Spectral models for turbulent pressure fluctuations are developed by directly Fourier transforming the integral solution to the Poisson equation for a homogeneous constantmean-shear flow. The turbulence-turbulence interaction is seen to possess the well-known k−7/3 inertial subrange and to dominate the high-wavenumber region. The turbulence–mean-shear contribution is seen to be dominant in the energy-containing range and falls off as $k^{-\frac{11}{3}}$ in the inertial subrange. The subrange constants and the mean-square pressure fluctuation are evaluated using a spectral model for the velocity. A spectral analysis of the velocity contamination of a pressure probe is also presented. Results are compared with spectral measurements with a static-pressure probe in the mixing layer of an axisymmetric jet.

The complete description of the lifetime of a droplet on a solid substrate evaporating in a ‘stick–slide’ mode is obtained. The unexpectedly subtle relationship between the lifetime of such a droplet and the lifetimes of initially identical droplets evaporating in the extreme modes (namely the constant contact radius and constant contact angle modes) is described and summarised in an appropriate master diagram. In particular, it is shown that the lifetime of a droplet is not, in general, constrained by the lifetimes of the extreme modes.

Direct numerical simulations (DNS) of steady and pulsatile flow through 75% (by area reduction) stenosed tubes have been performed, with the motivation of understanding the biofluid dynamics of actual stenosed arteries. The spectral-element method, providing geometric flexibility and high-order spectral accuracy, was employed for the simulations. The steady flow results are examined here while the pulsatile flow analysis is dealt with in Part 2 of this study. At inlet Reynolds numbers of 500 and 1000, DNS predict a laminar flow field downstream of an axisymmetric stenosis and comparison to previous experiments show good agreement in the immediate post-stenotic region. The introduction of a geometric perturbation within the current model, in the form of a stenosis eccentricity that was 5% of the main vessel diameter at the throat, resulted in breaking of the symmetry of the post-stenotic flow field by causing the jet to deflect towards the side of the eccentricity and, at a high enough Reynolds number of 1000, jet breakdown occurred in the downstream region. The flow transitioned to turbulence about five diameters away from the stenosis, with velocity spectra taking on a broadband nature, acquiring a -5/3 slope that is typical of turbulent flows. Transition was accomplished by the breaking up of streamwise, hairpin vortices into a localized turbulent spot, reminiscent of the turbulent puff observed in pipe flow transition, within which r.m.s. velocity and turbulent energy levels were highest. Turbulent fluctuations and energy levels rapidly decayed beyond this region and flow relaminarized. The acceleration of the fluid through the stenosis resulted in wall shear stress (WSS) magnitudes that exceeded upstream levels by more than a factor of 30 but low WSS levels accompanied the flow separation zones that formed immediately downstream of the stenosis. Transition to turbulence in the case of the eccentric stenosis was found to be manifested as large temporal and spatial gradients of shear stress, with significant axial and circumferential variations in instantaneous WSS.

Large-scale organized structures in the turbulent plane wake of a circular cylinder are investigated in air up to a downstream distance of 40d at a Reynolds number of Red = 1.3 × 104; d is the cylinder diameter. Velocity signals from a linear transverse rake of 8 X-wires are sampled simultaneously to calculate the instantaneous span wise vorticity. We have appropriately smoothed the temporal traces of vorticity to obtain time evolutions (including the transverse displacement, sign, strength and size distributions) of organized structures identified by vorticity contour maps. The periodicity of the initial structures is rapidly lost: dispersion in streamwise spacing, transverse displacement, strength and size of structures increases with increasing downstream distance.

Particular emphasis is placed on examining alternative general schemes for educing coherent structures in natural or unexcited turbulent shear flows, especially in their fully developed states. The optimal eduction scheme employed involves centring the rake at the most probable transverse location of centres of advected structures and accepting those structures that: (i) are centred at the midpoint of the rake, (ii) have a peak value of (smoothed) vorticity of a given sign above a specified level, and (iii) have streamwise and transverse extents of the (smoothed) vorticity contours above a specified size. From successive accepted structure signatures the instants of occurrence of structure centres (i.e. smoothed vorticity peaks) are identified. Un-smoothed signals are then time-aligned with respect to these instants and ensemble averaged to educe coherent structure and incoherent turbulence characteristics. Further enhancement is achieved by iteratively improving the time-alignment by maximizing the cross-correlation of individual structure vorticity with the ensemble-averaged vorticity and by discarding structures that require excessive time shifts or that produce significantly weak peak correlation values.

Following this optimal scheme, large-scale coherent structures have been educed in the fully turbulent wake. The average structure centre is found to be closer to the wake centreline than the half-width location, and the structure size does not increase in proportion to the wake width, suggesting that transverse wandering of structures (including their three-dimensionality) increases significantly with increasing downstream distance. The various flow properties associated with coherent and incoherent turbulence, and the coherent structure dynamics, in particular the role of vortex stretching (at the saddle) in turbulence production and mixing, are discussed.

Computational experiments based on direct numerical simulation of wall-bounded flow reveal that turbulence production can be suppressed by the action of a transverse travelling wave. Flow visualizations show that the near-wall flow structure is altered substantially, compared to other turbulence control techniques, leading to a large amount of shear stress reduction (i.e., more than 30%). The travelling wave can be induced by a spanwise force that is confined within the viscous sublayer, it has its maximum at the wall, and decays exponentially away from it. We demonstrate the robustness of this approach, and its application in salt water using arrays of electro-magnetic tiles that can produce the required travelling wave excitation. We also study corresponding results from spanwise oscillations using a similar force, which also leads to large drag reduction. Although the turbulence statistics for the two approaches are similar, the near-wall structures appear to be different: in the spanwise oscillatory excitation there is a clear presence of wall-streaks whereas in the travelling wave excitation these streaks have disappeared. From the fundamental point of view, the new finding of this work is that appropriate enhancement of the streamwise vortices leads to weakening of the streak intensity, as measured by the normal vorticity component, and correspondingly substantial suppression of turbulence production. From the practical point of view, our findings provide guidance for designing different surface-based actuation techniques including piezoelectric materials, shape memory alloys, and electro-magnetic tiles.

The linear growth of the spanwise correlation length scale with the distance from the wall in the logarithmic region of wall-bounded turbulent flows has been understood as a reflection of Townsend’s attached eddies. Based on this observation, in the present study, we perform a numerical experiment, which simulates energy-containing motions only at a given spanwise length scale in the logarithmic region, using their self-sustaining nature found recently. The self-sustaining energy-containing motions at each of the spanwise length scales are found to be self-similar with respect to the given spanwise length. Furthermore, their statistical structures are consistent with those of the attached eddies in the original theory, providing direct evidence on the existence of Townsend’s attached eddies. It is shown that a single self-sustaining attached eddy is composed of two distinct elements, one of which is a long streaky motion reaching the near-wall region, and the other is a relatively short vortical structure carrying all the velocity components. For the given spanwise length ${\it\lambda}_{z}$ between ${\it\lambda}_{z}^{+}=100$ and ${\it\lambda}_{z}\simeq 1.5h$, where $h$ is half the height of the channel, the former is found to be self-similar along $y\simeq 0.1{\it\lambda}_{z}$ and ${\it\lambda}_{x}\simeq 10{\it\lambda}_{z}$, while the latter is self-similar along $y\simeq 0.5{\it\lambda}_{z}\sim 0.7{\it\lambda}_{z}$ and ${\it\lambda}_{x}\simeq 2{\it\lambda}_{z}\sim 3{\it\lambda}_{z}$ where $y$ is the wall-normal direction. The scaling suggests that the smallest attached eddy would be a near-wall coherent motion in the form of a streak and quasi-streamwise vortices aligned to that, whereas the largest one would be an outer motion with a very-large-scale motion (VLSM) and large-scale motions (LSMs) aligned to that. The attached eddies in between, the size of which is proportional to their distance from the wall, contribute to the logarithmic region and fill the space caused by the length scale separation. The scaling is also found to yield behaviour consistent with the emergence of $k_{x}^{-1}$ spectra in a number of previous studies. Finally, a further discussion is provided, in particular on Townsend’s inactive motion and several recent theoretical findings.

The onset of Kármán-vortex shedding is studied experimentally in the wake of different two-dimensional bluff bodies, namely an oblong cylinder, circular cylinders and plates of rectangular cross-section. Different control measures, such as wake heating, transverse body oscillations and base bleed are investigated. As the steady-periodic Kármán shedding has previously been identified as a limit-cycle, i.e. as self-excited oscillations, the experiments are interpreted in the framework of the Stuart–Landau model. The coefficients of the Stuart–Landau equation for the characteristic vortex shedding amplitude, i.e. the linear temporal growth rate, linear frequency and the Landau constant, are fully determined for the two cylinders and in part for the plate. For this purpose transients are generated by suddenly switching transverse body oscillations or base bleed on or off. The analysis of these transients by a refined method based on complex demodulation provides reliable estimates of the model coefficients and yields an experimental validation of the concept that a global instability mode grows or decays as a whole. Also, it is demonstrated that the coefficients of the Stuart–Landau equation are independent of the experimental technique used to produce the transients.

Using the principles of continuum mechanics, a theory is developed for describing quantitatively the sedimentation of small particles in vessels having walls that are inclined to the vertical. The theory assumes that the flow is laminar and that the particle Reynolds number is small, but c0, the concentration in the suspension, and the vessel geometry are left arbitrary. The settling rate S is shown to depend upon two dimensionless groups, in addition to the vessel geometry: a sedimentation Reynolds number R, typically O(1)-O(10); and Λ, the ratio of a sedimentation Grashof number to R, which is typically very large. By means of an asymptotic analysis it is then concluded that, as Λ → ∞ and for a given geometry, S can be predicted from the well-known Ponder-Nakamura-Kuroda formula which was obtained using only kinematic arguments. The present theory also gives an expression for the thickness of the clear-fluid slit that forms underneath the downward-facing segment of the vessel walls, as well as for the velocity profile both in this slit and in the adjoining suspension.

The sedimentation rate and thickness of the clear-fluid slit were also measured in a vessel consisting of two parallel plates under the following set of conditions: c0 ≤ 0·1, R ∼ O(1), O(10)5 ≤ Λ ≤ O(107) and 0° ≤ α ≤ 50°, where α is the angle of inclination. Excellent agreement was obtained with the theoretical predictions. This suggests that the deviations from the Ponder-Nakamura-Kuroda formula reported in the literature are probably due to a flow instability which causes the particles to resuspend and thereby reduces the efficiency of the process.

Wing pitch reversal, the rapid change of angle of attack near stroke transition, represents a difference between hovering with flapping wings and with a continuously rotating blade (e.g. helicopter flight). Although insects have the musculature to control the wing pitch during flight, we show here that aerodynamic and wing inertia forces are sufficient to pitch the wing without the aid of the muscles. We study the passive nature of wing pitching in several observed wing kinematics, including the wing motion of a tethered dragonfly, Libellula pulchella, hovering fruitfly, hovering hawkmoth and simplified dragonfly hovering kinematics. To determine whether the pitching is passive, we calculate rotational power about the torsion axis owing to aerodynamic and wing inertial forces. This is done using both direct numerical simulations and quasi-steady fluid force models. We find that, in all the cases studied here, the net rotational power is negative, signifying that the fluid force assists rather than resists the wing pitching. To further understand the generality of these results, we use the quasi-steady force model to analyse the effect of the components of the fluid forces at pitch reversal, and predict the conditions under which the wing pitch reversal is passive. These results suggest the pitching motion of the wings can be passive in insect flight.

Turbulent entrainment at the density interface of a stable two-layer stratified fluid is studied in the laboratory, a constant surface stress being applied at the free surface. Conservation of mass requires that the overall Richardson number Ri = Dgδρ/ρu*2 is constant in each experiment, where D is the depth of the mixed layer, gδρ/ρ the buoyancy difference and u* the friction velocity. If the entrainment rate E = ue/u* is a function only of Ri, it is therefore constant in each experiment and can be measured with a greater accuracy than has previously been attained. The functional dependence of ue/u* on Ri is established over the range 30 < Ri < 1000; it is found not to follow any simple power law. The entrainment rates are considerably higher than those measured by Kato & Phillips (1969), for which the fluid below the mixed layer was linearly stratified. Such a condition allows internal gravity waves to be radiated downwards and the reduction in entrainment rate is consistent with that found by Linden (1975).

The linear response to stochastic and optimal harmonic forcing of small coherent perturbations to the turbulent channel mean flow is computed for Reynolds numbers ranging from Reτ = 500 to 20000. Even though the turbulent mean flow is linearly stable, it is nevertheless able to sustain large amplifications by the forcing. The most amplified structures consist of streamwise-elongated streaks that are optimally forced by streamwise-elongated vortices. For streamwise-elongated structures, the mean energy amplification of the stochastic forcing is found to be, to a first approximation, inversely proportional to the forced spanwise wavenumber while it is inversely proportional to its square for optimal harmonic forcing in an intermediate spanwise wavenumber range. This scaling can be explicitly derived from the linearized equations under the assumptions of geometric similarity of the coherent perturbations and of logarithmic base flow. Deviations from this approximate power-law regime are apparent in the pre-multiplied energy amplification curves that reveal a strong influence of two different peaks. The dominant peak scales in outer units with the most amplified spanwise wavelength of λz ≈ 3.5h, while the secondary peak scales in wall units with the most amplified λz+ ≈ 80. The associated optimal perturbations are almost independent of the Reynolds number when, respectively, scaled in outer and inner units. In the intermediate wavenumber range, the optimal perturbations are approximatively geometrically similar. Furthermore, the shape of the optimal perturbations issued from the initial value, the harmonic forcing and the stochastic forcing analyses are almost indistinguishable. The optimal streaks corresponding to the large-scale peak strongly penetrate into the inner layer, where their amplitude is proportional to the mean-flow profile. At the wavenumbers corresponding to the large-scale peak, the optimal amplifications of harmonic forcing are at least two orders of magnitude larger than the amplifications of the variance of stochastic forcing and both increase with the Reynolds number. This confirms the potential of the artificial forcing of optimal large-scale streaks for the flow control of wall-bounded turbulent flows.

Well-resolved large-eddy simulations (LES) are performed in order to investigate flow phenomena and turbulence structure of the boundary layer along a supersonic compression ramp. The numerical simulations directly reproduce an available experimental result. The compression ramp has a deflection angle of $\beta\,{=}\,25^\circ$. The mean free-stream Mach number is $M_\infty\,{=}\,2.95$. The Reynolds number based on the incoming boundary-layer thickness is $Re_{\delta_0}\,{=}\,63\,560$ in accordance with the reference experiment. These simulations overcome deficiencies of earlier direct numerical simulations (DNS) and LES in terms of ramp-deflection angle, Reynolds number and spanwise size of the computational domain which is required for capturing the essential flow phenomena. The filtered conservation equations for mass, momentum and energy are solved with a high-order finite-difference scheme. The effect of subgrid scales is modelled by the approximate deconvolution model. About $18.5\,{\times}\,10^6$ grid points are used for discretizing the computational domain. To obtain mean flow and turbulence structure the flow is sampled 1272 times over 703 characteristic time scales of the incoming boundary layer. Statistical data are computed from these samples. An analysis of the data shows good agreement with the experiment in terms of mean quantities such as shock position, separation and reattachment location, skin-friction and surface-pressure distributions, and turbulence structure. The computational data confirm theoretical and experimental results on fluctuation amplification across the interaction region. In the wake of the main shock a shedding of shocklets is observed. The temporal behaviour of the coupled shock–separation system agrees well with experimental data. Unlike previous DNS the present simulation data provide indications of a large-scale shock motion. Also, evidence for the existence of three-dimensional large-scale streamwise structures, commonly referred to as Görtler-like vortices, is found.

The generation of Aeolian tones is interpreted in terms of the theroy of aerodynamic sound. To do this, the intensity of the radiated sound is expressed in terms of the fluctuations in force upon a moving rigid body using the approximation that, at low mach numbers, these forces can be calculated assuming incompressibility of the flow. The fluctuations in lift and drag upon a circular cylinder at Reynolds numbers between 40 and 160 are calculated by integrating the fluctuations in momentum in the eddying wake, using the experimental data of Kovasznay (1949). It is found that the fluctuating lift per unit length is approximately $f_l = 0\cdot 38\rho U^2dcos2\pi nt$ where ρ is the density of the fluid and t the time, and that the magnitude of the fluctuations in drag is about 10% of this.

The axial length scale of these force fluctuations was found by an auxiliary experiment to be very large for Reynolds numbers below 100. When 100 < R < 160, the lenth scale is approximately 17d, the transition apparently occurring as a result of the instability to three-dimensional disturbances of the laminar eddying wake. Using this datum, the mean square acoustic pressure generated by the motion about the cylinder at these Reynolds numbers is found to be $\overline{P^2}(r) \sim 0\cdot 27 cos^2 \theta \frac {\rho ^2 ldU^6 S^2}{a^2r^2}$ where θ is the angle between the direction of observation and the incident stream, l the length of the cylinder and a the velocity of sound in the medium. Experiments undertaken to test this result directly, using sound intensity measurements from a whirling arm apparatus in an accoustically quiet room, gave very good agreement.

At higher Reynolds numbers, when the cylindg wake is turbulent, the theory leads to a, similar expression for $\overline{P^2}$ but with a smaller numerical factor. Analysis of previous experimental data on this basis gives good agreement, with a value of about 0·037 for the numerical constant. In view of this, it is likely that some of the earlier data which had appeared to give a Mach number dependence of M4, not M6, have been misinterpreted. Finally, mention is made of the conditions when the frequency of vortex generation is the same as the natural frequency of the wire, and the greatly increased intensity of the sound is seen to be the result of the greatly increased axial length scale of the force fluctuations.

The relative simplicity of this phenomenon makes possible perhaps the least ambiguous confirmation yet found of some of the essential ideas in the theory of aerodynamic sound.

An axisymmetric film bridge collapses under its own surface tension, disconnecting at a pair of pinchoff points that straddle a satellite bubble. The free-boundary problem for the motion of the film surface and adjacent inviscid fluid has a finite-time blowup (pinchoff). This problem is solved numerically using the vortex method in a boundary-integral formulation for the dipole strength distribution on the surface. Simulation is in good agreement with available experiments. Simulation of the trajectory up to pinchoff is carried out. The self-similar behaviour observed near pinchoff shows a ‘conical-wedge’ geometry whereby both principal curvatures of the surface are simultaneously singular – lengths scale with time as t2/3. The similarity equations are written down and key solution characteristics are reported. Prior to pinchoff, the following regimes are found. Near onset of the instability, the surface evolution follows a direction dictated by the associated static minimal surface problem. Later, the motion of the mid-circumference follows a t2/3 scaling. After this scaling ‘breaks’, a one-dimensional model is adequate and explains the second scaling regime. Closer to pinchoff, strong axial motions and a folding surface render the one-dimensional approximation invalid. The evolution ultimately recovers a t2/3 scaling and reveals its self-similar structure.

An experimental study is performed on the vortex induced vibrations of a rigid flexibly mounted circular cylinder placed in a crossflow. The cylinder is allowed to oscillate in combined crossflow and in-line motions, and the ratio of the nominal in-line and transverse natural frequencies is varied systematically. Experiments were conducted on a smooth cylinder at subcritical Reynolds numbers between 15 000 and 60 000 and on a roughened cylinder at supercritical Reynolds numbers between 320 000 and 710 000, with a surface roughness equal to 0.23 % of the cylinder diameter. Strong qualitative and quantitative similarities between the subcritical and supercritical experiments are found, especially when the in-line natural frequency is close to twice the value of the crossflow natural frequency. In both Reynolds number regimes, the test cylinder may exhibit a ‘dual-resonant’ response, resulting in resonant crossflow motion at a frequency fv, near the Strouhal frequency, and resonant in-line motion at 2 fv. This dual resonance is shown to occur over a relatively wide frequency region around the Strouhal frequency, accompanied by stable, highly repeatable figure-eight cylinder orbits, as well as large third-harmonic components of the lift force. Under dual-resonance conditions, both the subcritical and the supercritical response is shown to collapse into a narrow parametric region in which the effective natural-frequency ratio is near the value 2, regardless of the nominal natural-frequency ratio. Some differences are noted in the magnitudes of forces and the cylinder response between the two different Reynolds number regimes, but the dual-resonant response and the resulting force trends are preserved despite the large Reynolds number difference.

Electro-osmotic flow is a convenient mechanism for transporting fluid in microfluidic devices. The flow is generated through the application of an external electric field that acts on the free charges that exist in a thin Debye layer at the channel walls. The charge on the wall is due to the particular chemistry of the solid–fluid interface and can vary along the channel either by design or because of various unavoidable inhomogeneities of the wall material or because of contamination of the wall by chemicals contained in the fluid stream. The channel cross-section could also vary in shape and area. The effect of such variability on the flow through microfluidic channels is of interest in the design of devices that use electro-osmotic flow. The problem of electro-osmotic flow in a straight microfluidic channel of arbitrary cross-sectional geometry and distribution of wall charge is solved in the lubrication approximation, which is justified when the characteristic length scales for axial variation of the wall charge and cross-section are both large compared to a characteristic width of the channel. It is thereby shown that the volume flux of fluid through such a microchannel is a linear function of the applied pressure drop and electric potential drop across it, the coefficients of which may be calculated explicitly in terms of the geometry and charge distribution on the wall. These coefficients characterize the ‘fluidic resistance’ of each segment of a microfluidic network in analogy to the electrical ‘resistance’ in a microelectronic circuit. A consequence of the axial variation in channel properties is the appearance of an induced pressure gradient and an associated secondary flow that leads to increased Taylor dispersion limiting the resolution of electrophoretic separations. The lubrication theory presented here offers a simple way of calculating the distortion of the flow profile in general geometries and could be useful in studies of dispersion induced by inhomogeneities in microfluidic channels.

An unusually low-frequency oscillation in the flow over an airfoil is studied experimentally as well as computationally. Wind-tunnel measurements are carried out with two-dimensional airfoil models in the chord Reynolds number (Rc) range of 0.15 × 105−3.0 × 105. During deep stall, at α [gsim ] 18°, the usual ‘bluff-body shedding’ occurs at a Strouhal number, Sts ≈ 0.2. But at the onset of static stall around α = 15°, a low-frequency periodic oscillation is observed, the corresponding Sts being an order of magnitude lower. The phenomenon apparently takes place only with a transitional state of the separating boundary layer. Thus, on the one hand, it is not readily observed with a smooth airfoil in a clean wind tunnel, while on the other, it is easily removed by appropriate ‘acoustic tripping’. Details of the flow field for a typical case are compared with a case of bluff-body shedding. The flow field is different in many ways from the latter case and does not involve a Kármán Vortex street. The origin of the flow fluctuations traces to the upper surface of the airfoil and is associated with a periodic switching between stalled and unstalled states. The mechanism of the frequency selection remains unresolved, but any connection to blower instabilities, acoustic standing waves or structural resonances has been ruled out.

A similar result has been encountered computationally using a two-dimensional Navier–Stokes code. While with the assumption of laminar flow, wake oscillation akin to the bluff-body shedding has been observed previously, the Sts is found to drop to about 0.03 when a ‘turbulent’ boundary layer is assumed. Details of the flow field and unsteady forces, computed for the same conditions as in the experiment, compare reasonably well with the experimental data.

The phenomenon produces intense flow fluctuations imparting much larger unsteady forces to the airfoil than that experienced in bluff-body shedding, and may represent the primary aerodynamics of stall flutter of blades and wings.

We study the capillary rise of wetting liquids in the corners of different geometries and show that the meniscus rises without limit following the universal law: h(t)/a ≈ (γt/ηa)1/3, where γ and η stand for the surface tension and viscosity of the liquid while is the capillary length, based on the liquid density ρ and gravity g. This law is universal in the sense that it does not depend on the geometry of the corner.

Starting from a kinetic theory (KT) model for monodisperse granular flow, the exact Reynolds-averaged (RA) equations are derived for the particle phase in a collisional fluid–particle flow. The corresponding equations for a constant-density fluid phase are derived from a model that includes drag and buoyancy coupling with the particle phase. The fully coupled macroscale/hydrodynamic model, rigorously derived from a kinetic equation for the particles, is written in terms of the particle-phase volume fraction, the particle-phase velocity and the granular temperature (or total granular energy). As derived from the hydrodynamic model, the RA turbulence model solves for the RA particle-phase volume fraction, the phase-averaged (PA) particle-phase velocity, the PA granular temperature and the PA turbulent kinetic energy of the particle phase. Thus, unlike in most previous derivations of macroscale turbulence models for moderately dense granular flows, a clear distinction is made between the PA granular temperature $\Theta _\textit {p}$, which appears in the KT constitutive relations, and the particle-phase turbulent kinetic energy $k_\textit {p}$, which appears in the turbulent transport coefficients. The exact RA equations contain unclosed terms due to nonlinearities in the hydrodynamic model and we briefly discuss the available closures for these terms. Finally, we demonstrate by comparing model predictions with direct numerical simulation results that even for non-collisional fluid–particle flows it is necessary to provide separate models for $\Theta _\textit {p}$ and $k_\textit {p}$ in order to correctly account for the effect of the particle Stokes number and mass loading.

A small drop placed on a horizontal surface will spread under the action of capillary forces until it reaches an equilibrium position. The rate at which it spreads provides a means for testing certain hypotheses about moving contact lines; namely that there must be slip between the fluid and the solid boundary near the rim of the drop to avoid a force singularity there, and that the contact angle measured at the rim itself does not show the dynamic behaviour observed by measurements that ignore rapid changes in slope in the immediate vicinity of the rim but remains equal to its static value.

By the use of matched asymptotic expansions, an equation for the rate of spread of a drop as a function of the radius of the contact circle is obtained. Experiments on the spreading of small drops of molten glass allow a comparison to be made between the spreading of a drop determined experimentally and that predicted theoretically, which supports the use of the proposed hypotheses as appropriate for the study of fluid motions containing moving contact lines.