We use lubrication theory and matched asymptotic expansions to model the quasi-steady propagation of a liquid plug or bolus through an elastic tube. In the limit of small capillary number, asymptotic expressions are found for the pressure drop across the bolus and the thickness of the liquid film left behind, as functions of the capillary number, the thickness of the liquid lining ahead of the bolus and the elastic characteristics of the tube wall. These results generalize the well-known theory for the low capillary number motion of a bubble through a rigid tube (Bretherton 1961). As in that theory, both the pressure drop across the bolus and the thickness of the film it leaves behind vary like the two-thirds power of the capillary number. In our generalized theory, the coefficients in the power laws depend on the elastic properties of the tube.

For a given thickness of the liquid lining ahead of the bolus, we identify a critical imposed pressure drop above which the bolus will eventually rupture, and hence the tube will reopen. We find that generically a tube with smaller hoop tension or smaller longitudinal tension is easier to reopen. This flow regime is fundamental to reopening of pulmonary airways, which may become plugged through disease or by instilled/aspirated fluids.

In this paper, the transient settling dynamics of a spherical particle sedimenting in a linearly stratified fluid is investigated by performing fully resolved direct numerical simulations. The settling behaviour is quantified for different values of Reynolds, Froude and Prandtl numbers. It is demonstrated that the transient settling dynamics is correlated to the induced Lagrangian drift of flow around the settling particle. A simplified model is provided to predict the maximum velocity of the settling particle in linearly stratified fluids. The peak velocity can be followed by the oscillation of the settling velocity and the particle can even reverse its direction of motion before reaching to its neutrally buoyant level. The frequency of oscillation of settling velocity scales with the Brunt–Väisälä frequency and the motion of the particle can lead to the formation of secondary and tertiary vortices following the primary vortex.

The adjustment under gravity of a fluid containing a horizontal density gradient is described.’ The fluid is initially at rest and the resulting motion is calculated as the flow accelerates, driven by the baroclinic density field. Two forms of the initial density distribution are considered. In the first the initial horizontal gradient is constant. A purely horizontal motion develops as the isopycnals rotate towards the horizontal. The vertical density gradient increases continually with time but the horizontal density gradient remains unchanged. The horizontal velocity has a uniform vertical shear, and the gradient Richardson number is constant in space and decreases monotonically with time to ½. The second density distribution consists of a piecewise constant gradient with a jump in the gradient along a vertical isopycnal. The density is continuous. In this case frontogenesis is predicted to occur on the isopycnal between the two constant-density-gradient regions, and the timescale for the formation of a front is determined. Laboratory experiments are reported which confirm the results of these calculations. In addition, lock exchange experiments have been carried out in which the horizontal mean gradient is represented by a series of step density differences separated by vertical gates.

The yield conditions for the displacement of three-dimensional fluid droplets from solid boundaries are studied through a series of numerical computations. The study considers low-Reynolds-number shear flows over plane boundaries and includes interfacial forces with constant surface tension. A comprehensive study is conducted, covering a wide range of viscosity ratio γ, capillary number Ca and advancing and receding contact angles, θA and θR. This study seeks the optimal shape of the contact line which yields the maximum flow rate (or Ca) for which a droplet can adhere to the surface. The critical shear rates are presented as functions Ca(γ, θA, Δθ) where Δθ=θA−θR is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on solid surfaces. Additional constraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While good agreement is found in the joint asymptotic limits Δθ[Lt ]θA[Lt ]1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets.

In many applications it is advantageous to construct effective slip boundary conditions, which could fully characterize flow over patterned surfaces. Here we focus on laminar shear flows over smooth anisotropic surfaces with arbitrary scalar slip , varying in only one direction. We derive general expressions for eigenvalues of the effective slip-length tensor, and show that the transverse component is equal to half of the longitudinal one, with a two times larger local slip, . A remarkable corollary of this relation is that the flow along any direction of the one-dimensional surface can be easily determined, once the longitudinal component of the effective slip tensor is found from the known spatially non-uniform scalar slip.

Fully developed air-flows through an equilateral triangular duct of 12·7 cm sides were investigated over a Reynolds number range of 53 000 to 107 000. Based on equivalent hydraulic diameter, friction factors were found to be about 6% lower than for pipe flow. Mean axial velocity distributions near the wall were describable by the inner law of the wall (when based on local wall shear stress) but the constants differ slightly from those for pipe flow. As expected, the secondary flow pattern was found to consist of six counter-rotating cells bounded by the corner bisectors. Maximum secondary velocities of about 1 ½% of the bulk velocity were observed. The effects of secondary currents were evident in the cross-sectional distributions of mean axial velocity, wall shear stress and Reynolds stresses, and very prominent in the turbulent kinetic energy distribution. For the flow prediction, the vorticity production terms were expressed by modelling the Reynolds stresses in the plane of the cross-section in terms of gradients in the mean axial velocity and a geometrically calculated turbulence length scale. The experimental and predicted characteristics of the flow are shown to be in good agreement.

Large-eddy and interface simulation using an interface tracking-based multi-fluid flow solver is conducted to investigate the breaking of steep water waves on a beach of constant bed slope. The present investigation focuses mainly on the ‘weak plunger’ breaking wave type and provides a detailed analysis of the two-way interaction between the mean fluid flow and the sub-modal motions, encompassing wave dynamics and turbulence. The flow is analysed from two points of views: mean to sub-modal exchange, and wave to turbulence interaction within the sub-modal range. Wave growth and propagation are due to energy transfer from the mean flow to the waves, and transport of mean momentum by these waves. The vigorous downwelling–upwelling patterns developing at the head and tail of each breaker are shown to generate both negative- and positive-signed energy exchange contributions in the thin sublayer underneath the water surface. The details of these exchange mechanisms are thoroughly discussed in this paper, together with the interplay between three-dimensional small-scale breaking associated with turbulence and the dominant two-dimensional wave motion. A conditional zonal analysis is proposed for the first time to understand the transient mechanisms of turbulent kinetic energy production, decay, diffusion and transport and their dependence and/or impact on surface wrinkling over the entire breaking process. The simulations provide a thorough picture of air–liquid coherent structures that develop over the breaking process, and link them to the transient mechanisms responsible for their local incidence.

We treat the onset of convective motions for the case in which the base-state density profile is evolving in time. The formulation is in terms of random forcing which we take to be thermodynamic in origin, following our earlier work (see Jhaveri & Homsy 1980). Experimental evidence is reviewed which clearly demonstrates the need for such a stochastic formulation. The randomly forced initial-value problem is solved numerically at high Rayleigh numbers in the mean-field approximation for both a step change and linear temporal increase in surface temperature. The numerical results give both an expected value for the onset time for which convection is measurable and the variance of that expected value. The results are in good agreement with available experiments.

A rigid cylinder is initially at relative rest in a uniformly rotating, inviscid, incompressible fluid, with its generators perpendicular to the axis of rotation. The fluid is accelerated suddenly to a small constant velocity parallel to the axis of rotation, which is maintained thereafter. The growth of the subsequent disturbance due to the cylinder is interpreted in terms of plane inertial waves, the disturbance energy propagating with the local group velocity, which is in the plane of the wave front and proportional to the wavelength. Taylor columns, in which the fluid moves with the cylinder rather than round it, grow indefinitely in both directions parallel to the rotation axis, the head of the column moving with finite speed.

In a slightly viscous fluid, an ultimate steady state is reached, in which the columns are of finite length.

If a cylinder is moved horizontally in a non-rotating uniformly stratified Boussinesq liquid, an identical analysis may be applied with a similar interpretation in terms of internal gravity waves rather than inertial waves.

The conservative evolution of weakly nonlinear narrow-banded gravity waves in deep water is investigated numerically with a modified nonlinear Schrödinger equation, for application to wide wave tanks. When the evolution is constrained to two dimensions, no permanent shift of the peak of the spectrum is observed. In three dimensions, allowing for oblique sideband perturbations, the peak of the spectrum is permanently downshifted. Dissipation or wave breaking may therefore not be necessary to produce a permanent downshift. The emergence of a standing wave across the tank is also predicted.

This note corrects and augments the results of Blumen, Drazin & Billings (1975) on the linear instability of a shear layer between two streams of an inviscid compressible fluid at the same temperature. The main conclusion of these authors, that the shear layer is unstable to two-dimensional disturbances however large the Mach number, is confirmed in the present paper. Also a third mode of instability is discovered and the character of all three modes revealed by a series of accurate calculations.

A study on the generation and development of high-amplitude steady streamwise streaks in a flat-plate boundary layer is presented. High-amplitude streamwise streaks are naturally present in many bypass transition scenarios, where they play a fundamental role in the breakdown to turbulence process. On the other hand, recent experiments and numerical simulations have shown that stable laminar streamwise streaks of alternating low and high speed are also capable of stabilizing the growth of Tollmien–Schlichting waves as well as localized disturbances and to delay transition. The larger the streak amplitude is, for a prescribed spanwise periodicity of the streaks, the stronger is the stabilizing mechanism. Previous experiments have shown that streaks of amplitudes up to 12 % of the free stream velocity can be generated by means of cylindrical roughness elements. Here we explore the possibility of generating streaks of much larger amplitude by using a row of miniature vortex generators (MVGs) similar to those used in the past to delay or even prevent boundary layer separation. In particular, we present a boundary layer experiment where streak amplitudes exceeding 30 % have been produced without having any secondary instability acting on them. Furthermore, the associated drag with the streaky base flow is quantified, and it is demonstrated that the streaks can be reinforced by placing a second array of MVGs downstream of the first one. In this way it is possible to make the control more persistent in the downstream direction. It must be pointed out that the use of MVGs opens also the possibility to set up a control method that acts twofold in the sense that both transition and separation are delayed or even prevented.

We investigate the dynamics of drop impact on a thin liquid film at short times in order to identify the mechanisms of splash formation. Using numerical simulations and scaling analysis, we show that it depends both on the inertial dynamics of the liquid and the cushioning of the gas. Two asymptotic regimes are identified, characterized by a new dimensionless number $J$: when the gas cushioning is weak, the jet is formed after a sequence of bubbles are entrapped and the jet speed is mostly selected by the Reynolds number of the impact. On the other hand, when the air cushioning is important, the lubrication of the gas beneath the drop and the liquid film controls the dynamics, leading to a single bubble entrapment and a weaker jet velocity.

We present a theoretical and experimental study of viscous flows injected into a porous medium that is confined vertically by horizontal impermeable boundaries and filled with an ambient fluid of different density and viscosity. General three-dimensional equations describing such flows are developed, showing that the dynamics can be affected by two separate contributions: spreading due to gradients in hydrostatic pressure, and that due to the pressure drop introduced by the injection. In the illustrative case of a two-dimensional injection of fluid at a constant volumetric rate, the injected fluid initially forms a viscous gravity current insensitive both to the depth of the medium and to the viscosity of the ambient fluid. Beyond a characteristic time scale, the dynamics transition to being dominated by the injection pressure, and the injected fluid eventually intersects the second boundary to form a second moving contact line. Three different late-time asymptotic regimes can emerge, depending on whether the viscosity of the injected fluid is less than, equal to or greater than that of the ambient fluid. With a less viscous injection, the flow undergoes a slow decay towards a similarity solution in which the two contact lines extend linearly in time with differing prefactors. Perturbations from this long-term state are shown to decay algebraically with time. Equal viscosities result in both contact lines approaching the same leading-order asymptotic position but with a first-order correction to the distance between them that expands as $t^{1/2}$ due to gravitational spreading. For a more viscous injection, the distance between the contact lines approaches a constant value, with perturbations decaying exponentially. Data from a new series of laboratory experiments confirm these theoretical predictions.

An experimental investigation of the sedimentation of monodisperse colloidal silica spheres with grafted octadecyl chains with three different interaction potentials is presented. Small particles (0.27 μm) behaved as hard spheres in cyclohexane, but larger ones (0.60 and 0.94 μm) are weakly flocculated by van der Waals attractions. The smallest particles (0.08 μm) in hexadecane are strongly flocculated by attractions between the octadecyl layers. A medical computer tomography (CT) scanner provided an accurate and absolute density measurement without disrupting the process. For the hard spheres and the weakly flocculated systems, the kinetics of sedimentation for the dispersed phase could readily be predicted utilizing the flux curve. For flocculated networks, we found a power-law relationship between compressive yield stresses and solids fractions comparable with other experimental systems.

The propagation of weak shock waves (Ms = 1.007, 1.03 and 1.1) through a statistically uniform random medium has been investigated experimentally in a shock tube. The wave-from geometry, rise time and amplitude of initially plane shocks which have propagated through a random mixture of helium and refrigerant 12 are measured. The effect of shock propagation on the properties of the random medium is visualized with schlieren and shadow photography. The pressure histories of the distorted shock waves reflecting from a normal end wall are observed to be both peaked and rounded. In the rounded case the perturbed shock is found to be made up of a succession of weak, slightly curved fronts with a total effective rise time orders of magnitude greater than the classical Taylor thickness. The radius of curvature of the weakest shocks after propagating through the random medium is inferred from observations at two downstream stations to be about 7 times the integral scale of the gas inhomogeneities. It is concluded that the observed distortions of the wave fronts can best be explained in terms of random focusing and defocusing of the front by the inhomogeneities in the medium. A ray-tracing calculation has been used to interpret the experimental observations. It is found that geometrical considerations are sufficient to account for many of the effects observed on the shocks.

The paper presents a method that removes the effects of all irregular frequencies in boundary-integral equations governing the interaction of regular waves with floating bodies of general geometry. A modified integral equation is obtained by the linear superposition of the classical Green equation and its normal derivative with respect to the field point. The selection of a purely imaginary constant of proportionality ensures the removal of all irregular frequencies in the continuous problem and the appropriate selection of its magnitude eliminates their undesirable effects in its numerical implementation. Computations are presented of the added-mass and damping coefficients and exciting forces on a sphere and a truncated vertical cylinder, illustrating the effectiveness of the method.

Melting from an electrically heated horizontal cylinder embedded in a paraffin (n-octadecane, fusion temperature 301·3 °K) has been studied experimentally. The shape of the solid-liquid interface has been determined photographically, and the local heat transfer coefficients have been measured using a shadowgraph technique. The experiments provide conclusive evidence of the important role played by natural convection in melting a solid due to an embedded cylindrical heat source. The four distinct pieces of quantitative evidence which contribute to this conclusion are the melt shape, surface temperature, local and average heat transfer coefficients and their variation with time.

The experimental findings prove the importance of natural convection in phase change problems involving melting and indicate that continued practice of neglecting the effects in the analysis of such problems does not appear reasonable. Natural convection should be considered in analysis and design of systems involving phase change.

When a stably stratified density interface is embedded in a region of strong velocity shear, hydrodynamic instabilities result. Here we generalize the stratified shear layer to allow an offset between the centre of the shear layer and the density interface. By including this asymmetry, and keeping the density interface thin with respect to the shear layer, the asymmetric Holmboe (AH) instability emerges. This study examines the evolution and mixing behaviour of AH instabilities, and compares the results to the well-known Kelvin–Helmholtz (KH) and Holmboe instabilities. This is done by performing a series of direct numerical simulations (DNS). The simulation results show that there are two different mixing mechanisms present. The first is a feature of KH instabilities and leads to the mixing and production of intermediate density fluid. The second mixing mechanism is found in AH and Holmboe instabilities and consists of regions of mixing and turbulence production that are located on one or both sides of the density interface. Since the Holmboe-type instabilities do not generate a large-scale overturning of the central isopycnal, the density interface is able to retain its identity throughout the mixing event. The amount of mixing that takes place is found to be strongly dependent on the degree of asymmetry in the flow.

This study investigates the mean flow structure of two shock-wave boundary-layer interactions generated by moderately swept compression ramps in a Mach 2 flow. The ramps have a compression angle of either $19^{\circ }$ or $22.5^{\circ }$ and a sweep angle of $30^{\circ }$. The primary diagnostic methods used for this study are surface-streakline flow visualization and particle image velocimetry. The shock-wave boundary-layer interactions are shown to be quasi-conical, with the intermittent region, separation line and reattachment line all scaling in a self-similar manner outside of the inception region. This is one of the first studies to investigate the flow field of a swept ramp using particle image velocimetry, allowing more sensitive measurements of the velocity flow field than previously possible. It is observed that the streamwise velocity component outside of the separated flow reaches the quasi-conical state at the same time as the bulk surface flow features. However, the streamwise and cross-stream components within the separated flow take longer to recover to the quasi-conical state, which indicates that the inception region for these low-magnitude velocity components is actually larger than was previously assumed. Specific scaling laws reported previously in the literature are also investigated and the results of this study are shown to scale similarly to these related interactions. Certain limiting cases of the scaling laws are explored that have potential implications for the interpretation of cylindrical and quasi-conical scaling.