Dimensionless Surface Tension Research Articles

Linear theory of gravity waves on a Voigt viscoelastic medium

Linear surface gravity waves on a semi-infinite incompressible Voigt medium are studied in this paper. Three dimensionless parameters, the dimensionless viscoelastic parameter ϑ, the dimensionless wave number and the dimensionless surface tension are introduced. A dimensionless characteristic equation describing the waves is derived. This is a sixth order complex algebraic equation which is solved to give the complex dispersion relation. Based on the numerical solution, two critical values of ϑ, ϑA=0.607 and ϑB=2.380, which represent the appearance of the cutoff region and the disappearance of the strong dispersion region, are found. The effects of ϑ on the characteristic equation and the properties of the waves are discussed.

Viscous fingering in liquid crystals: anisotropy and morphological transitions

We show that a minimal model for viscous fingering with a nematic liquid crystal in which anisotropy is considered to enter through two different viscosities in two perpendicular directions can be mapped to a twofold anisotropy in the surface tension. We numerically integrate the dynamics of the resulting problem with the phase-field approach to find and characterize a transition between tip splitting and side branching as a function of both anisotropy and dimensionless surface tension. This anisotropy dependence could explain the experimentally observed (reentrant) transition as temperature and applied pressure are varied. Our observations are also consistent with previous experimental evidence in viscous fingering within an etched cell and simulations of solidification.

Negative Pressure in the Oil Film of Journal Bearing. 1st Report: Modelling of Negative Pressure Development and Influences of Parameters on Pressure Distribution.

Using a bubble theory which takes into account the surface dilatational viscosity κ, influences of eccentricity ε, 4 bubble and 4 bearing parameters on pressure distribution are studied for the oil film in journal bearing : When κ is neglected, the maximum negative pressure is given by a formula containing only a dimensionless surface tension. For small ε, pressure distribution scarcely depends on bubble parameters, and coincides nearly with that calculated by conventional theory under Reynolds condition. For large ε, it changes also with bubble parament values. Whether ε is small or large, the maximum pressure hardly depends on bubble parameters, but depends only on bearing parameters. It is nearly equal to that calculated by conventional theory under Reynolds condition. For increasing ε, κ and shaft speed, the pressure distribution tends to that of Sommerfeld type.

Two-finger selection theory in the Saffman-Taylor problem.

We find that solvability theory selects a set of stationary solutions of the Saffman-Taylor problem with coexistence of two unequal fingers advancing with the same velocity but with different relative widths lambda1 and lambda2 and different tip positions. For vanishingly small dimensionless surface tension d0, an infinite discrete set of values of the total filling fraction lambda=lambda1+lambda2 and of the relative individual finger width p=lambda1/lambda are selected out of a two-parameter continuous degeneracy. They scale as lambda-1/2 approximately d0(2/3) and p-1/2 approximately d0(1/3). The selected values of lambda differ from those of the single finger case. Explicit approximate expressions for both spectra are given.

Mass, Momentum, and Energy Transport in a Molten Pool When Welding Dissimilar Metals

Velocity and temperature fields, and shapes of the fusion zone in welding dissimilar metals are systematically investigated. Fluid flow in the molten pool is driven by the Marangoni force in different directions and magnitudes on the flat free surface. To interpret clearly without loss of generality, the three-dimensional quasi-steady welding is simulated by an unsteady two-dimensional process. Transport process on the maximum cross section of the molten region (or a cross section of the fusion zone) therefore is predicted. Interfaces between immiscible dissimilar metals and solid and liquid are, respectively, calculated by the volume of fluid and enthalpy methods. The computed results show different flow and thermal fields and molten regions of dissimilar metals as functions of dimensionless surface tension coefficients, viscosities, melting temperatures, and thermal conductivities of dissimilar metals and distinct phases, and beam power, welding speed, and the energy distribution parameter. The predicted shapes of fusion zones in welding pure irons, and aluminum to iron, agree with the experimental results.

Dissipation of energy by finite-amplitude surface waves

Direct numerical simulations of the Navier–Stokes equations are used to study energy dissipation for deep-water finite-amplitude traveling waves. The effects of the dimensionless surface tension coefficient, the Reynolds number, the density ratio and the kinematic viscosity ratio on energy dissipation and wave profiles are investigated. It is found that the dimensionless surface tension plays an important role for the wave profiles and energy dissipation rates; energy dissipation occurs mostly near the crest for gravity waves, but mostly near the trough for capillary waves. Waves of lower Reynolds numbers have smoother profiles than those at higher Reynolds numbers. When the Reynolds number is low enough, the energy dissipation rate follows closely the viscous linear theory, even if the initial wave amplitude is large.

Liquid-vapor interface of an ionic fluid

We investigate the liquid-vapor interface of the restricted primitive model (RPM) for an ionic fluid using a density-functional approximation based on correlation functions of the homogeneous fluid as obtained from the mean-spherical approximation (MSA). In the limit of a homogeneous fluid our approach yields the well-known MSA (energy) equation of state. The ionic interfacial density profiles, which for the RPM are identical for both species, have a shape similar to those of simple atomic fluids in that the decay towards the bulk values is more rapid on the vapor side than on the liquid side. This is the opposite asymmetry of the decay to that found in earlier calculations for the RPM based on a square-gradient theory. The width of the interface is, for a wide range of temperatures, approximately four times the second moment correlation length of the liquid phase. We discuss the magnitude and temperature dependence of the surface tension, and argue that for temperatures near the triple point the ratio of the dimensionless surface tension and critical temperature is much smaller for the RPM than for simple atomic fluids.

Cylindrical solitary waves and their interaction in Bénard-Marangoni layers

We consider a motionless Newtonian and incompressible shallow liquid layer bounded below by a bottom plate where temperature is held fixed and above with a free deformable surface whose tension is linearly temperature dependent and on which the heat flux is fixed. When the dimensionless surface tension gradient, measured by a Marangoni number, slightly exceeds its critical value, radially symmetric, long-wavelength excitations obey a dissipative cylindrical Korteweg--de Vries (DCKdV) equation. A dissipative cylindrical Kadomtsev-Petviashvili (DCKP) equation is also derived for nearly radially symmetric disturbances. Exact solitary wave solutions of the DCKP equation are found and the solitary wave solutions of the DCKdV equation are also discussed. Finally the head-on collision between two concentric cylindrical solitary waves is considered and its solitonic character is displayed.

The effect of surface tension on nonpropagating hydrodynamic solitons

The distribution of breather and kink solitons in the kd versus σ plane is measured experimentally, where kd is the dimensionless wavenumber and σ the dimensionless surface tension coefficient of the liquid. The surface tension shows little effect on the distribution. An analysis is made on the effect of surface tension on the distribution, as well as on the stability region for the existence of breather solitons.

On parasitic capillary waves generated by steep gravity waves: an experimental investigation with spatial and temporal measurements

An experimental investigation of steep, high-frequency gravity waves (∼ 4 to 5 Hz) and the parasitic capillary waves they generate is reported. Spatial, as well as temporal, non-intrusive surface measurements are made using a new technique. This technique employs cylindrical lenses to magnify the vertical dimension in conjunction with an intensified, high-speed imaging system, facilitating the measurement of the disparate scales with a vertical surface-elevation resolution on the order of 10 μm. Thus, high-frequency parasitic capillary waves and the underlying gravity wave are measured simultaneously and accurately in space and time. Time series of spatial surface-elevation measurements are presented. It is shown that the location of the capillary waves is quasi-stationary in a coordinate system moving with the phase speed of the underlying gravity wave. Amplitudes and wavenumbers of the capillaries are modulated in space; however, they do not propagate with respect to the gravity wave. As capillary amplitudes are seen to decrease significantly and then increase again in a recurrence-like phenomenon, it is conjectured that resonance mechanisms are present. Measured surface profiles are compared to the theories of Longuet-Higgins (1963) and Crapper (1970) and the exact, two-dimensional numerical formulation of Schwartz & Vanden-Broeck (1979). Significant discrepancies are found between experimental and theoretical wavetrains in both amplitude and wavenumber. The theoretical predictions of the capillary wave amplitudes are much smaller than the measured amplitudes when the measured phase speed, amplitude, and wavelength of the gravity wave are used in the Longuet-Higgins model. In addition, this theory predicts larger wavenumbers of the capillaries as compared to experiments. The Crapper model predicts the correct order-of-magnitude capillary wave amplitude on the forward face of the gravity wave, but predicts larger amplitudes on the leeward face in comparison to the experiments. Also, it predicts larger capillary wavenumbers than are experimentally determined. Comparison of the measured profiles to multiple solutions of the stationary, symmetric, periodic solutions determined using the Schwartz & Vanden-Broeck numerical formulation show similar discrepancies. In particular, the assumed symmetry of the waveform about crest and trough in the numerical model precludes a positive comparison with the experiments, whose underlying waves exhibit significantly larger capillaries on their forward face than on their leeward face. Also, the a priori unknown multiplicity of numerical solutions for the same dimensionless surface tension and steepness parameters complicates comparison. Finally, using the temporal periodicity of the wave field, composite images of several successive wavelengths are constructed from which potential energy and surface energy are calculated as a function of distance downstream.

SPINODAL DECOMPOSITION WITH SURFACE TENSION DRIVEN FLOWS

Abstract The Cahn-Hilliard theory of phase separation by spinodal decomposition has been extended to include hydrodynamics caused by surface tension. Numerical solutions for phase separation in binary polymer mixtures in two spatial dimension are presented. The surface tension influences the growth rate of the average domain size and the morphology for near critical quenches that would otherwise form cocontinuous networks. The growth rate exponent depends on the value of dimensionless surface tension, γ = [(ρ0 m 3/2)/(Dμ)][(κυ s )/(RgT)]½, and ranges from 0.30 ±0.03, for γ = 0 (no hydrodynamics) to 0.69 ±0.04 for γ = 1 for a critical quench. For an off-critical quench in which a dispersed phase would be formed by diffusion alone, the scaling exponent shows little effect of surface tension and ranges from 0.28 ±0.02 for γ = 0 to 0.32 ±0.02 for γ = l. While not conclusive, the results suggest that it is infeasible to prepare bulk samples of cocontinuous polymer phases by the process known as compositional q...

Gradient theory of surface tension of pure fluids: Effect of molecular shape and polarity

In this paper, we investigate the effect of molecular shape and multipolarity on interfacial properties using a modified van der Waals theory of interfaces and an equation of state developed by Mohanty and Davis for attracting, nonspherical particles with a rigid, convex core. The equation of state contains, in addition to the classical van der Waals parameters, molecular shape and multipolar parameters. Our results indicate that the dimensionless surface tension of pure fluids varies linearly with the multipolar parameter and exponentially with the molecular shape parameter. The agreement between our predictions and experimental data is good.

Linear theory of gravity waves on a maxwell fluid

Surface gravity waves on a semi-infinite incompressible Maxwell fluid are studied by means of linear theory. A dimensionless (memory) time-number (Θ), different from the Deborah number, is introduced, together with a dimensionless wave-number and a dimensionless surface tension. A characteristic equation describing the waves is derived. This is an 8-degree complex polynomial which is solved to give the complex dispersion relation. Two critical time-numbers are found, Θ A = 0.321 and Θ c = 0.906. These are important for the number of pure decay solutions and propagating waves. The dispersion relation is shown for Θ in the range from 0 to 2. A similarity to gravity waves on inviscid fluids is seen for small wave numbers. For large wave numbers these are solutions of the Rayleigh wave type.

Time-Dependent Aerosol Models and Homogeneous Nucleation Rates

Two types of numerical models for homogeneous nucleation and particle growth are compared: models in which the time rate of change in the stable aerosol population is given by an analytical expression for the nucleation rate, and discrete models, in which an expression for the nucleation rate is not required nor explicitly calculated. The classical expression for the homogeneous nucleation rate, coupled to a moment model, is found to produce poor agreement with a discrete-sectional model except for very low values of dimensionless surface tension. A new expression for the homogeneous nucleation rate is proposed, which when coupled to a moment model produces excellent agreement with a discrete-sectional model over a wide range of dimensionless surface tensions. The new expression is also consistent with experimental data on homogeneous nucleation of dibutylphthalate.

Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension γ, including γ=0. The method makes it possible to predict a number of bifurcations that occur as γ decreases. The existence of numerous families of stationary traveling waves when γ ≫ 1 was demonstrated in [6]. The present study shows that as γ→1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various γ. When γ=0 and the wave number α is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point ϕ=140–150. The method proposed can be used to calculate other wavy film flows.

Hele-Shaw scaling properties of low-contrast Saffman-Taylor flows.

We have measured variations of Saffman-Taylor flows by changing dimensionless surface tension B alone and by changing B in conjunction with changes in dimensionless viscosity contrast A. Our low-aspect-ratio cell permits close study of the linear- and early nonlinear-flow regimes. Our critical binary-liquid sample allows study of very low values of A. The predictions of linear stability analysis work well for predicting which length scales are important, but discrepancies are observed for growth rates. We observe an empirical scaling law for growth of the Fourier modes of the patterns in the linear regime. The observed front propagation velocity for side-wall disturbances is constantly 2\ifmmode\pm\else\textpm\fi{}1 in dimensionless units, a value consistent with the predictions of Langer and of van Saarloos. Patterns in both the linear and nonlinear regimes collapse impressively under the scaling suggested by the Hele-Shaw equations. Violations of scaling due to wetting phenomena are not evident here, presumably because the wetting properties of the two phases of the critical binary liquid are so similar; thus direct comparison with large-scale Hele-Shaw simulations should be meaningful.

The influence of surface tension on the diffusion-controlled growth or dissolution of spherical gas bubbles

Surface tension can markedly affect the growth or dissolution of small gas bubbles but, even when spherical symmetry is maintained and the interfacial concentration assumed constant, generally valid analytical solutions for the change of size with time cannot be obtained; approximations of limited validity are therefore often used. However, accurate and efficient methods for computing the diffusion-controlled growth or dissolution of spheres in such conditions have recently been developed; the results obtained when interfacial solute concentration is assumed constant have been published and validated by comparing them, where possible, with the equivalent analytical solutions. This paper extends that work to include the effects of a constant surface tension, for both Henry’s and Sievert’s laws. The numerical results are valid for as wide a range of parameters as is likely to be needed and are compared with the few analytical solutions available for particular cases. It is shown that, when surface tension is introduced, a complete description of the problem requires an additional saturation parameter as well as the dimensionless surface tension. It is also shown that Henry’s and Sievert’s laws can have very different effects on dissolving bubbles: when Sievert’s law applies a higher surface tension can increase time for a bubble to dissolve but this paradox can be resolved by examining the competing effects involved.

Saffman-Taylor finger width at low interfacial tension

High-accuracy numerical results are shown which go against current conjecture that dimensionless finger width \ensuremath{\lambda} in the Saffman-Taylor problem is a monotonically increasing function of dimensionless surface tension \ensuremath{\tau}, asymptotically approaching 1/2 as \ensuremath{\tau}\ensuremath{\rightarrow}0. Specifically, quasi steady-state fingers are shown for finite \ensuremath{\tau} with \ensuremath{\lambda} significantly below 0.5. These solutions are consistent with published experimental results obtained in a Hele-Shaw cell, indicating that the Saffman-Taylor equations are sufficient to explain this phenomenon.

Prediction of aerosol concentrations resulting from a burst of nucleation

The number of particles formed by a burst of homogeneous nucleation in a closed system is predicted approximately and found to depend, in the absence of initial aerosol, only on the dimensionless source rate of vapor, a dimensionless surface tension, and an appropriately defined Knudsen number. The effect of a seed aerosol on the ultimate number of particles formed is also studied. For the purposes of exploring the nature of systems in which a competition for vapor exists between nucleation and condensation, a closed system with a steady source of vapor is considered. It is shown that a narrow window exists within which seed particles can be used to control the ultimate number of particles formed in such a system.

Droplet theory of correlations in ordered phases

The explicit droplet theory of low-dimensional phase transitions, developed recently, is extended to yield a description of the pair correlation function for the universality class of q-state Potts models in the ordered phase and in zero external field. It is shown that, in d=1+ epsilon dimensions, the short distance behaviour of the correlation function and in particular, the exponent eta , are controlled by nested nearly spherical droplets. In contrast, it is argued that the spatial dependence of the large distance behaviour, in d=2 or above, is controlled by highly anisotropic droplets, thus illuminating the Widom relation, sigma xi d-1 approximately= constant, linking the dimensionless surface tension sigma and the bulk correlation length xi . In the particular case of d=2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings, in the spirit of recent independent, arguments by Abraham and by Fisher: a non-Ornstein-Zernike correlation function prefactor, and the result of sigma xi =1/2 follow in accord with exact results for the q=2 (Ising) case, and with implications for the q to 1 (percolation) problem.