Shape and stability of a liquid drop moving at low Weber numbers

The equilibrium cross sectional shape and stability of a liquid drop moving in a gaseous medium is studied analytically. Such liquid drops appear as the final product in numerous industrial spraying and atomisation processes. Raindrops falling at their terminal speed can also be described by the present model. The equilibrium shape is formed by the interaction of two main factors; the dynamic pressure distribution in the gaseous medium which tends to deform the liquid drop into an oblate shape and the surface tension which tends to restore the spherical shape. The meridional shape of the liquid drop is obtained as a power series in the Weber number. The linear stability of the deformed shapes described above to small surface disturbances is studied. The stability analysis shows the effect of the surrounding gas flow on the natural frequencies of oscillation (vibration) of the liquid drop. The liquid drop is found to be stable in the region of low Weber numbers studied with a decrease in oscillation frequency proportional to the Weber number. This is in agreement with existing experimental data. Extrapolation of the results here lead to a Weber number of W=5.33 for breakup, again in agreement with experimental correlations.     

Gravity-induced motion of a uniformly heated solid particle in a gaseous medium

The steady motion of a uniformly heated spherical aerosol particle in a viscous gaseous medium is analyzed in the Stokes approximation under the condition that the mean temperature of the particle surface can be substantially different from the ambient temperature. An analytical expression for the drag force and the velocity of gravity-induced motion of the uniformly heated spherical solid particle is derived with allowance for temperature dependences of the gaseous medium density, viscosity, and thermal conductivity. It is numerically demonstrated that heating of the particle surface has a significant effect on the drag and velocity of gravity-induced motion. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 74–80, January–February, 2008.     

Motion of a drop with allowance for thermocapillary forces

A study is made of the motion of a drop in a viscous medium of nonuniform temperature when the dependence of the surface tension on the temperature at the interface of the two media gives rise to additional tangential stresses leading to thermocapillary convection in fluids. The motions of a spherical drop and a deformed drop (when the surface of the drop is determined in the process of finding the solution) are considered. The shape of the drop surface is calculated for different values of the parameters of the problem. Dependences are obtained for the Reynolds number of the thermocapillary drift of drops in the absence of body forces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 80–86, July–August, 1982.     

Thermophoretic motion of large heated aerosol spherical particles

The stationary motion of a large spherical aerosol particle in the external field of a temperature gradient in zero gravity is theoretically described using the Stokes approximation and the assumption that the average temperature of the particle surface differs considerably from the temperature of the surrounding gaseous medium. The gas dynamics equations are solved taking into account the power-law temperature dependence of the molecular transport coefficients (viscosity, thermal conductivity) and the density of the gaseous medium. Numerical estimates show that the dependence of the thermophoretic force and velocity on the average temperature of the particle surface is nonlinear.     

Development of a high performance flexible porous burner(FPMB) with an adjustable cooling effect

A high performance flexible porous medium burner that can burn gaseous and liquid fuel with different type of flames(premixed and non-premixed) is proposed. The merit of the combustion within porous medium is that heat is recirculated from the combustion gas to porous medium at upstream wherein vaporization is taken place(in case of liquid fuel) or preheated(in case of gaseous fuel) before mixing with the combustion air followed by combustion within another porous medium at downstream. In a former version of the high performance flexible porous medium burner, the upstream porous medium is incorporated with a cooling system using the combustion air as a coolants to prevent thermal decomposition of fuels and thus the burner clogging caused by carbon deposit within the porous medium can be avoided. However, the cooling effect cannot be properly controlled such that the boiling point of the liquid fuel is maintained at suitable value irrespective of the volume flow rate of the combustion air,which is linearly varied with the firing rate of the burner. In particular at the lean burn condition, where high air flow rate is required with high cooling effect with porous medium. This can result in the porous medium temperature lower than the corresponding boiling point of the liquid fuel and thus evaporation of the fuel is failed and the combustion is ceased. Therefore, method of controlling the cooling air flow rate in the porous medium is proposed and studied in order to appropriately control the porous medium temperature and maintain it at above the boiling point irrespective of the combustion conditions. In this research, experimental and computation analysis are used to design the flexible porous burner(FPMB),with adjustable cooling effect. The result shows that, the new design of FPMB which has temperature in the upstream porous medium is higher than boiling point and lower than thermal decomposition temperature of fuel(kerosene) at all conditions and can be operated at a wide range of equivalence ratio without fuel decomposition and fuel non-vaporization problem.     

Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium

The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.     

Convection in a porous medium with velocity slip and temperature jump boundary conditions

The onset of convection in a rarefield gas saturating a horizontal layer of a porous medium has been investigated using both Darcy and Brinkman models. It is assumed that due to rarefaction both velocity slip and temperature jump exist at the boundaries. The results show that (i) when the degree of rarefaction increases the critical Rayleigh number as well as the critical wave number for the onset of convection increases, (ii) stabilizing effect of temperature jump is more than that of velocity slip, (iii) Darcy model is seen to be the most stable one when compared to Brinkman model or the pure gaseous layer (i.e. in the absence of porous medium).     

横向气流穿过垂直下落液滴场的二维流动数值计算

对Cl2/He混合气体横向穿过垂直下落的BHP(按重量25%的KOH,25%的H2O2及50%的H2O)液滴场的化学反应流动作了数值计算.模拟的流场是气体/液滴两相流场,在气相方程中,考虑了液滴与Cl2反应产生及释放O2(1Δ)的质量源项及表示液滴对气流阻碍作用的动量源项.由于气相动量小,液滴在下落过程横向偏移小,下落...     

Motion of a gas mixture through a porous medium with sorption

Solutions are investigated of a system of linear partial differential equations describing the motion of a gaseous (liquid) mixture through an undeformable homogeneous porous medium with sorption at interfaces between gaseous (liquid) and solid phases, the kinetics of which are described by a linear equation. If the porous medium consists of spherical granules, the problem is solved in quadratures. For the case of symmetric granules with arbitrary symmetry parameter, various approximate solutions are obtained; first and central moments are used as criteria for the accuracy of the approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–100, September–October, 1970.     

Coalescence and break-up of drops in two-phase flows

The systematic experimental study of physical phenomena taking place at collisions of drops of water, water-glycerine solutions and transformer oil moving with moderate and high relative velocities has been carried out. The cases of drops interaction of one fluid and various fluids are considered. The regularities of drops collisions both in the quiescent and the moving gaseous medium have been studied. It has been stated that interaction is almost always accompanied by breaking a large drop with forming a certain amount of polydisperse fragments. The generalizing formulae are obtained for the parameter of coalescence and break-up Φ_ji, as well as for the function of fragments distribution by their size and initial velocity.     

Mass transfer from a pulsating drop

The primary difficulty in solving the problem of mass transport through an isolated drop (or bubble) moving in a fluid medium at high Reynolds numbers lies in the extreme complexity of the hydrodynamic pattern of the phenomenon. For sufficiently high velocities a separation of the external flow will occur in the rear portion of the drops and bubbles, which leads to the appearance of a turbulent wake and a sharp increase of the hydrodynamic resistance. Beginning with those dimensions for which the resistance force acting per unit surface of the drop or bubble from the external medium becomes greater than the capillary pressure, the surface of the drops and bubbles begins to deform and pulsate. The local variations of the surface tension, resulting either from the process of convective diffusion or from adsorption of surface-active substances, have a large effect on the hydrodynamics of drops and bubbles (particularly on the deformation of their surface) [1, 2], The presence of vortical, and possibly even turbulent, motion within the drops and bubbles may, under certain conditions [1], lead to their fractionation.Naturally, at the present time such complex hydrodynamics cannot be described by exact quantitative relations. Several authors have attempted to solve this problem approximately within the framework of certain assumptions. In particular [3–6], a theory was developed for the boundary layer on the surface of spherical and ellipsoidal gaseous bubbles moving in a liquid, studies were made [7, 8] of the hydrodynamics of drops located in a gas flow and the conditions were found for which fractionation of such drops takes place. Of considerable practical interest is the development of the theory of mass transfer in pulsating drops and bubbles and finding in explicit form the dependence of the mass transfer coefficients on the hydrodynamic characteristics of these systems. Until this relationship is established, every theory which ignores the effect of hydrodynamics on the mass transfer rate from an individual drop or bubble cannot be considered in any way well-founded. This relates particularly to the theories [9, 10] which consider mass transfer in systems with concentrated streams of drops and bubbles. The present paper is devoted to the study of mass transport through the surface of an isolated drop in an irrotational gas or liquid stream for large Peclet numbers P.In conclusion the authors wish to thank V. G. Levich for his helpful discussions.     

Effect of interface deformability on thermocapillary motion of a drop in a tube

The effect of an externally imposed axial temperature gradient on the mobility and deformation of a drop in an otherwise stagnant liquid within an insulated cylindrical tube is investigated. In the absence of bulk transport of momentum and energy, the boundary integral technique is used to obtain the flow and temperature fields inside and outside the deformable drop. The steady drop shapes and the corresponding migration velocities are examined over a wide range of the dimensionless parameters. The steady drop shape is nearly spherical for dimensionless drop sizes <0.5, but becomes slightly elongated in the axial direction for drop sizes comparable to tube diameter. The adverse effect of drop deformation on the effective temperature gradient driving the motion is slightly more pronounced than its favorable effect of reducing drag, thereby leading to a slight reduction in drop mobility with increasing drop deformation. Increasing the viscosity ratio reduces drop deformation and leads to a slight enhancement in the relative mobility (with respect to free thermocapillary motion) of confined drops. When the drop fluid has a lower thermal conductivity than the exterior phase, the presence of the thermally-insulating wall increases the thermal driving force for drop motion (compared to that for the same drop in unbounded domain) by causing more pronounced bending of the isotherms toward the drop. However, the favorable thermal effect of the confining wall is overwhelmed by its retarding hydrodynamic effect, causing the confined drop to always move slower than its unbounded counterpart regardless of the value of the thermal conductivity ratio.     

Critical radius of explosive gaseous mixtures with initial concentration gradients

A synthetic review of works concerning detonation of non-uniform explosive mixtures allows to retain a criterion of detonation onset. This criterion is due to Zel'dovich and then Makhviladze. The present study proposes to adapt Makhviladze's criterion to define the formation point of a quasi-stationary detonation in spatially non-uniform, non-stationary and non-preheated gaseous mixtures. The temperature increase of gaseous mixture is due to the travelling shock. This point is compared to the critical radius defining the distance for which the Chapman-Jouguet conditions are attained. The detonability of a non-uniform medium (propane/oxygen mixtures) is characterized by a small scale experimental investigation and by application of a simple advection-diffusion model. Accepted 3 March 1996     

Transfer of energy from an evaporating drop in a vapor medium

The article considers the temperature distribution around an evaporating drop in a vapor medium. The transfer of energy is effected by molecular thermal conductivity, convection, and radiation. The mean length of the free flight path of the radiation considerably exceeds the characteristic distance at which the temperature changes. The times required for relaxation of the temperature to a steady-state value are determined, as well as the characteristic distances at which the temperature distribution changes.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 74–78, January–February, 1972.The authors thank V. G. Levich for his evaluation of the results obtained.     

Rigorous derivation of the effective model describing a non-isothermal fluid flow in a vertical pipe filled with porous medium

This paper reports an analytical investigation of non-isothermal fluid flow in a thin (or long) vertical pipe filled with porous medium via asymptotic analysis. We assume that the fluid inside the pipe is cooled (or heated) by the surrounding medium and that the flow is governed by the prescribed pressure drop between pipe’s ends. Starting from the dimensionless Darcy–Brinkman–Boussinesq system, we formally derive a macroscopic model describing the effective flow at small Brinkman–Darcy number. The asymptotic approximation is given by the explicit formulae for the velocity, pressure and temperature clearly acknowledging the effects of the cooling (heating) and porous structure. The theoretical error analysis is carried out to indicate the order of accuracy and to provide a rigorous justification of the effective model.     

Analysis of Nitrogen and Steam Injection in a Porous Medium with Water

We formulate conservation laws governing steam and nitrogen injection in a one-dimensional porous medium containing water. Compressibility, heat conductivity and capillarity are neglected. We study the condensation front and shock waves arising in the flow. We find that there are four possible types of solutions for the initial and boundary conditions of interest. We describe a simple construction in the temperature saturation plane that determines the complete solution for the given conditions. Applications of the theory developed here are in clean up of soil contaminated with nonaqueous phase liquids. We show that a substantial cold gaseous zone develops in all solutions of practical interest, thus counteracting downward migration of the pollutant.     

Effects of Temperature-Dependent Viscosity on Forced Convection Inside a Porous Medium

Considering the exponential viscosity–temperature relation, effect of temperature-dependent viscosity on forced convection of a liquid through a porous medium, bounded by isoflux parallel plates, is investigated numerically based on the general model of momentum transfer. Local effects of viscosity variation on the distribution of velocity and temperature are analyzed. Moreover, global aspects of the problem are investigated where corrections are proposed for total pressure drop and the fully developed Nusselt number, in the form of out/in viscosity ratio. Results are obtained over a wide range of permeabilities from clear (of solid material) fluid to very low permeability, where for constant properties one expects a nearly slug flow.     

Experimental and theoretical studies of convective heat transfer in a cylindrical porous medium

Convective heat transfer at constant heat flux through unconsolidated porous media has been studied both experimentally and theoretically. Heat transfer measurements have been performed for convective heat transfer over a wide range of operational parameters at constant heat fluxes. In addition to heat transfer coefficients, pressure drop and temperature profiles both in radial and axial direction have been recorded. The equations of motion and energy which account for the non-Darcian effect are used to describe the flow and convective heat transfer through the porous medium. Mathematical models for the prediction of heat transfer coefficients and temperature profiles are presented which predict the experimental data with good accuracy.     

Stability and decay of a cylindrical film of liquid in a gaseous medium

Close to the orifice the film of liquid flowing from the nozzle of a pressure jet atomizer is approximately cylindrical in shape. Usually it also decays close to the nozzle and for a preliminary theoretical study it is convenient to formulate the problem of the stability of a cylindrical film of liquid moving in a stationary gaseous medium.     

Thermocapillary migration of a fluid droplet inside a drop in a space laboratory

The thermocapillary migration of a fluid droplet located inside a liquid drop in a space laboratory is analyzed. The quasi-static momentum and energy equations are solved at the instant when the droplet passes the center of the drop. Results are presented for prescribed axisymmetric distributions of temperature on the drop surface.