Convective diffusion to a drop under arbitrary conditions of absorption. Diffusion boundary layer approximation

The diffusion boundary layer approximation is used to investigate the stationary convective diffusion of substance dissolved in a flow to a spherical drop under arbitrary conditions of absorption on its surface, in particular when a chemical reaction of arbitrary order takes place on the surface. An integral equation is obtained for the local diffusion flux to the surface of the drop. It is shown that 1) the total Sherwood number increases with increasing rate of the reaction and decreases with increasing exponent of the reaction rate; 2) with increasing Péclet number, saturation occurs (i.e., the total diffusion flux to the surface of the drop tends to a limiting value, which depends only on the reaction kinetics). The case of total absorption of diffusing, substance on the surface of reacting solid and liquid particles in a homogeneous Stokes flow at large Péclet numbers was investigated in [1]. The problem of convective diffusion to the surface of a solid spherical particle in the case of mixed kinetics was considered in [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 64–69, November–December, 1979.     

Mass transfer between particles and a flow in the presence of a volume chemical reaction

The exterior problem of the mass transfer between a spherical drop and a linear shear flow in the presence of a first-order volume reaction is solved in the diffusion boundary layer approximation. A simple approximate expression for calculating the average Sherwood number for a drop or solid particle of arbitrary shape is proposed. At large Péclet numbers this expression is applicable to any type of flow over the entire range of variation of the reaction rate constant. The problem of diffusion to a spherical drop in a translational Stokesian flow in the presence of a first-order volume reaction was investigated in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 109–113, November–December, 1987.     

The Onset of Bioconvection in a Horizontal Porous-Medium Layer

In this note the problem of the onset of bioconvection in a horizontal layer occupied by a saturated porous medium is analyzed. Gyrotactic effects are incorporated in the analysis. The Darcy flow model is employed, and it is assumed that the bioconvection Péclet number is not greater than unity. Critical values of the bioconvection Rayleigh number and the corresponding critical Rayleigh number are obtained for various values of the bioconvection Péclet number, the gyrotaxis number and the cell eccentricity.     

Mass transfer problem for particles,drops and bubbles in a shear flow

A numerical solution of the axisymmetric steady heat and mass transfer problem for spherical particles, drops and bubbles in a linear Stokes shear flow is obtained for the entire range of Péclet numbers. Simple approximate expressions for the average Sherwood number in good agreement with the results of the numerical calculations are proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–141, July–August, 1990.     

Motion of a bubble in a viscous liquid

The discussion concerns steady-state flow of a viscous fluid around a spherical bubble at small Reynolds number R. Asymptotic matching [1] provides a way of calculating the resistance force, which agrees well with the measured force for R < 5. The rate of growth or dissolution of the bubble is calculated on the assumption that the Péclet number is large.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 107–111, January–February, 1971.We are indebted to V. G. Levich for a discussion.     

Operating regimes of a chemical reactor with longitudinal and transverse mixing in the case of large Péclet numbers

A two-dimensional model of a chemical reactor with longitudinal and transverse mixing is investigated in the case of large Péclet numbers calculated from the effective thermal conductivity in the transverse direction. For this model the existence of at least one steady-state regime has been demonstrated [1], sufficient criteria of its uniqueness have been determined, an asymptotic expansion of the solution has been constructed in the case of small Péclet numbers, and the critical ignition and quenching parameters have been found. In this paper the other limiting case of the model, in which heat is propagated in the transverse direction much more slowly than it is transported by the flow along the reactor (large Péclet numbers), is analyzed in detail. An asymptotic expansion of the solution which closely coincides with the data of numerical calculations is constructed. The critical quenching and ignition conditions of the process are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–127, January–February, 1987.     

Surface activity and bubble motion

This paper reviews recent progress in the theories of the surface boundary conditions of adsorbed solutes in liquids, and of the effects of those solutes on the steady motion of a bubble or drop in the liquid. Both singular perturbation theory and numerical solutions have useful roles in this problem, and their relationship is explored. In addition, analytical solutions are given to two problems concerning a spherical bubble rising steadily at low Reynolds number in a viscous fluid. One of these is displacement of the internal vortex centre from its position in the absence of surface activity when there is a small stagnant cap of surfactant at the rear. The results agree with experimental data in the direction of that displacement but give only about half its amount. The other problem is the velocity perturbation all round the surface caused by a very dilute solution of a weak surfactant at high Péclet number. This compares quite well with the numerical solution for a Péclet number of 60, having relative errors of order (60)^–1/2 as would be expected.     

Steady thermocapillary motion in a horizontal layer of liquid metal locally heated from above

The article considers stationary thermocapillary convection in a thin horizontal layer of fluid with Prandtl number Pr < 1 when it is being locally heated from above in conditions in which the curvature of the free surface is small. It is shown that the motion has a cellular structure. The size of the convective cell is determined from the solution to the spectral problem to which the integration of the free convection system of equations reduces. If the Maragoni (Péclet) number is sufficiently high, the length of the convective cell turns out to be large in comparison with the thickness of the layer. The convection picture is considered and an expression obtained for the velocity of the developing flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 146–152, November–December, 1984.     

Macroscopic simulation of relaxation mass transport in a porous medium

The problem of the macroscopic simulation of the motion of a viscous fluid and mass transport in a porous medium is considered under the assumption that the mass transport can locally be described by the Fick relaxation law. Several cases determined by the local inertia number of the mass flow and the Péclet number are investigated. The macroscopic transport models are analyzed and compared with well-known phenomenological models.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 21–30.Original Russian Text Copyright © 2004 by Khuzhayorov.     

Some general invariance relations in problems of convective heat and mass transfer at large péclet numbers

Some general invariance relations are obtained for the integral diffusion fluxes of the reactant on the surface of one or several reacting particles of arbitrary shape in Stokes flow of a viscous incompressible fluid around the particles at large Péclet numbers. The case of irrotational flow is also considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 92–97, October–December, 1981.     

Mass transfer to particle clusters and bubbles in a fluidized bed at low Reynolds numbers

The process of mass transfer to a particle cluster or bubble rising in a developed fluidized bed rapidly enough for a region of closed circulation of the fluidizing agent (cloud) to be formed is investigated in the Stokes approximation on the basis of a model of the steady-state motion of the fluid and solid phases near the cluster or bubble [1]. Within the cloud surroundinga local inhomogeneity of the fluidized bed intense mixing of the fluid phase takes place and the mass transfer between the cloud and the surrounding medium is determined by diffusion. The method of matched asymptotic expansions is used to obtain an analytic solution of the problem of the concentration field and the diffusion mass flux to the surface of the cloud at small and large values of the Péclet number. The latter is determined from the relative velocity of the cluster, the radius of the cloud, and the effective diffusion coefficient. In the limiting case of zero concentration of the solid phase within the cluster the solution obtained describes the mass transfer to a bubble in the fluidized bed. A comparison is made with the corresponding results previously obtained within the framework of a model of the solid phase as an inviscid fluid [2]. It is shown that the effect of viscosity on the mass transfer to the bubble is most important at large Péclet numbers, and that the correction to the total diffusion flux to the surface of the closed circulation zone due to viscosity effects may reach 40%.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 60–67, July–August, 1986.     

Maximum dimension of ice-rock bodies formed in liquid flow past a linear chain of cold sources

An asymptotic analysis of the limiting situation of liquid flow past a linear chain of cold sources when an ice-rock body with constant cross section grows without limit along the flow is carried out. The dependence of the critical heat flow rate of the cold sources and the transverse dimension of the body on the Péclet number is determined.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 179–181, July–August, 1996.     

Electrization of dielectric liquids flowing in tubes

The methods of the mechanics of continuous media [1] are used to consider the problem of electrization of dielectric liquids flowing in tubes [2–6]. According to modern ideas [2–6], there is always dissolved in such liquids a slight admixture of an electrolyte, whose molecules in such a dilute solution dissociate to a certain extent into positively and negatively charged ions. On the walls, oxidizing and reducing reactions take place, as a result of which the negative and positive ions, respectively, give up to the wall surplus electrons or take missing electrons from it. Thus, a positive (respectively, negative) total electric charge is induced in the liquid by the flow. We consider in this paper the electrization of a dielectric liquid in laminar flow in a circular cylindrical tube. We find the distribution of the electric charge in the liquid, the maximal electric current, and the dependence of the length over which the distribution of the electric charge in the tube is established on the tube radius, the Debye radius of the liquid, and the Péclet diffusion number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 41–47, November–December, 1979.We thank V. V. Gogosov for helpful comments made in a discussion of thwe work.     

Dispersion coefficient with the displacement of a gas from a porous medium by a gas

The article gives the dependence of the dispersion coefficient on the Péclet number, obtained with the analysis of experimental data on the displacement of a gas by a gas from a porous medium of varying permeability.Translated from Zhurnal Prikladkoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 142–145, July–August, 1975.The authors are grateful to N. P. Baturina and L. N. Demidova for their aid in carrying out and analyzing the experiments, and to V. M. Ryzhik for evaluating the results of the work.     

Heat transfer in an MHD channel flow with boundary conditions of the third kind

The heat transfer equation for a two-dimensional magnetohydrodynamic channel flow has been solved using boundary conditions of the third kind considering a discontinuity in the ambient temperature. The boundary conditions of the third kind indicate that the normal temperature gradient at a particular point in the boundary is assumed to be proportional to the difference between the fluid temperature and the externally prescribed ambient temperature. The presence of an external circuit is also considered to permit the flow of an electric current in the direction perpendicular to the plane of analysis. The resistance of the external circuit is varied from zero (closed circuit) to infinity (open circuit). Temperature fields far away from and near to the discontinuity are found separately and then added in order to obtain the temperature in the whole flow region. The solutions in the limits where the boundary conditions become first (Dirichlet) or second (Neumann) kind are discussed and the influence of the external resistance and the Hartmann, Péclet and Biot numbers on the temperature distribution is investigated.     

Diffusion flux to a distorted gas bubble at large Reynolds numbers

The diffusion flux to a distorted gas bubble situated in a uniform viscous incompressible fluid flow is determined for large Reynolds and Péclet numbers and finite Weber numbers. The bubble has the shape of an ellipsoid of revolution, oblate in the flow direction, making it possible to use the flow field derived by Moore [1] in the form of a two-term expansion with respect to the flow parameter =R^–1/2 (R is the Reynolds number; the zeroth term of the expansion corresponds to potential flow). The dependence of the diffusion flux onto the bubble surface on the Weber and Reynolds numbers is determined. The results of Winnikow [2] and Sy and Lightfoot [3] are thus generalized to the case of finite Weber numbers and a broader range of Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 70–76, July–August, 1976.     

Diffusion in a nonuniform velocity field

The effect of a nonuniform velocity field on the diffusion process is examined. When the local passive admixture transport equation is averaged over the channel cross section, the differential equation for the average concentration over the cross section is obtained in the form of an infinite asymptotic series whose terms are linear combinations of the derivatives of the average concentration with respect to the coordinate and time, while the coefficients depend on the degree of transverse nonuniformity of the velocity field and the radial Péclet number. Estimates show that in most of the cases encountered in practice to ensure that the calculations have the necessary accuracy the series must include derivatives up to the third order. An approximate solution of the averaged equation is found by the method of asymptotic expansions and the initial moments of the residence time distribution function are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 122–128, September–October, 1989.     

Asymptotic Analysis of the Joining of an Ice-Rock Body in Flow Through Porous Media

The problem of determining the equilibrium configuration of a plane, doubly connected ice-rock body formed about a system of two freezing columns traversing a flow through a porous medium is asymptotically analyzed in the limit of small Péclet numbers. Two terms of the asymptotic expansion are retained. It is shown that in this approximation the criterion of joining of the doubly connected body coincides with the criterion of non-disjoining of the simply connected body. However, the solution structure is such that taking the third asymptotic term into account can lead to a second solution when the ice-rock body is close to joining. This means that the size of the joining-disjoining hysteresis loop is of at least the second order in the Péclet number.     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.