Axisymmetric shapes and stability of nonlinearly polarizable dielectric (ferrofluid) drops of fixed volume which are pendant/sessile on one plate of a parallel-plate capacitor and are subjected to an applied electric (magnetic) field are determined by solving simultaneously the free boundary problem comprised of the Young-Laplace equation for drop shape and the Maxwell equations for electric (magnetic) field distribution. Motivated by the desire to explain certain experiments with ferrofluids, a constitutive relation often used to describe the variation of polarization with applied field strength is adopted here to close the set of equations that govern the distribution of electric field. Specifically, the nonlinear polarization, P, is described by a Langevin equation of the form P = α[coth (τE) −1/(τE)], where E is the electric field strength. As expected, the results show that nonlinearly polarizable drops behave similarly to linearly polarizable drops at low field strengths when drop deformations are small. However, it is demonstrated that at higher values of the field strength when drop deformations are substantial, nonlinearly polarizable supported drops whose contact lines are fixed, as well as ones whose contact angles are prescribed, display hysteresis in drop deformation over a wide range of values of the Langevin parameters α and τ. Indeed, properly accounting for the nonlinearity of the polarization improves the quantitative agreement between theory and the experiments of Bacri et al. (1982) and Bacri & Salin (1982, 1983). Detailed examination of the electric fields inside nonlinearly polarizable supported drops reveals that they are very non-uniform, in contrast to the nearly uniform fields usually found inside linearly polarizable drops.

A theory is developed to describe the conformation change of polymers in flow through dilute, random fixed beds of spheres or fibres. The method of averaged equations is used to analyse the effect of the stochastic velocity fluctuations on polymer conformation via an approach similar to that used in our previous analysis of particle orientation in flow through these beds (Shaqfeh & Koch 1988a, b). The polymers are treated as passive tracers, i.e. the polymeric stress in the fluid is neglected in calculating the stochastic flow field. Simple dumbbell models (either linear or FENE) are used to model the polymer conformation change. In all cases we find that the long-range interactions provide the largest contribution (in the limit of vanishingly small bed volume fraction) to an evolution equation for the probability density of conformation. These interactions create a conformation-dependent diffusivity in such an equation. Solutions for the second moment of the distribution demonstrate that there is a critical pore-size Deborah number beyond which the radius of gyration of a linear dumbbell will grow indefinitely and that of the FENE dumbbell will grow to a large fraction of its maximum extensibility. This behaviour is shown to be related to the development of ‘algebraic tails’ in the distribution function. The physical reasons for this critical condition are examined and its dependence on bed structure is analysed. These results are shown to be equivalent to those which we derive by the consideration of a polymer in a class of anisotropic Gaussian flow fields. Thus, our results are explicitly related to recent work regarding polymer stretch in model turbulent flows. Finally, the effect of close interactions and their modification of our previous results is discussed.

It is proposed that an array of Helmholtz resonators connected to a tunnel in its axial direction will suppress the propagation of sound generated by a travelling train and especially the emergence of shock waves in the far field. Under the approximation that the resonators may be regarded as continuously distributed, quasi-one-dimensional formulation is given for nonlinear acoustic waves by taking account of not only the resonators but also the wall friction due to the presence of a boundary layer and the diffusivity of sound. For a far-field propagation, the spatial evolution equation coupled with the equation for the response of the resonator is then derived. The linear dispersion relation suggests that the resonators, if appropriately designed, enhance the dissipation and give rise to the dispersion as well. By solving initial-value problems for the evolution equation, the array of resonators is proved to be very effective in suppressing shock waves in the far field. The resonators themselves fail to counteract shock waves once formed, but rather prevent their emergency by rendering acoustic waves dispersive. By this dispersion, it becomes possible, in a special case, for an acoustic soliton to be propagated in place of a shock wave.

This theoretical study describes how three-dimensional nonlinear distortion may soon take effect, following a small initial input disturbance that is nearly planar, in an otherwise two-dimensional boundary layer at high Reynolds number. The mechanism involved is a form of vortex-wave interaction, the first such to be examined in the so-called high-frequency range. The interaction is powerful, in that three-dimensional disturbances of relatively low amplitude (the wave part) interact nonlinearly with the three-dimensional corrections to the mean flow (the vortex part) at a stage where the purely two-dimensional case alone would still be linear. A coupled nonlinear partial-differential system is derived, governing the vortex and wave parts. Computations and analysis of the system are then presented. These point to a finite-time singularity arising in the solution, involving blow-up of both the vortex and the wave amplitudes (but particularly the former), accompanied by spanwise focusing into streets. This is believed to be the first nonlinear interaction in the high-frequency range to produce a finite-time (or-distance) blow-up. The blow-up is such that the local flow soon enters a strongly nonlinear three-dimensional stage in which the total mean flow is altered. The implications of this blow-up and focusing for one of the classic paths of boundary-layer transition are also discussed, and here quantitative and/or order-of-magnitude comparisons suggest that the theory is in line with the findings of Klebanoff & Tidstrom (1959) and later experiments.

The mean flow and turbulent statistics obtained from the numerical simulation of the fully developed turbulent flow through a straight duct of square cross-section are reported. The Reynolds number based on the bulk velocity and hydraulic diameter is 4410. Spatial and temporal approximations of the equations of motion were derived from standard finite-difference techniques. To achieve sufficient spatial resolution 16.1 × 106 grid nodes were employed. Turbulent statistics along the wall bisectors show good agreement with plane channel data despite the influence of the sidewalls in the former flow. The mean secondary flow field consists of two counter-rotating cells symmetrically placed about the corner bisectors with their common flow towards each corner with strong evidence for the existence of a smaller and much weaker pair situated about the wall bisectors. The mean streamwise vorticity of each corner cell is found to be associated with a stronger vorticity distribution of the opposite sign having an absolute maximum on the nearest duct wall.

A theoretical investigation of similarity solutions for interactive laminar boundary layers is presented. The questions of uniqueness and of the appearance of homogeneous eigensolutions are discussed. The similarity solutions yielding the asymptotic behaviour of the nonlinear triple-deck equations in the far field can be used either to improve the development of computational schemes or to check the accuracy of numerical results. A special similarity solution governed by a modified Falkner-Skan boundary-value problem determines the shape of a wall generating the largest possible deflection of a laminar boundary layer in supersonic flow if separation is to be avoided. Increasing the controlling parameter of this special pressure distribution (for both supersonic and subsonic flows) beyond a cutoff value leads to a global breakdown of the interacting laminar-boundary-layer approach which cannot be removed or avoided.

The paper is devoted to a theoretical analysis of nonlinear two-dimensional waves on the surface of a liquid film freely falling down a vertical plane. A bifurcation analysis of the wave regimes found in Part 1 of this work (Tsvelodub & Trifonov 1991), and of the new wave families obtained here in Part 2, has been carried out. It is demonstrated that there is a great number of different steady-state travelling wave classes which are parameterized by wavenumber at a fixed Reynolds number for a given liquid. It is shown that some of them quantitatively agree with experimental results. The question of stability of various wave regimes with respect to two-dimensional infinitesimal disturbances is examined and it is shown that one particular wave family is found. The most amplified disturbances are evaluated.

The evolution of small-amplitude finite-rate waves in fluids having high specific heats is studied adopting the assumption that the unperturbed state varies in the propagation direction. It is shown that this not only leads to quantitative changes of the results holding for homogeneous media but also gives rise to new phenomena. Most interesting, shocks are found to terminate at a finite distance from the origin if the fundamental derivative changes sign along the propagation path.

Consider a two-dimensional bubble moving with speed U through an unbounded, inviscid fluid. Let all lengths be normalized by T/ρU2 where T is the surface tension. Then the shape of the bubble depends on a single parameter Γ = 2Δp/ρU2 − 1, where Δp = pb − p∞ is the difference between the bubble pressure and the ambient pressure. We obtain solutions for the bubble shape over the whole range of Γ-values that are physically relevant. The formulation involves a mapping from an auxiliary circle plane ζ where the flow field is known. The problem then reduces to solving an infinite set of nonlinear algebraic equations for the coefficients in the mapping function.

To a first approximation, when Γ → ∞, the bubble takes an elliptical shape of aspect ratio $(1 + {\textstyle\frac{2}{3}}\Gamma^{-1})/(1 - {\textstyle\frac{2}{3}}\Gamma^{-1})$ flattened in the flow direction. The solution correct to order Γ−5 is then obtained which is fairly accurate for Γ as low as 2. When Γ = 0 the exact, nonlinear solution for the bubble shape is given by $x = \frac{1}{3}(\frac{1}{3}\cos\phi - \frac{1}{27}\cos 3\phi), y=\frac{1}{3}(\frac{5}{3}\sin\phi + \frac{1}{27}\sin 3\phi)$. We can then obtain a perturbation solution for Γ → 0 correct to order Γ6. This solution, useful in the range 0.75 > Γ > − 0.4537, even gives reasonable descriptions of non-convex bubble shapes for Γ < 0 down to the pinch-off limit Γ* when the bubble ceases to be simply connected. It is remarkable that a simple analytical representation correct to order Γ2 analytically yields a value for Γ* of − 0.4548, i.e. within 0.3% of the correct value; naturally, the higher-order approximations are even more accurate. While the present results eliminate the need for direct numerical computations over most of the range of Γ, such results, too, are presented. Finally, the dependence of the bubble geometrical parameters, Weber number and added mass on Γ is determined.

A liquid-metal flow impinging upon a region of non-uniform d.c. magnetic field experiences a certain amount of braking owing to the effect of Lorentz forces acting on the metal. Practical electromagnetic flow control devices utilize this property to alter the flow rate at which a liquid metal emerges from a receptacle. As a preliminary step to understanding the three-dimensional behaviour a numerical model is constructed which examines the two-dimensional flow of liquid metal passing through a quadrupole magnetic field generated by four line currents. In the vicinity of the local neutral point it is found that the nonlinear flow becomes unidirectional and linear. This linear behaviour agrees well with analytic solutions for flow through an infinitely extended neutral point. The generalized forms of the magnetic fields which permit unidirectional flows to exist are investigated in both axisymmetric and two-dimensional geometries. Examples of these fields include both the extended neutral point and the uniform transverse magnetic field present in Hartmann flow. The optimum conditions for braking the flow with a specified field are characterized by the pressure and volume data. These variables are derived from the model for a range of values of field strengths and Reynolds numbers and allow a comparison to be made with the asymptotic results obtained from the linear theory for two-dimensional flows. The numerical scheme may be adapted for any type of magnetic field and also permits extensions to the more realistic axisymmetric case.

A two-dimensional method was used to observe the interactions of plane shock waves with single cavities. This allowed study of processes occurring within the cavity during collapse. Results were obtained from high-speed framing photography. A variety of collapse shock pressures were launched into thin liquid sheets either by firing a rectangular projectile or by using an explosive plane-wave generator. The range of these shock pressures was from 0.3 to 3.5 GPa. Cavities were found to collapse asymmetrically to produce a high-speed liquid jet which was of approximately constant velocity at low shock pressures. At high pressures, the jet was found to accelerate and crossed the cavity faster than the collapse-shock traversed the same distance in the liquid. In the final moments of collapse, high temperatures were concentrated in two lobes of trapped gas and light emission was observed from these regions. Other cavity shapes were studied and in the case of cavities with flat rear walls, multiple jets were observed to form during the collapse.

The fully developed laminar flow in a helical circular pipe under the influence of both curvature and torsion is studied analytically. The solutions are obtained by the double series expansion method which perturbs the exact solution derived in this work for a twisted circular pipe. The perturbed parameters selected are dimensionless curvature k and dimensionless torsion τ. Since the expanded governing equations and series solutions have been arranged in a compact form, the complete solutions can be computed by a systematic procedure on computer. In addition, the accuracy of the solutions is only confined by the natural limitation of the series expansion method because no approximation was made in the governing equations. The ‘torsion number’ Tn which can be considered as the measure of the torsion effect that swirls the flow is defined Tn = 2τ[Rscr ], where [Rscr ] is the Reynolds number. The characteristics of the flow in the helical circular pipe are thus controlled by three parameters: [Rscr ], Dean number K and Tn. The flow rate solution of the extended Dean equations of Germano (1989) is then found. The effects of finite curvature and torsion on the flow rate, axial velocity and secondary flow are also found. The inconsistency of torsion effect on the secondary flow between Wang (1981) and Germano (1982, 1989) is also quantitatively explained by the different coordinate systems used.

Flow and bed topography in a regular sequence of meanders is shown to be strongly influenced by nonlinear effects within a fairly wide range of aspect ratios of the channel and meander wavenumbers. This finding is associated with the behaviour of meanders as nonlinear resonators in a neighbourhood of the resonance conditions discovered by Blondeaux & Seminara (1985). A weakly nonlinear approach valid for relatively small measures of channel curvature and within a neighbourhood of the resonant conditions displays all the typical features of nonlinear resonators, including non-uniqueness of the channel response. The nonlinear structure of forced bars close to resonance is also shown to be related to that of nonlinear free steady bars spatially developing in a straight channel from a non-uniform initial condition. Finally we show how to reconcile the intrinsic nonlinearity of the near-resonant channel response with traditional bend stability theories. Some comparison with a systematic set of experimental observations of Colombini, Tubino & Whiting (1990) provides qualitative support for the present theory but also suggests that strongly nonlinear effects may play a non-negligible role for fairly small values of channel curvature. The main implication of this work is the clear need to revisit the literature on the modelling of flow and bed topography in river meanders, which is mostly based on linear theories.

As shown in Foster & Smith (1989), at large flow force M, Long's self-similar vortex is in the form of a swirling ring-jet, whose axial velocity profile is of sech2 form. At azimuthal wavenumber n of comparable order to the axial wavenumber, linear helical modes of instability are essentially those of the Bickley jet varicose and sinuous modes. However, at small axial wavenumbers, the three-dimensionality of the vortex is important, and the instabilities depend heavily on the effects of the swirl. We explore here the effects of finite Reynolds number Re on these long-wave inertial modes. It is shown that, because the radial velocity scales with Re−1M, the non-parallelism of the flow is more important than the viscous terms in determining the finite-Re behaviour. The three-layer structure of the parallel-flow instability modes remains, but with a critical layer considerably modified by radial velocity. In investigating the critical range Re = O(M3), we find the following: for n > 1, the non-parallelism stabilizes the unstable inertial modes, leading to determination of neutral curves; for n < − 1, the non-parallel effects always destabilize the vortex to these helical modes. Determination of the unstable modes and neutral curves for the n > 1 case requires a computational scheme that accounts for the presence of viscosity. It turns out that the n < 1 (n > − 1) modes are prograde (retrograde) with respect to the rotation of the main vortex.

An analysis of the motion and deformation of red blood cells between two parallel flat plates is presented. The motion is driven by an imposed pressure gradient in the surrounding fluid. Mammalian red cells are highly flexible, but deform at constant volume because the contents of the cell are incompressible, and at nearly constant surface area because the membrane strongly resists dilatation. Consequently, a minimum spacing between the plates exists, below which passage of intact cells is not possible. We consider spacings slightly larger than this minimum. The shape of the cell in this case is a disk with a rounded edge. The flow of the surrounding fluid is described using lubrication theory. Under the approximation that the distance between the plates is small compared with the cell diameter, cell shapes, pressure distributions, membrane stresses and cell velocities are deduced as functions of geometrical parameters. It is found that the narrow gaps between the cell and the plate are not uniform in width, and that as a result, membrane shear stresses are generated which increase in proportion to flow velocity. This contrasts with axisymmetric configurations, in which membrane shear stress remains bounded as cell velocity increases. The variation of cell velocity with spacing of the plates is similar to that previously demonstrated for rigid disk-shaped particles of corresponding dimensions.

We consider the extensional flow and twist of a finite, slender, nearly straight, Newtonian viscous fibre when its ends are drawn apart at prescribed velocity. The initial cross-section of the fibre may be arbitrary and may vary gradually in the axial direction. We derive the leading-order equations for the fibre's free surface and its flow velocity from a regular perturbation expansion of the full Stokes’ flow problem in powers of the aspect ratio. In order to obtain these equations systematically, it is necessary to consider terms beyond the leading order in the perturbation expansion, because those obtainable from the leading-order terms give an indeterminate set of equations. Our results are a pair of well-known hyperbolic equations for the area and axial velocity, together with (i) straightness of the line of centres of mass of the cross-section and (ii) a new hyperbolic evolution equation for the twist of the cross-section. It is only through this hyperbolic equation that history effects are manifest.

An experimental and numerical study investigating the flow development and fully developed flows of an incompressible Newtonian fluid in a curved duct of square cross section with a curvature ratio of 15.1 is presented. Numerical simulations of flow development from a specified inlet profile were performed using a parabolized form of the steady three-dimensional Navier-Stokes equations. No symmetry conditions were imposed. In general there was good agreement between the numerical predictions of the developing axial velocity profiles and LDV measurements. In addition, for computational expediency, the two-dimensional solution structure was calculated by imposing fully developed conditions together with symmetry conditions along the horizontal duct centreline.

Laser-Doppler measurements of axial velocity and flow visualization at Dean number Dn = 125, 137 and 150, revealed a steady and symmetric two-vortex flow at Dn = 125, and a steady and symmetric four-vortex flow at both Dn = 137 and 150 (Dn = Re/(R/a)½, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension). Axial velocity measurements showed that the four-vortex flow at Dn = 150 developed to the solution predicted by the two-dimensional numerical simulation. However, the four-vortex flow at Dn = 137 was still developing when the flow had reached the end of the 240° axial length of the duct. A numerical investigation for Dean numbers in the range of 50 to 175 revealed that at the limit point of the two-cell to four-cell transition the development length appeared to be infinite, and thereafter decreased for increasing Dean numbers. The behaviour of decreasing development length of the four-vortex flow with increasing Dean number has not been reported previously.

Using a symmetrically positioned pin at θ = 5° to induce the four-cell flows, the two-dimensional solution structure for Dn [les ] 150 was experimentally observed for the first time. Experiments were consistent with the prediction by Winters (1987) that four-vortex flows are stable to symmetric perturbations, but unstable to asymmetric perturbations. Experimental and numerical investigations suggested that, when perturbed asymmetrically, the four-vortex flow might evolve to flows with sustained spatial oscillations farther downstream.

One of the most significant mechanisms of noise generation in prototype counter-rotation propeller systems is the unsteady interaction between the rear row and the wake and trailing tip vortices shed by the front row. A crucial part of predicting this noise is the determination of the resulting unsteady lift distribution on the rear row; since much of the interaction occurs in the vicinity of the rear-row blade tips, however, two-dimensional airfoil response theory cannot be applied exclusively, and some account of the presence of the blade tip must be taken. With this in mind, we solve the model problem of the unsteady interaction between a convected harmonic velocity gust and a quarter-plane, for the case of mean flow Mach number in the transonic range. The detailed lift distribution near the leading edge and corner is analysed, revealing the complicated nature of the lift singularity at the corner, and allowing the lift distribution throughout a narrow region along the leading edge to be determined. A closed-form expression for the practically important acoustically weighted lift is derived, which could easily be incorporated into existing noise prediction schemes in order to correct the rear-row blade response calculations for the presence of the blade tips. The radiation in this quarter-plane problem is also considered, and the field is seen to possess two components, one arising from the interaction between the gust and the semi-infinite leading edge, and the other from the interaction between the gust and the blade corner. The acoustic energy associated with this second term, corresponding to the conversion of vortical energy into sound by the corner, is considered in detail, and the directivity and parametric scaling determined. Our exact solution to this model problem is also used to assess the accuracy of a strip-theory approximation, which is seen to be accurate in this case over only a restricted range of observer positions.

The structure of a premixed flame front propagating in a region of two-dimensional turbulence is investigated using full numerical simulation including heat release, variable properties, and one-step Arrhenius chemistry. The influence of reactant Lewis number (Le = ratio of thermal to species diffusivity) is reported for Le = 0.8, Le = 1.0, and Le = 1.2 flames. Local flame behaviour is described by comparing the local instantaneous turbulent flame structure (local consumption rate of reactants) to the steady one-dimensional laminar flame structure for the same thermochemical parameters. Statistics of flame front strain rates and curvature are calculated and global quantities of interest in modelling (flame surface area, mean reactant consumption rate per unit area of flame, and turbulent flame speed) are reported. Principal findings are: that probability density functions (p.d.f.s) of flame curvature are nearly symmetric about a near-zero mean; that the flame tends to align preferentially with extensive tangential strain rates; that the local flame structure of the non-unity Lewis number flames correlates more strongly with local flame curvature than with tangential strain rate; that the mean consumption rate per unit area is relatively insensitive to curvature and is controlled by the mean tangential strain rate; and, that more flame area is generated for Le < 1 than for Le > 1. Implications of the results for flamelet models of turbulent premixed combustion are discussed.

In this paper we are concerned with the unsteady plane liquid motion due to the penetration of a blunt undeformable contour through the free surface. Initially the liquid is at rest, and the contour touches its free surface at a single point. At the initial stage of the process the liquid motion is described within the framework of the acoustic approximation. It is known that, just behind the shock front which is generated under the impact, the liquid motion does not depend on the presence of the free surface for all time. The pressure distribution and the velocities of liquid particles inside this region are calculated analytically for an arbitrary contour. It is shown that liquid motion close to the contact points just before the shock wave escapes onto the free surface is self-similar; the singularity of the pressure is analysed. The focusing of the shock wave generated by the impact of a body with a shallow depression in the front surface is discussed.