimensional time-dependent simulations of steady and unsteady breakdown have been performed. Steady numerical and experimental flow fields obtained at Reynolds number 2200 for a gap ratio of 2 show notable agreement. At critical Reynolds numbers of approximately 3095, for a gap ratio of 2, and 2435, for a gap ratio of 3, the flow was observed becoming unsteady. The periodic behaviour exhibited by the unsteady flow suggested the occurrence of a supercritical Hopf bifurcation. This conjecture was confirmed by the evolution of the oscillation amplitude as a function of criticality, measured for a gap ratio of 3. The dynamical behaviour of unsteady vortex breakdown structures is depicted by numerical simulation of two distinct oscillatory regimes, at Reynolds numbers 2700 and 3100. A thorough analysis of the numerical results has shown that whereas the former regime is characterized by the steady oscillation of closely axisymmetric breakdowns, the latter displays precession of breakdown structures about the central axis. Additionally, it was observed that the mode bringing about the Hopf bifurcation is non-axisymmetric, with azimuthal periodicity of π/2 radians. From examination of measured velocity power spectra at higher Reynolds numbers, a transition scenario was also educed. In the present case, the Ruelle–Takens–Newhouse theorem has been shown to apply.

Direct numerical simulations of the motion of two- and three-dimensional finite Reynolds number buoyant bubbles in a periodic domain are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The rise Reynolds numbers are around 20–30 for the lowest volume fraction, but decrease as the volume fraction is increased. The rise of a regular array of bubbles, where the relative positions of the bubbles are fixed, is compared with the evolution of a freely evolving array. Generally, the freely evolving array rises slower than the regular one, in contrast to what has been found earlier for low Reynolds number arrays. The structure of the bubble distribution is examined and it is found that while the three-dimensional bubbles show a tendency to line up horizontally, the two-dimensional bubbles are nearly randomly distributed. The effect of the number of bubbles in each period is examined for the two-dimensional system and it is found that although the rise Reynolds number is nearly independent of the number of bubbles, the velocity fluctuations in the liquid (the Reynolds stresses) increase with the size of the system. While some aspects of the fully three-dimensional flows, such as the reduction in the rise velocity, are predicted by results for two-dimensional bubbles, the structure of the bubble distribution is not. The magnitude of the Reynolds stresses is also greatly over-predicted by the two-dimensional results.

The small grains in a bidisperse porous medium have the greater influence on the permeability, while the large grains are more effective in dispersing chemical tracers. We compute the dispersion induced by a dilute array of large spheres in a Brinkman medium whose permeability is determined by the radii and volume fraction of the small spheres. The effective diffusivity contains a purely hydrodynamic contribution proportional to Ua1ϕ1 and an O(Ua1ϕ1 ln (Ua1/D)) contribution from the mass transfer boundary layers near the spheres. Here, U is the mean velocity in the medium, a1 and ϕ1 are the radii and volume fraction of the large spheres and D is the molecular diffusivity. The boundary-layer dispersion is small when the Brinkman screening length κ (or square root of permeability) is much smaller than a1, but is important for κ[ges ]O(a1). Experimental results for the dispersion due to flow through a bidisperse packed bed are reported and compared with the theoretical predictions. In addition to its application to bidisperse porous media, the present calculation allows an extension of Koch & Brady's (1985) analysis of monodisperse fixed beds to include higher-order terms in the expansion for small particle volume fraction.