A model of two-velocity particles transport in a porous medium

A one-dimensional flow of suspension with two types of solid particles moving with different velocities in a porous medium is considered. A mathematical model of deep bed filtration which generalizes the known equations of mass balance and particle capture kinetics for a flow of fluid with identical particles is developed. The exact solution is evaluated at the filter inlet and on the concentration front of fast suspended and retained particles, asymptotic solutions are provided in certain vicinities of these lines. A global asymptotic solution to the problem with a small limit deposit is constructed. The asymptotics rapidly converges to the numerical solution.     

Mass and heat transfer during solid sphere dissolution in a non-uniform fluid flow

We studied a nonisothermal dissolution of a solvable solid spherical particle in an axisymmetric non-uniform fluid flow when the concentration level of the solute in the solvent is finite (finite dilution of solute approximation). It is shown that simultaneous heat and mass transfer during solid sphere dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with a parabolic velocity profile can be described by a system of generalized equations of convective diffusion and energy. Solutions of diffusion and energy equations are obtained in an exact analytical form. Using a general solution the asymptotic solutions for heat and mass transfer problem during spherical solid particle dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with parabolic velocity profile are derived. Theoretical results are in compliance with the available experimental data on falling urea particles dissolution in water and for solid sphere dissolution in a shear flow.     

Projectile motion in a resistant medium Part II: approximate solution and estimates

The motion of a particle which is projected into a resistant medium and subjected to a uniform gravitational field is considered. The drag force that acts upon the particle within the medium is proportional to the particle speed squared. The problem is formulated in terms of particle-speed and local-path-angle variables, and the equations of motion that result are non-linear and coupled. An exact solution to these equations can be obtained but involves quadratures which cannot be analytically evaluated in terms of standard functions. An approximate solution that is remarkably accurate is presented. This solution is based upon the so-called cubic law, which is motivated by certain properties of the exact solution. This solution is also utilized to obtain estimates for the maximum projectile range, optimal projection angle, and other quantities of interest related to the particle motion.     

Combined heat and mass transfer by hydromagnetic natural convection over a cone embedded in a non-Darcian porous medium with heat generation/absorption effects

 The problem of combined heat and mass transfer by natural convection over a permeable cone embedded in a uniform porous medium in the presence of an external magnetic field and internal heat generation or absorption effects is formulated. The cone surface is maintained at either constant temperature and constant concentration or uniform heat and mass fluxes. In addition, the cone surface is assumed permeable in order to allow for possible fluid wall suction or blowing. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved numerically by an implicit, iterative, finite-difference method. Comparisons with previously published work are performed and excellent agreement between the results is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the temperature and concentration profiles as well as the local Nusselt number and the Sherwood number is illustrated graphically to show special trends of the solutions. Received on 5 June 2000 / Published online: 29 November 2001     

Lie group analysis for thermophoretic and radiative augmentation of heat and mass transfer in a Brinkman-Darcy flow over a flat surface with heat generation

A boundary layer analysis is presented to investigate numerically the effects of radiation,thermophoresis and the dimensionless heat generation or absorption on hydromagnetic flow with heat and mass transfer over a flat surface in a porous medium.The boundary layer equations are transformed to non-linear ordinary differential equations using scaling group of transformations and they are solved numerically by using the fourth order Runge-Kutta method with shooting technique for some values of physical parameters.Comparisons with previously published work are performed and the results are found to be in very good agreement.Many results are obtained and a representative set is displayed graphically to illustrate the influence of the various parameters on the dimensionless velocity,temperature and concentration profiles as well as the local skin-friction coefficient,wall heat transfer,particle deposition rate and wall thermophoretic deposition velocity.The results show that the magnetic field induces acceleration of the flow,rather than deceleration(as in classical magnetohydrodynamics(MHD) boundary layer flow) but to reduce temperature and increase concentration of particles in boundary layer.Also,there is a strong dependency of the concentration in the boundary layer on both the Schmidt number and mass transfer parameter.     

Electrically charged two-phase flow past a sphere with a high electrostatic surface potential

A two-phase medium with a carrier phase in the form of an incompressible electrically neutral fluid and a dispersed phase in the form of inertial charged particles flows past an electrically charged sphere. It is assumed that the electrohydrodynamic interaction parameter is insignificant and that the flow conditions correspond to potential unseparated flow of the carrier medium over the sphere. The motion of the dispersed phase is described by continuum dynamic equations incorporating the electric field, which is the sum of the external field created by the sphere and the field induced by the dispersed particles. The electric field is determined by means of the equations of electrodynamics, which must be considered together with the dynamic equations. In the case considered a large electrostatic potential is applied to the sphere. This prevents the particles striking the surface of the sphere and leads to the intersection of the particle trajectories. In order to solve this problem within the framework of the two-velocity continuum we introduce a surface of discontinuity of the parameters to replace the zone of multiphase flow. The location of the surface of discontinuity, the distribution of the velocity and density of the dispersed phase and the distribution of electrostatic potential are found as a result of solving a system of elliptic and hyperbolic equations in two regions separated by the surface of discontinuity. The results of numerically integrating the system of equations formulated are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 90–95, March–April, 1987.     

Two-way coupling model for shock-induced laminar boundary-layer flows of a dusty gas

The present paper describes a numerical two-way coupling model for shock-induced laminar boundary-layer flows of a dust-laden gas and studies the transverse migration of fine particles under the action of Saffman lift force. The governing equations are formulated in the dilute two-phase continuum framework with consideration of the finiteness of the particle Reynolds and Knudsen numbers. The full Lagrangian method is explored for calculating the dispersed-phase flow fields (including the number density of particles) in the regions of intersecting particle trajectories. The computation results show a significant reaction of the particles on the two-phase boundary-layer structure when the mass loading ratio of particles takes finite values.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

A stochastic model for filtration of particulate suspensions with incomplete pore plugging

A population balance model for particulate suspension transport with capture of particles by porous medium accounting for complete and incomplete plugging of pores by retained particles is derived. The model accounts for pore space accessibility, due to restriction on finite size particle movement through the overall pore space, and for particle flux reduction, due to transport of particles by the fraction of the overall flux. The novel feature of the model is the residual pore conductivity after the particle retention in the pore and the possibility of one pore to capture several particles. A closed system of governing stochastic equations determines the evolution of size distributions for suspended particles and pores. Its averaging results in the closed system of hydrodynamic equations accounting for permeability and porosity reduction due to plugging. The problem of deep bed filtration of a single particle size suspension through a single pore size medium where a pore can be completely plugged by two particles allows for an exact analytical solution. The phenomenological deep bed filtration model follows from the analytical solution.     

Impact of thermophoresis particle deposition and chemical reaction on unsteady non-Darcy mixed convective flow over a porous wedge in the presence of temperature-dependent viscosity

An analysis is presented to investigate the effect of thermophoresis particle deposition and temperature dependent viscosity on unsteady non-Darcy mixed convective heat and mass transfer of a viscous and incompressible fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform non-Darcian porous medium in order to allow for possible fluid wall suction or injection. The governing partial differential equations of the problem, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary layer analysis. Numerical calculations are carried out for different values of dimensionless parameters arising in the problem. The results are compared with available ones in the literature and excellent agreement is obtained. An analysis of the obtained results shows that the flow field is influenced appreciably by the chemical reaction and thermophoresis particle deposition.     

Gleichzeitiger Wärme- und Stofftransport in katalytischen Festbettreaktoren

In order to optimize a fixed bed reactor and to consider the heat and mass transfer in the reactor it is necessary to apply a model taking into account the physical and chemical processes in the packing and in the particle. By the aid of a disperse two-phase model the essential equations for conservation of energy and mass are formulated and the equations for relevant transport phenomena are provided. An example of numerical solution shows steep radial gradients of concentration, therefore the neglect of radial dispersion causes considerable errors. By comparison of theoretical results with experiments the utility of the disperse two-phase model is demonstrated.     

Flow and temperature field measurements of thermal convection in a small vertical gap using liquid crystals

Thermal convection in a small vertical gap is studied experimentally applying digital particle image velocimetry/thermometry. This optical method enables the simultaneous measurement of two-dimensional flow and temperature fields in a liquid. The principle is based on seeding the liquid flow medium with thermochromic liquid crystal particles. The temperature is measured by the crystal particles which change their reflected colour as function of temperature. The flow velocity is measured by using the same particles as flow tracers. The investigation shall contribute to the understanding of the fluid mechanical behaviour of biological liquids within micro reactor systems. However, the problem is also of fundamental interest as far as heat and mass transfer is concerned. Measured temperature and flow velocity fields are presented and discussed. Presented in part at the 4th Chemnitz/Hamburger Colloquium (CHC) on Microflows, Hamburg, Germany, November 2004.     

Freezing around a finite heat sink immersed in an infinite phase change medium

Freezing around a spherical heat sink immersed in an infinite phase change medium — a free boundary problem involving growth and decay of the free boundary — is analysed here. A one-dimensional conduction model is formulated and the resulting partial differential equations are solved by finite difference methods. The energy discharged from the phase change medium during the heat transfer process is analysed for latent heat thermal energy storage applications. Results are presented for a wide range of parameters that are encountered in energy storage devices. The cases of slab/cylindrical heat sink are reexamined for a range of parameters not covered by the earlier investigators     

A Brief Account of G.D.Birkhoff’s Problem in the Problem of Three Bodies

The present article gives a historical survery of G.D.Birkhoff’s seventh problem which is an inquiry about the topological structure of the set of definition of the reduced differential equations of motion.Recent advances in the problem and their meaning have been briefly indicated.The classical 3-body problem concerns how the three particles should move under their mutual Newtonian attraction.By a particle we mean a goometrical point endorsed with a constant positive number m which is called mass.Expressed mathematically,the problem appears as to solving of the following system of differential equations:     

Extremal nozzle contours for gas flows with particle lag

The determination of the extremal nozzle contour for gas flow without foreign particles has been carried out in several studies [1–6], based on the calculation of the flow field using the method of characteristics.In [7, 8] the equations are derived for the characteristics and the relations along the streamlines which are required for calculating two-dimensional gas flow with foreign particles. The variational problem for two-phase flow in the two-dimensional formulation may be solved by the method of Guderley and Armitage [9] with the use of equations given in [7] or [8]; however this method is very tedious, even with the use of high-speed computers.In [10, 11] studies are made of two-phase one-dimensional flows by expanding the unknown functions in series in a small parameter, defined by the particle dimensions. In [12] a solution is given for the variational problem (in the one-dimensional formulation) of designing the contour of a nozzle with maximal impulse. However that study does not take account of the static term appearing in the impulse and the solution is obtained in relative cumbersome form. Moreover, the question of account for the losses due to nonparallelism and nonuniformity of the discharge was not considered.The present paper considers in the one-dimensional formulation the flow of a two-phase medium in a Laval nozzle with small particle lags (in velocity and temperature). The variational problem of determining the maximal nozzle impulse is formulated along the nozzle contour for fixed geometric expansion ratio. The impulse losses due to nonparallelism of the discharge are simulated by a function which depends on the ordinates which are variable along the contour and on the slope of the tangent to the contour.The author wishes to thank Yu. D. Shmyglevskii and A. N. Kraiko for helpful discussions and V. K. Starkov for carrying out the calculations on the computer.     

Two-way coupling model for shock-induced laminar boundary-layer flows of a dusty gas

The present paper describes a numerical two-way coupling model for shock-induced laminar boundary-layer flows of a dust-laden gas and studies the transverse migration of fine particles under the action of Saffman lift force. The governing equations are formulated in the dilute two-phase continuum framework with consideration of the finiteness of the particle Reynolds and Knudsen numbers. The full Lagrangian method is explored for calculating the dispersed-phase flow fields (including the number density of particles) in the regions of intersecting particle trajectories. The computation results show a significant reaction of the particles on the two-phase boundary-layer structure when the mass loading ratio of particles takes finite values. The project supported by the National Natural Science Foundation of China (90205024) and Russian Foundation for Basic Research (RFBR and (RFBR-NSFC-39004) The English text was polished by Yunming Chen     

Radiative and convective heat transfer during blowing into a compressed hypersonic layer of two-phase products from the ablation of a covering

The study examines the screening of the radiative heat flux in conditions of hypersonic flow around blunt bodies with ablated carbon-based coverings. In contrast to the studies already known [1–3], allowance is made for the presence of condensed microscopic particles in the products of ablation. In [4] the problem of radiative transfer is considered in a layer of two-phase ablation products with parametrically prescribed dimensions, particle temperature, and layer thickness. The present study uses a closed system of equations which describes the processes of heat and mass transfer. This gives rise to considerable differences in the numerical results, according to the degree of screening.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–166, November–December 1985.deceased     

Nonstationary Plane Elastic Contact Problem for Matched Cylindrical Surfaces

An approach is proposed to study a collision of a long cylinder with the inside surface of a circular cylindrical cavity in an elastic medium. The problem is solved in plane formulation. A nonstationary mixed initial–boundary-value problem with unknown boundaries moving with a variable velocity is formulated and then reduced to an infinite system of Volterra integral equations of the second kind or, in a simplified formulation, to a sequence of Volterra integral equations. The penetration velocity is determined as a function of the cylinder mass and initial conditions. It is established that the reaction force peaks instantaneously and then dies out     

A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions

A phase-transitional flow takes place during the filling stage by injection molding of short-fiber reinforced thermoplastics. The mechanical properties of the final product are highly dependent on the flow-induced distribution and orientation of particles. Therefore, modelling of the flow which allows to predict the formation of fiber microstructure is of particular importance for analysis and design of load bearing components. The aim of this paper is a discussion of existing models which characterize the behavior of fiber suspensions as well as the derivation of a model which treats the filling process as a phase-transitional flow of a binary medium consisting of fluid particles (liquid constituent) and immersed particles-fibers (solid-liquid constituent). The particle density and the mass density are considered as independent functions in order to account for the phenomenon of sticking of fluid particles to fibers. The liquid constituent is treated as a non-polar viscous fluid, but with a non-symmetric stress tensor. The state of the solid-liquid constituent is described by the antisymmetric stress tensor and the antisymmetric moment stress tensor. The forces of viscous friction between the constituents are taken into account. The equations of motion are formulated for open physical systems in order to consider the phenomenon of sticking. The chemical potential is introduced based on the reduced energy balance equation. The second law of thermodynamics is formulated by means of two inequalities under the assumption that the constituents may have different temperatures. In order to take into account the phase transitions of the liquid-solid type which take place during the flow process a model of compressible fluid and a constitutive equation for the pressure are proposed. Finally, the set of governing equations which should be solved numerically in order to simulate the filling process are summarized. The special cases of these equations are discussed by introduction of restricting assumptions.Received: 6 May 2002, Accepted: 16 December 2002, Published online: 29 July 2003PACS: 83.10.Ff, 83.70.Hg, 83.50.Cz Correspondence to: H. Altenbach