Plane Liquid Surfaces Research Articles

Linear oscillations of constrained drops, bubbles, and plane liquid surfaces

The small-amplitude oscillations of constrained drops, bubbles, and plane liquid surfaces are studied theoretically. The constraints have the form of closed lines of zero thickness which prevent the motion of the liquid in the direction normal to the undisturbed free surface. It is shown that, by accounting explicitly for the singular nature of the curvature of the interface and the force exerted by the constraint, methods of analysis very close to the standard ones applicable to the unconstrained case can be followed. Weak viscous effects are accounted for by means of the dissipation function. Graphical and numerical results for the oscillations of constrained drops and bubbles are presented. Examples of two- and three-dimensional gravity-capillary waves are treated by the same method. A brief consideration of the Rayleigh-Taylor unstable configuration shows that the nature of the instability is not affected, although its growth rate is decreased.

Hydrodynamic interactions of evaporating droplet with plane liquid surface

The hydrodynamic interactions between freely evaporating or growing droplet suspended in gas and the infinite plane liquid layer is theoretically studied with allowance for effects that are linear with respect to the Knudsen number. It is assumed that substances comprising droplet and liquid layer are identical, and it is taken into account that these substances can contain dissolved nonvolatile component whose concentration is so low that its presence should necessarily be accounted for only in the calculation of the concentration of saturated vapor. Vapor pressure at large distance from the droplet was assumed to be equal to the pressure of saturated vapor above the plane liquid surface. Results of numerical calculations of the rate of steady motion of water droplet evaporating or growing in the air are reported.

Observation of the focusing of liquid surface waves

The transmission properties of liquid surface waves through a biconvex array of rigid cylinders, arranged in a two-dimensional periodic lattice, are studied experimentally and numerically. Both experimental and numerical results show that this biconvex array of rigid cylinders can act as a lens, which can focus plane liquid surface waves in a wide range of frequencies.

混相流研究の進展 ４ 液滴・液膜衝突に及ぼす液膜厚さおよび衝突角度の影響

Two-dimensional interface tracking simulations and photographic experiments were carried out to investigate the effects of film thickness and impingement angle on the splashing event during single-drop impact onto a plane liquid surface. For sufficiently small values of liquid film depth, liquid originally included in the inner region of liquid crown was not transported to the outer region. As a result, the pressure rise at the impact neck became significant, a thin liquid crown was formed, and the crown wall was broken into many secondary drops. Numerical results showed that the pressure rise induced on the bottom wall by drop impact is significantly mitigated if the impacted wall is covered with the liquid film whose thickness is comparable to the primary drop size. With respect to the effect of impingement angle, pressure rise in liquid was equally dependent on the normal and tangential components of impact velocity for slightly and moderately oblique impacts; however, the pressure rise was mitigated for very oblique impacts.

Air entrainment upon liquid impact

The impact of round and plane liquid jets on plane liquid surfaces is simulated numerically on the basis of a potential flow model. It is found that the mass of the jet is as important as its momentum in determining the flow. For sufficiently long jets and small Froude numbers air cavities are entrained. Entrainment is more unlikely in two dimensions than in three.

Systematics of pure and doped 4He clusters.

Optimized variational calculations have been carried out for pure and doped clusters of $^{4}\mathrm{He}$ atoms up to a cluster size of N=1000 particles. For small cluster sizes with less than or equal to 112 particles, where comparisions with existing diffusion Monte Carlo results are possible, we find excellent agreement for the ground-state energy, correlation, and structure functions. For larger clusters our ground-state energies extrapolate smoothly toward a bulk limit of -7.2 K with a surface energy of 0.272 K A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}2}$. The resulting ground-state densities show unmistakable oscillations, confirming our earlier conclusions based on diffusion Monte Carlo studies. The present study of large clusters allows us to bridge the gap between finite systems and the bulk limit. Specifically, we show how the bulk limit of collective energies is reached as well as how the bulk Feynman spectrum is reproduced in the S-wave component of the dynamic structure function in large droplets. By plotting the collective excitation energy of higher multipole modes as a function of an effective wave number k=\ensuremath{\surd}scrl(scrl+1)/R, we show that the resulting spectrum can be directly compared with experimental excitation energies determined for plane liquid surfaces and films. By summing up to scrl=50 partial wave components, we show that the full dynamic structure function simultaneously displays the phonon-roton and the ripplon excitation spectrum. In the case of helium droplets doped with impurities such as rare gas atoms or the ${\mathrm{SF}}_{6}$ molecule, we show that the dipole collective mode becomes unstable with increased droplet size, strongly indicating that these impurities are delocalized inside large droplets. The microscopic character of the instability is revealed in the excitation functions and transition densities of the dipole mode. The introduction of impurities also profoundly alters the dynamic structure function, severely ``fragments'' the Feynman spectrum, and obliterates landmark structures such as the maxon and the roton.

Surface excitations of helium droplets.

We apply an optimized variational theory to the study of high angular vibrational modes of helium droplets. Collective excitation energies for angular modes $\ensuremath{\ell}>2$ are computed for cluster sizes $N=20$ to $N=1000$ on the basis of Feynman's ansatz. We show that the resulting spectrum, when plotted as a function of an effective wave number $k=\frac{\sqrt{\ensuremath{\ell}(\ensuremath{\ell}+1)}}{R}$, forms a universal excitation curve insensitive to the size of the droplet and in excellent agreement with theoretical and experimental excitation energies determined for plane liquid surfaces and films. This agreement, in turn, allows us to use experimental film and surface data to infer about the excitations of helium droplets.

Steady-state heat and matter exchange at a phase interface

Equations are derived for the steady-state heat and matter fluxes through the boundary layer at a general phase interface; their range of validity is discussed, and the importance of the timescale of the observations is emphasized. Where only diffusive matter fluxes and conductive heat fluxes are involved, the results derived from irreversible thermodynamics remain valid for large departures from equilibrium. The theory includes the case where the location of the boundary layer is marked by an obstruction such as a membrane: the properties of the obstruction control the magnitudes of the fluxes but do not affect the thermodynamic forces. The theory is applied to transport processes at a plane liquid surface and at the surface of an aerosol droplet.

On the temperature distribution in liquid–vapor phase change between plane liquid surfaces

The problem of temperature distribution in liquid-vapor phase change between plane liquid surfaces is considered both analytically and experimentally. Classical kinetic theory calculations indicate that for a monatomic vapor, large temperature jumps exist at the interfaces, large enough possibly to cause the temperature distribution to be anomalous. Surprisingly, no experimental results have been reported so far. A continuum analysis, with special consideration of the Knudsen layers, clearly establishes the adverse roles played by polyatomic molecules and the presence of a contaminant gas. The need to work with monatomic vapors under low levels of contamination is clearly indicated. An experimental setup was constructed to measure temperature distributions in the vapor between an evaporating liquid surface and a cooler condensing surface. Temperature distributions were measured using Chromel–Alumel thermocouples. Measurements were made in water, Freon 113, and mercury. The temperature profiles obtained in mercury, over a wide range of temperatures, clearly show large temperature jumps at the interfaces, as large as almost 50% of the applied temperature difference. These results appear to be the first to confirm this important phenomenon.

Further verification of a theory for mass and heat transfer from evaporating pools

In an earlier paper (by the present author), a theory was described for predicting evaporation rates from plane liquid surfaces, or heat transfer rates from a region of uniform temperature rise in a smooth or rough plane surface, into a neutral turbulent boundary layer. In this paper the theory is compared with seven further sets of data. Without any modifications, the theory works for laboratory conditions to a good degree of accuracy. Application to the conditions of accidental chemical spills in the open air introduces more uncertainty for various reasons, but gives results that should be acceptable for hazard assessment purposes, provided the terrain is flat and unobstructed.

Evaporation from a plane liquid surface into a turbulent boundary layer

A new analytic solution is presented for predicting evaporation rates from plane liquid surfaces into a neutral turbulent boundary layer. Conditions of passive dispersion are assumed. Molecular diffusivity is incorporated into the boundary conditions. Both smooth and rough surfaces are considered. A comparison with a wide variety of experimental data is made; this tends to reveal inadequacies and inconsistencies in the data, rather than test the theory. The effects of a roughness change at the boundary of the liquid surface and of high vapour pressures can be included for practical purposes by simple formulae. A criterion is derived for the validity of the neglect of buoyancy effects.

Nozzle design for the generation of plane liquid surfaces.

Measurements besides total fluorescence intensity of more elaborate optical techniques such as light scattering, extinction as well as fluorescence polarization and slit scan profiles have been reported in various cell sorter design studies. However the cylindrical liquid stream in air, common to many research system, limits the optical properties of the system. We report the successful application of three new types of nozzles, all having in common the fact that the main optical axes intersect plane liquid surfaces. This has been achieved on the basis of a double sheath arrangement, which we have reported previously. The nozzles generate fluid streams with either rectangular, square or triangular cross-sections at the location of the optical axis. The latter facilitates the alignment of two beams with different wavelengths on the same optical axis within the liquid. Square or rectangular cross-sections improve light scattering and extinction measurements and may well be better suited to more complicated geometrical analysis of cells and particles in flow streams.

Concentration jump on the plane surface in a binary gas mixture

From the solution of linearized Boltzmann equations with the collision operators in the Gross and Krook form the authors have obtained an expression for the concentration jump of a vapor evaporating from a plane liquid surface into a stationary gas, for which the liquid surface is impermeable. Langmuir [1] was the first to point out that at the surface of an evaporating liquid the vapor pressure differs from saturated vapor pressure, i.e., a jump occurs in vapor concentration. An expression was obtained by Brock [2] for the concentration jump at the plane surface in the binary gas mixture with identical molecular masses of components by gas kinetic analysis. No solution has yet been obtained to the problem at different molecular masses of components.

A Theory of Liquid Surface Rupture by a Uniform Electric Field

Surface distortion and rupture permits field emission from liquid surfaces at field strengths less than those effective for equally smooth solid surfaces. An approximate mathematical theory of the rupture of a plane liquid surface in a uniform electric field has been developed. The relation between initial distortion, rupture time, and field strength has been calculated for fields large compared to that which just renders the surface unstable. This critical field is ${E}_{m}=2{\ensuremath{\pi}}^{\frac{1}{2}}{(\ensuremath{\rho}gT)}^{\frac{1}{4}}=53$ kv ${\mathrm{cm}}^{\ensuremath{-}1}$ for mercury, where $\ensuremath{\rho}$ is density and $T$ is surface tension. Typically, the theory shows that a hump initially 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ cm high and of diameter 9\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ cm will lead to rupture in 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ sec. in a field of ${10}^{6}$ v ${\mathrm{cm}}^{\ensuremath{-}1}$. Relative to initial humps in the surface whose linear dimensions vary inversely as the square of the field, the time to rupture varies inversely as the cube of the field strength. This calculation shows that a lowered sparking potential to liquid mercury can be ascribed to surface rupture and shows that it is possible that surface rupture plays a part in Beams' low field emission from liquid mercury. Possible application of the theory to the high field condition at the cathode spot of the Hg arc is not clear.