A kinetic model for droplet growth in the transition regime

A variant of the moment expansion method, used in an earlier paper to describe the flow of a gas toward an absorbing sphere, is applied to a more realistic model of a droplet condensing from a supersaturated vapor. In the simplest version a spherical droplet absorbs all incoming vapor molecules, but spontaneously emits molecules with a Maxwellian distribution at the droplet temperature and with the corresponding saturated vapor density. From a solution of the stationary linearized Boltzmann equation with these boundary conditions we obtain expressions for the heat and mass currents toward the sphere as a function of the supersaturation and the temperature difference between the droplet and the vapor at infinity. For small droplet radii the known free flow limit is obtained in a natural way. From the calculated expressions for the heat and mass current we derive evolution equations for the radius and temperature of the droplet. The temperature evolves more rapidly and can thus be eliminated adiabatically; the resulting growth curve for the radius shows a sharp transition from a kinetically controlled regime for small radii to a regime dominated by heat conduction for large radii. The effect of incomplete absorption at the surface is also studied. The actual calculations are carried out for Maxwell molecules, with parameters corresponding to argon at 0.65T _c and 100% supersaturation.     

The influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.     

Kinetic boundary layers in gas mixtures: Systems described by nonlinearly coupled kinetic and hydrodynamic equations and applications to droplet condensation and evaporation

We consider a mixture of heavy vapor molecules and a light carrier gas surrounding a liquid droplet. The vapor is described by a variant of the Klein-Kramers equation, a kinetic equation for Brownian particles moving in a spatially inhomogeneous background; the gas is described by the Navier-Stokes equations; the droplet acts as a heat source due to the released heat of condensation. The exchange of momentum and energy between the constituents of the mixture is taken into account by force terms in the kinetic equation and source terms in the Navier-Stokes equations. These are chosen to obtain maximal agreement with the irreversible thermodynamics of a gas mixture. The structure of the kinetic boundary layer around the sphere is then determined from the self-consistent solution of this set of coupled equations with appropriate boundary conditions at the surface of the sphere. For this purpose the kinetic equation is rewritten as a set of coupled moment equations. A complete set of solutions of these moment equations is constructed by numerical integration inward from the region far away from the droplet, where the background inhomogeneities are small. A technique developed in an earlier paper is used to deal with the severe numerical instability of the moment equations. The solutions so obtained for given temperature and pressure profiles in the gas are then combined linearly in such a way that they obey the boundary conditions at the droplet surface; from this solution source terms for the Navier-Stokes equation of the gas are constructed and used to determine improved temperature and pressure profiles for the background gas. For not too large temperature differences between the droplet and the gas at infinity, self-consistency is reached after a few iterations. The method is applied to the condensation of droplets from a supersaturated vapor, where small but significant corrections to an earlier, not fully consistent version of the theory are found, as well as to strong evaporation of droplets under the influence of an external heat source, where corrections of up to 40 % are obtained.     

The theory of droplet diffusiophoresis for concentrated solutions

The theoretical diffusiophoresis velocity is obtained for a droplet of a concentrated solution suspended in a binary gaseous mixture. The droplet is characterized by a high thermal conductivity. The droplet radius is assumed to be much greater than the mean free path for gaseous-mixture molecules. One of the gaseous-mixture molecular components is the vapor of the droplet solvent. According to the formula obtained in this study, the droplet is driven toward lower concentration of the volatile gaseous-mixture component by diffusive slip and in the opposite direction by phase transition. An increase in the relative mass concentration of the volatile solvent in the droplet enhances effects associated with the dependence of surface tension on the volatile-component concentration and the reactive transport due to the surface nonuniformity of phase transition. As the relative mass concentration of the volatile solvent in the droplet approaches unity, the effect of diffusive slip tends to vanish.     

Light-induced evaporation and condensation growth of aerosol particles

We analyze the possibility of resonant optical radiation inducing evaporation and condensation growth of aerosol particles suspended in a vapor-gas mixture. The molecules of the vapor absorb radiation selectively as to velocity. We examine the Knudsen regime, in which the particle radius is much smaller than the mean free path of the molecules. Zh. éksp. Teor. Fiz. 113, 1036–1047 (March 1998)     

Heat transfer between a small sphere and a dilute gas; the influence of the kinetic boundary layer

The transfer of heat between an object not too large compared to a mean free path and a gas surrounding it is influenced considerably by the structure of the kinetic boundary layer around the object. We calculate this effect for a spherical object in a dilute gas by applying a recently developed variant of the moment method to solve the stationary linearized Boltzmann equation for the gas surrounding the sphere, From the solution we determine the temperature jump coefficient, which occurs in the boundary condition to be used for the heat conduction equation at the surface of the sphere. We study the dependence of this quantity on the radius and on the thermal accomodation coefficient. We find that typical boundary layer effects become less important as the accomodation coefficient decreases, and propose a simple approximate formula, which describes the results for large spheres to within about half a percent.     

The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary

The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance x_M beyond the wall. We find x_M=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.     

Diffusive growth of a single droplet with three different boundary conditions

We study a single, motionless three-dimensional droplet growing by adsorption of diffusing monomers on a 2D substrate. The diffusing monomers are adsorbed at the aggregate perimeter of the droplet with different boundary conditions. Models with both an adsorption boundary condition and a radiation boundary condition, as well as a phenomenological model, are considered and solved in a quasistatic approximation. The latter two models allow particle detachment. In the short time limit, the droplet radius grows as a power of the time with exponents of 1/4, 1/2 and 3/4 for the models with adsorption, radiation and phenomenological boundary conditions, respectively. In the long time limit a universal growth rate as is observed for the radius of the droplet for all models independent of the boundary conditions. This asymptotic behaviour was obtained by Krapivsky [#!krapquasi!#] where a similarity variable approach was used to treat the growth of a droplet with an adsorption boundary condition based on a quasistatic approximation. Another boundary condition with a constant flux of monomers at the aggregate perimeter is also examined. The results exhibit a power law growth rate with an exponent of 1/3 for all times. Received 19 July 1999     

Influence of the Prandtl and Knudsen numbers on heat-transfer process in the problem of planar Poiseuille flow

The analytic solution (in the form of the Neumann series) has been derived for the problem of computing the heat flux in a planar channel in the presence of a pressure gradient parallel with the walls (in the problem of planar Poiseuille flow) within the framework of the kinetic approach for arbitrary values of the Prandtl number. The ellipsoidal-statistical model of the Boltzmann kinetic equation is used as the governing equation, and the model of diffuse reflection is used as the boundary condition. The conducted numerical analysis of final expressions obtained in the present work showed a substantial dependence of the heat flux on the value of the Prandtl number of gas for channels whose thickness is comparable with the mean free path of gas molecules.     

Critical droplets near coexistence of wet and nonwet surface states

In first-order wetting transitions the decay of metastable states starts with a nucleation process on the wall of the system. For a planar wall the critical droplet has cylindrical symmetry and therefore should be described by two different critical lengths. We discuss these quantities and also the excess free energy of the critical droplet as functions of the spreading coefficient near coexistence of the wet and the nonwet state of the wall.     

The kinetic boundary layer for the Fokker-Planck equation: Selectively absorbing boundaries

We study the joint distribution function for position and velocity of a Brownian particle near a wall. The wall absorbs all particles that hit it with sufficiently high velocity and reflects all slower ones, either specularly or diffusely. We determine in particular stationary distributions in the absence of external forces. Appreciable deviations from local equilibrium occur in a kinetic boundary layer near the wall; its details depend strongly on the way in which the slow particles are reflected. The resulting effective absorption rate is calculated and compared with the result of approximations analogous to the transition state theory of chemical reactions. The method used is a generalization of the one used in an earlier paper for the case of a completely absorbing wall; a numerical algorithm based on an expansion of the distribution function in terms of a presumably complete set of boundary layer solutions.     

On the vortex formation at the moving front of lightweight granular particles

The formation of vortices at a moving front of lightweight granular particles is investigated experimentally. The particles used in this study are made of polystyrene foam with three different diameters of nearly uniform size. Pairs of vortices are found to emerge at the moving front at regular intervals, thereby forming a wavy pattern. Once the vortices are produced, the flow velocity tends to increase. A simple analysis suggests the existence of a velocity boundary layer at the moving front, whose thickness increases with increasing particle diameter. The frontal radius of each vortex pair is about the size of this boundary layer; when the radius exceeds this size, the front tends to bifurcate into a train of vortices with the size of the boundary layer. The formation of twin vortices leads to a reduction in the air drag force exerted on the system, and thereby the system attains a higher flow velocity, i.e., a higher conversion rate of gravitational potential energy to the kinetic energy of the particle motion. The higher conversion rate of potential energy thus feeds back to the development of the vortex motion, resulting in the twin vortex formation.     

Kinetic effects in an optically excited gas

A resonance transition in a one-dimensional layer of gas contained between transparent parallel plates and optically excited by external radiation has been treated using kinetic theory. A perturbation method has been used to obtain the “first scattering” results for the velocity distribution and number density of excited atoms and the intensity of radiation at any point in the gas. Two special cases are discussed in detail: broad band excitation with inhomogeneous broadening and monochromatic excitation with homogeneous broadening. The effects of particle streaming and wall quenching are shown to produce boundary layer behavior in the excited level density which scales with the particle mean free path. In addition, line reversal of the radiation reemited from the gas is shown to occur and to be a direct result of particle streaming. Numerical and asymptotic results are presented which show these effects. These results should be pertinent to many laboratory and industrial devices in which the particle and photon mean free paths are comparable and to diagnostic techniques which use resonance fluorescence to infer excited level densities.     

Asymptotic Behavior for Rayleigh Problem Based on Kinetic Theory

We investigate the dynamics of the gas bounded by an infinite flat plate which is initially in equilibrium and set at some instant impulsively into uniform motion in its own plane. We use the Boltzmann equation to describe intermolecular collisions and assume the diffuse reflection to describe the interaction of the gas with the boundary. The Mach number of the plate is assumed to be small so that we can linearize the Boltzmann equation as well as the boundary condition. We show that the asymptotic behavior of the gas represents a perturbation to the free molecular gas when the time is much less than the mean free time. On the other hand, if the time is much greater than the mean free time, we show that the gas dynamics is governed by the linearized Navier–Stokes equation with a slip flow on the boundary and establish a boundary layer correction with thickness of the order of the mean free path. We also establish the singularity of velocity distribution function along the particle trajectory near the boundary.     

高温壁面上静止液滴的Leidenfrost温度模型研究

本文以高温壁面上的静止液滴为研究对象,对其蒸发特性开展了理论研究.不同计算工况下得到的液滴蒸发过程中半径和蒸汽膜厚度变化与实验值吻合良好.结果表明随着壁面温度的降低,蒸汽膜厚度逐渐减小.结合表面粗糙度的影响,研究中提出了 Leidenfrost温度的触发机制:当蒸汽膜厚度足够小时,会极易被加热表面的不平整突起贯穿,蒸汽...     

Velocity statistics in excited granular media

We present an experimental study of velocity statistics for a partial layer of inelastic colliding beads driven by a vertically oscillating boundary. Over a wide range of parameters (accelerations 3-8 times the gravitational acceleration), the probability distribution P(v) deviates measurably from a Gaussian for the two horizontal velocity components. It can be described by P(v) approximately exp(-mid R:v/v(c)mid R:(1.5)), in agreement with a recent theory. The characteristic velocity v(c) is proportional to the peak velocity of the boundary. The granular temperature, defined as the mean square particle velocity, varies with particle density and exhibits a maximum at intermediate densities. On the other hand, for free cooling in the absence of excitation, we find an exponential velocity distribution. Finally, we examine the sharing of energy between particles of different mass. The more massive particles are found to have greater kinetic energy. (c) 1999 American Institute of Physics.     

过渡区微尺度流动的有效黏性多松弛系数格子Boltzmann模拟

为提高采用二维九速离散速度模型的格子Boltzmann方法 (LBM)模拟微尺度流动中非线性现象的精度和效率,引入Dongari等提出的有效平均分子自由程对黏性进行修正(Dongari N,Zhang Y H,Reese J M2011 J.Fluids Eng.133 071101);并针对以往研究微尺度流动时采用边界处理格式含有离散误差的问题,采用多松弛系数格子Boltzmann方法结合二阶滑移边界条件,对微尺度Couette流动和周期性Poiseuille流动进行模拟,并将速度分布以及质量流量等模拟结果与直接模拟蒙特卡罗方法模拟数据、线性Boltzmann方程的数值解以及现有的LBM模型模拟结果进行对比.结果表明,相对于现有的LBM模型,引入新的修正函数所建立的有效黏性多松弛系数LBM模型有效提高了LBM模拟过渡区的微尺度流动中的非线性现象的能力.