A composite hydrodynamic-diffusion model of the arterial wall is presented to describe the vesicular transport of relatively inert macromolecules across the inner endothelial lining of the larger arteries of humans and animals and their subsequent diffusion in the underlying tissue of the intima and media. This model is motivated by the highly specialized ultrastructure of the arterial wall observed in electron microscopic studies and the recent experimental measurements of the time-dependent uptake of labelled macromolecules in animal arteries under carefully controlled in vitro conditions. The proposed dynamic model for the vesicular transport across the endothelial cell layer considers the constrained Brownian diffusion of 700 Å vesicles subject to long-range hydrodynamic and short-range London-van der Waals force interactions with the plasmalemma membranes of the endothelial cell. Approximate solutions are developed for the motion and the steady-state vesicle density distribution near the plasmalemma and in the interior of the cell using boundary-layer-like methods. The model for the vesicular transport just described appears as a novel boundary condition in the basic diffusion model for the underlying tissue. The latter is treated as a two-phase medium comprised of an interstitial fluid continuum with a uniformly dispersed smooth muscle phase as first proposed by Hills (1968). This model for the underlying tissue assumes that the smooth muscle cells contribute insignificantly to the macromolecule diffusion across the arterial wall but act as the principal storage reservoir for the macromolecules for large diffusion times because of their large volume fraction. The dimensionless parameters that arise in the theoretical model are determined by comparing the solutions for the time-dependent total wall uptake with Fry's (1973) experimental data for canine carotid artery.

The linear stability of a thin liquid layer bounded from above by a free surface and from below by an oscillating plate is investigated for disturbances of arbitrary wavenumbers, a range of imposed frequencies and selective physical parameters. The imposed motion of the lower wall occurs in its own plane and is unidirectional and time-periodic. Long-wave instabilities occur only over certain bandwidths of the imposed frequency, as determined by a long-wavelength expansion. A fully numerical method based on Floquet theory is used to investigate solutions with arbitrary wavenumbers, and a new free-surface instability is found that has a finite preferred wavelength. This instability occurs continuously once the imposed frequency exceeds a certain threshold. The neutral curves of this new finite-wavelength instability appear significantly more complex than those for long waves. In a certain parameter regime, folds occur in the finite-wavelength stability limit, giving rise to isolated unstable regions. Only synchronous solutions are found, i.e. subharmonic solutions have not been detected. In Appendix A, we provide an argument for the non-existence of subharmonic solutions.

We study the motion which results when a fluid film is squeezed between two parallel plane surfaces in relative motion. Particular attention is given to the special case where one surface is fixed and the other is rapidly accelerated from a state of rest to a state of uniform motion. The analysis is based in part on linear theory and in substance on a finite-difference analysis of the full nonlinear equations of motion.

The mixed-Eulerian–Lagrangian method using high-order boundary elements, described in Xue et al. (2001) for the simulation of fully nonlinear three-dimensional wave–wave and wave–body interactions, is here extended and applied to the study of two nonlinear three-dimensional wave–body problems: (a) the development of bow waves on an advancing ship; and (b) the steep wave diffraction and nonlinear high-harmonic loads on a surface-piercing vertical cylinder. For (a), we obtain convergent steady-state bow wave profiles for a flared wedge, and the Wigley and Series 60 hulls. We compare our predictions with experimental measurements and find good agreement. It is shown that upstream influence, typically not accounted for in quasi-two-dimensional theory, plays an important role in bow wave prediction even for fine bows. For (b), the primary interest is in the higher-harmonic ‘ringing’ excitations observed and quantified in experiments. From simulations, we obtain fully nonlinear steady-state force histories on the cylinder in incident Stokes waves. Fourier analysis of such histories provides accurate predictions of harmonic loads for which excellent comparisons to experiments are obtained even at third order. This confirms that ‘ringing’ excitations are directly a result of nonlinear wave diffraction.

We study the shear-induced self-diffusion of both a liquid tracer and a tagged spherical particle along the directions perpendicular to the ambient flow in a dilute suspension of neutrally buoyant spheres undergoing a simple shearing motion in the absence of inertia and Brownian motion effects. The calculation of the liquid diffusivity requires the velocity of a fluid point under the influence of two spheres, which was determined via Lamb's series expansion; conversely, the calculation of the particle diffusivity involves the trajectories of three spheres, which were determined using far-field and near-field asymptotic expressions. The displacements of the liquid tracer and of the tagged sphere were then computed analytically when the spheres and the tracer are all far apart, and numerically for close encounters. After summing over all possible encounters, the leading terms of the lateral liquid diffusion coefficients, both within and normal to the plane of shear, were thereby found to be 0.12γac and 0.004γac, respectively, where γ is the applied shear rate, a the radius of the spheres and c their volume fraction. The analogous coefficients of the lateral particle diffusivity were found to be 0.11γac, and 0.005γac, respectively. Also, liquid and particle diffusivities in a monolayer, with the liquid tracer and all the particle centres lying on the same plane of shear, were found to be 0.067γyac, and 0.032γac, respectively, with c denoting the areal fraction occupied by the spheres on the plane.

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.

Experiments of Andereck et al. (1986) with corotating cylinders, show that Taylor-vortex flow (TVF) can bifurcate into one of the following cellular flows: wavy vortices (WV), twisted vortices (TW), wavy inflow boundaries (WIB), wavy outflow boundaries (WOB). We describe here the structure of these different flows, showing how they result from simple symmetry breaking. Moreover we consider the codimension-two situation where WIB and WOB interact, since this is an observed physical situation.

The method used in this paper is based on symmetry arguments. It differs notably from the Liapunov-Schmidt reduction used in particular by Golubitsky & Stewart (1986) on the same problem with counter-rotating cylinders. Here we take into account all the dynamics, instead of restricting the study to oscillating solutions. In addition to the standard oscillatory modes, we have a translational mode due to the indeterminacy of TVF under the shifts along the axis. We derive an amplitude-expansion procedure which allows the translational mode to depend on time. Our amplitude equations have nevertheless a simple structure because the oscillatory modes have a precise symmetry. They break, in general, the rotational invariance and they are either symmetric or antisymmetric with respect to the plane z = 0. Moreover, the most typical cases are when either of these modes has the same axial period as TVF or when their axial period is double this. This leads to four different cases which are shown to give WV, TW, WIB or WOB, all these flows being ‘rotating waves’, i.e. they are steady in a suitable rotating frame.

Finally we consider the interaction between WIB and WOB that occurs when, at the onset of instability, the two critical modes arise simultaneously. In this case we show in particular that there may exist a stable quasi-periodic flow bifurcating from WIB or WOB. The two main frequencies are those of underlying WIB and WOB, while there may exist a third frequency corresponding to a slow superposed travelling wave in the axial direction.

The method was used in the counter-rotating case for interacting non-axisymmetric modes (see Chossat et al. 1986). One of the original contributions here is not only to clarify the origin of all observed bifurcations from TVF, but also to handle the translational mode which may not stay small. This technique combined with centre-manifold and equivariance techniques may be helpful for many problems starting with orbits of solutions, such as the TVF considered here.

This paper continues an investigation of the effects of surface tension on steep water waves in deep water begun in Hogan (1979a). A Stokes-type expansion method is given which can be applied to most wavelengths. For capillary waves (2 cm or less) it is found that the surface of the highest wave encloses a bubble of air, as was found for pure capillary waves by Crapper (1957). For intermediate waves (20 cm) the wave profiles are similar to those of pure gravity waves and the wave properties increase monotonically. For gravity waves (200 cm) the wave properties all exhibit a maximum just short of the maximum wave height obtained by the method. The integral properties for all the waves are drawn and given in numerical form in the appendix.

The radial distribution function (RDF, a statistical measure of preferential concentration), and the relative velocity measured along the line-of-centres of two particles are the key statistical inputs to the collision kernel for finite-inertia particles suspended in a turbulent flow Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, p. 75). In this paper, we investigate the behaviour of these two-particle statistics using direct numerical simulation (DNS) of homogeneous isotropic turbulence. While it is known that the RDF for particles of any Stokes number (St) decreases with separation distance Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, p. 75), Reade & Collins (Phys. Fluids, vol. 12, 2000, p. 2530), Salazar et al. (J. Fluid Mech., vol. 600, 2008, p. 245), we observe that the peak in the RDF versus St curve shifts to higher St as we increase the separation distance. Here, St is defined as the ratio of the particle's viscous relaxation time to the Kolmogorov time-scale of the flow. Furthermore, as found in a previous study Wang, Wexler, & Zhou (J. Fluid Mech., vol. 415, 2000, p. 117), the variance of the radial relative velocity (wr) is found to increase monotonically with increasing separation distance and increasing Stokes number; however, we show for the first time that the parameteric variation of the skewness of wr with St and r/η is qualitatively similar to that of the RDF, and points to a connection between the two. We then apply low-pass filters (using three different filter scales) on the DNS velocity field in wavenumber space in order to produce ‘perfect’ large-eddy simulation (LES) velocity fields without any errors associated with subgrid-scale modelling. We present visual evidence of the effect of sharp-spectral filtering on the flow structure and the particle field. We calculate the particle statistics in the filtered velocity field and find that the RDF decreases with filtering at low St and increases with filtering at high St, similar to Fede & Simonin (Phys. Fluids, vol. 18, 2006, p. 045103). We also find that the variation of the RDF with St shifts towards higher St with filtering at all separation distances. The variance of wr is found to decrease with filtering for all St and separation distances, but the skewness of wr shows a non-monotonic response to filtering that is qualitatively similar to the RDF. We consider the variation of the RDF and moments of wr with filter scale and find that they are approximately linear in the inertial range. We demonstrate that a simple model consisting of a redefinition of the St based on the time-scale of the filtered velocity field cannot recover the unfiltered statistics. Our findings provide insight on the effect of subgrid-scale eddies on the RDF and wr, and establish the requirements of a LES model for inertial particles that can correctly predict clustering and collisional behaviour.

We examine a three-dimensional rivulet flowing down a vertical plane. There exists a basic state with fully developed, unidirectional flow and straight contact lines. In the absence of contact-angle hysteresis the slope of the contact angle versus contact-line speed relationship measures the mobility of these contact lines. The stability characteristics of flat rivulets subject to long wave disturbances are examined using lubrication theory. We find that kinematic-wave instabilities are predicted for wide rivulets or rivulets with rather immobile contact lines, while capillary break-up is predicted for narrower rivulets with mobile contact lines. We find for all cases that the expression for the growth rate depends weakly on slip between the liquid and solid near the contact line, but strongly on the shape of the rivulet and the mobility of the contact lines. We discuss in detail the mechanisms by which the contact lines affect the instabilities.

The weakly reactive state leading to the ignition of a steady laminar boundary-layer flow of a combustible mixture over a hot, isothermal, non-permeable, non-catalytic flat plate is studied both numerically and using matched asymptotic analysis in the realistic limit of large activation energy. It is shown that the flow consists of a locally similar diffusive-reactive region next to the plate and a non-similar diffusive-convective region outside it; that the analytic solution obtained reproduces the lower half of the S-shaped ignition-extinction response curve such that ignition is expected to occur when a suitably defined Damköhler number, which increases with the streamwise distance, reaches unity; and that at the point of ignition the heat transfer from the wall vanishes identically. An explicit expression for the minimum distance for ignition to occur is also derived.

Direct numerical simulations of statistically steady homogeneous isotropic turbulence in viscoelastic fluids described by the FENE-P model, such as those laden with polymers, are presented. It is shown that the strong depletion of the turbulence dissipation reported by previous authors does not necessarily imply a depletion of the nonlinear energy cascade. However, for large relaxation times, of the order of the eddy turnover time, the polymers remove more energy from the large scales than they can dissipate and transfer the excess energy back into the turbulent dissipative scales. This is effectively a polymer-induced kinetic energy cascade which competes with the nonlinear energy cascade of the turbulence leading to its depletion. It is also shown that the total energy flux to the small scales from both cascade mechanisms remains approximately the same fraction of the kinetic energy over the turnover time as the nonlinear energy cascade flux in Newtonian turbulence.

We present an experimental study of a buoyancy-driven, low-Reynolds-number (Re < 1) exchange flow of two Newtonian fluids in a vertical cylindrical pipe (length 1 m and diameter 38.4 mm) connecting two fluid reservoirs. The denser, more viscous fluid was golden syrup and the less dense, less viscous fluid was a golden syrup–water solution; the ratio of the viscosities of the two fluids (β) ranged from 2 to 1180. Flows were initiated by removing a bung in the base of the upper reservoir or sliding out a gate positioned at the top, middle or bottom of the pipe. We observe the flows over long time durations (up to 356 h), and define the development of the flow with reference to a non-dimensional time (τ). The initial transient development of the flow was dependent on which of the two fluids initially filled the pipe, but this did not systematically affect the flow regime observed at τ ≫ 1. Two distinct flow regimes were observed: axisymmetric core-annular flow (CAF), in which the less viscous fluid occupies a cylindrical core and the denser fluid flows downwards in an annulus, and side-by-side (SBS) flow where both fluids are in contact with the pipe and there is a single interface between them. CAF formed at β ≥ 75 and SBS flow at β ≤ 117. In several experiments, for 5 ≤ β ≤ 59, a slowly developing transitional SBS (TSBS) flow was observed where SBS flow and CAF occurred simultaneously with SBS in the lower portion of the pipe; SBS existed throughout most of the pipe and in one case grew with time to entirely fill the pipe. Velocity profiles determined by tracking tracer particles show that the observed CAFs are adequately described by the formulation of Huppert & Hallworth (J. Fluid Mech., vol. 578, 2007, pp. 95–112). Experimental SBS velocity profiles are not well produced by the formulation of Kerswell (J. Fluid Mech., 10.1017/jfm.2011.190), possibly because the latter is restricted to flows whose cross-section has an interface of constant curvature. Despite the variations in flow regime, volume fluxes can be described by a power-law function of β, Q1 = 0.059 β−0.74. A comparison of experimental data with the theoretical approaches of Huppert Hallworth (2007) and Kerswell (2011) indicates that fluids are not arranged in the regime that maximises volume flux (e.g. SBS or CAF), nor do they adopt the geometry that maximises volume flux within that particular regime.

A system of shallow water equations and a bed evolution equation are used to examine the evolution of perturbations on an erodible, initially plane beach subject to normal wave incidence. Both a permeable (under Darcy's law) and an impermeable beach are considered. It is found that alongshore-periodic morphological features reminiscent of swash beach cusps form after a number of incident wave periods on both beaches. On the permeable (impermeable) beach these patterns are accretional (erosional). In both cases flow is ‘horn divergent’. Spacings of the cusps are consistent with observations, and are close to those provided by a standing synchronous linear edge wave. An analysis of the processes leading to bed change is presented. Two physical mechanisms are identified: concentration gradient and flow divergence, which are dominant in the lower and upper swash respectively, and their difference over a wave cycle leads to erosion or deposition on an impermeable beach. Infiltration enters this balance in the upper swash. A bed wave of elevation is shown to advance up the beach at the tip of the uprush, with a smaller wave of depression on the backwash. It is found that cusp horns can grow by a positive feedback mechanism stemming from decreased (increased) backwash on positive (negative) bed perturbations.

Free convection in liquid metals is of significant practical interest to the crystal-growing community since the adverse effects of convective instabilities in the melt phase can be frozen into the solid product. Here, we present the results of a combined numerical and experimental study of steady convective flows in a sample of liquid gallium which is heated at one end and cooled at the other. Experimental measurements of temperature distributions in the flow are compared with the standard Hadley-cell solution and with the numerical results obtained from a two-dimensional model. Excellent quantitative agreement is found between all three for low Grashof numbers but a systematic divergence between the results is seen as this parameter is increased.

In many polymer processing applications, the generation of heat by viscous losses in the flowing molten polymer is highly significant. The heating reduces the viscosity of the melt sharply and the flow patterns are different from the isothermal case. In this paper, a finite element scheme based on the Galerkin method is developed and is used to explore the effects of thermally induced property changes in extrusion.

In the program we solve simultaneously for the flow and temperature fields at each iteration. To check the program for accuracy and correctness, some simple problems were first attempted. A solution for viscous heating in Poiseuille flow was used to check the variable-viscosity part of the program. The crucial convection (‘radiation’) boundary condition was checked using the solution for cooling of a moving rod. Finally, the swelling of extruded jets with self-heating was investigated. A new phenomenon, thermal extrudate swell, was thereby discovered. We have found extrudate expansions up to 70% of the die diameter in a Newtonian fluid with thermal properties similar to those of low density polyethylene. It is clear that this phenomenon will affect many experimental interpretations of extrudate swelling.

When homogeneous turbulence acts upon a non-uniform magnetic field B0(x, t), a mean electromotive force [Escr ](x, t) is in general established owing to the correlation between the velocity field and the fluctuating magnetic field that is generated. The relation between [Escr ] and B0 is linear, and it is generally believed that, when the scale of inhomogeneity of B0 is sufficiently large, it can be expressed as a series involving successively higher spatial derivatives of B0: \[ {\cal E}_i = \alpha_{il} B_{0l}+\beta_{ilm}\partial B_{0l}/\partial x_m+\ldots, \] where the tensors αil, βilm,… are determined (in principle) by the statistical properties of the turbulence and the magnetic diffusivity Λ of the fluid. These tensors are of crucial importance in turbulent dynamo theory. In this paper the question of their asymptotic form in the limit Λ → 0 is considered. By putting Λ = 0 and assuming that the field is non-random at an initial instant t = 0, the developing form of the tensors αl(t) and βm(t) is determined. The expressions involve time integrals of Lagrangian correlation functions associated with the velocity field, which are comparable in structure with the eddy diffusion tensor in the analogous turbulent diffusion problem (Taylor 1921). Some doubts are expressed concerning the convergence of the time integrals as t → ∞, and it is concluded that a satisfactory treatment of the problem will require the inclusion of weak diffusion effects (as recognized by Parker 1955).

The variances of the fluid-particle acceleration and of the pressure-gradient and viscous force are given. The scaling parameters for these variances are velocity statistics measureable with a single-wire anemometer. For both high and low Reynolds numbers, asymptotic scaling formulas are given; these agree quantitatively with DNS data. Thus, the scaling can be presumed known for all Reynolds numbers. Fluid-particle acceleration variance does not obey K41 scaling at any Reynolds number; this is consistent with recent experimental data. The non-dimensional pressure-gradient variance named λT/λP is shown to be obsolete.