A Decomposition Method for Solving Coupled Multi–Species Reactive Transport Problems

Concerns over the problems associated with mixed waste groundwater contamination have created a need for more complex models that can represent reactive contaminant fate and transport in the subsurface. In the literature, partial differential equations describing solute transport in porous media are solved either for a single reactive species in one, two or three dimensions, or for a limited number of reactive species in one dimension. Those solutions are constrained by many simplifying assumptions. Often, it is desirable to simulate transport in two or three dimensions for a more practical system that might have multiple reactive species. This paper presents a decomposition method to solve the partial differential equations of multi–dimensional, multi–species transport problems that are coupled by linear reactions. A matrix method is suggested as a tool for describing the reaction network. In this way, the level of complexity required to solve the multi–species reactive transport problem is significantly reduced.     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

Averaging of stochastic evolution equations of transport in porous media

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.     

A method for solving linearized problems of the transfer theory for spherical geometry at arbitrary Knudsen numbers

A method of calculating the bracket integrals containing a discontinuous two-sided Maxwellian distribution function is developed. This method makes it possible to obtain analytic solutions for different linearized systems of moment equations for all Knudsen numbers. The particular problem of heat transfer from a heated sphere is considered. New special functions associated with a specific moment system are introduced. For these functions a table of numerical values is given. Approximate analytic expressions for calculating the bracket integrals are also presented.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 181–186, November–December, 1994.     

Light-induced kinetic effects in a single-component gas with a small change in molecular cross section due to excitation

Linearized transport equations for describing the light-induced effects in a single-component gas are proposed. The equations were obtained by expanding the initial transport equations in the small parameter/ — the relative change in molecular cross section on excitation. The linear equations are solved numerically by the moment method.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 149–154, May–June, 1992.The author wishes to thank M. N. Kogan and N. K. Makashev for their interest in his work and for discussing the results.     

Analysis of the Applicability of Macroscopic Models of the Shock Wave Structure in a Binary Mixture of Inert Gases

The results of calculating the shock wave structure in Ne–Ar, He–Ar, He–Ne, and He–Xe mixtures by means of the relaxation method on the basis of the system of Navier-Stokes equations and complete and modified systems of Burnett equations are compared with the results of direct statistical simulation (Monte-Carlo method). The domain of applicability of these systems of equations for calculating gas dynamic variable profiles is analyzed as a function of both the molecular mass ratio and the initialconcentrations.     

Theory of slightly perturbed three-dimensional flows in nozzles

The theory of slightly perturbed flows in conical nozzles is used to determine the transverse force and moment generated in the presence of asymmetric perturbations. A system of ordinary differential equations is derived for finding the transverse force and moment. An approximate analytical solution of this system is constructed and its qualitative features are studied. A comparison is made with a numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1977.     

Conditional moment closure and large-scale fluctuations of scalar dissipation

The accuracy of the method of calculating turbulent reacting flows [1–3] which is based on the use of the conditional mean concentrations of the reacting components is analyzed. The effect of large-scale fluctuations of the dissipation of passive admixture concentration [4–6] on the accuracy of the equations for the conditional means is considered and the corresponding corrections are computed. By means of numerical calculations the models for the conditional moment and traditional calculation methods are compared, as are the various approaches to the computation of the coefficients of the equations for the conditional means.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 50–59, September–October, 1993.The author wishes to express his wann thanks to A. B. Vatazhin for discussing the work and making useful comments.     

Numerical Analysis of the Branched Forms of Bending for a Rod

Nonlinear boundary–value problems of plane bending of elastic arches subjected to uniformly distributed loading are solved numerically by the shooting method. The problems are formulated for a system of sixth–order ordinary differential equations that are more general than the Euler equations. Four variants of rod loading by transverse and longitudinal forces are considered. Branching of the solutions of boundary–value problems and the existence of intersected and isolated branches are shown. In the case of a translational longitudinal force, the classical Euler elasticas are obtained. The existence of a unique (rectilinear) form of equilibrium upon compression of a rod by a following longitudinal force is shown.     

Nonlinear equations of elastic deformation of plates

A method for constructing nonlinear equations of elastic deformation of plates with boundary conditions for stresses and displacements at the face surfaces in an arbitrary coordinate system is proposed. The initial three–dimensional problem of the nonlinear theory of elasticity is reduced to a one–parameter sequence of two–dimensional problems by approximating the unknown functions by truncated series in Legendre polynomials. The same unknowns are approximated by different truncated series. In each approximation, a linearized system of equations whose differential order does not depend on the boundary conditions at the face surfaces which can be formulated in terms of stresses or displacements is obtained.     

Nonisothermal flow of a viscous incompressible fluid in a two-dimensional diffuser

The velocity and temperature distributions in a viscous incompressible fluid flow in a two-dimensional diffuser are analyzed. Fully developed flow is considered, i.e., the influence of the entrant section is disregarded. It is assumed that the diffuser walls are maintained at a temperature depending on the polar radius. The dynamic viscosity is considered to be an exponential function of the temperature. The problem is reduced to the solution of a system of ordinary differential equations, which is solved by the method of successive approximations. The convergence of the iterative scheme is proved.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 40–48, July–August, 1973.The author is indebted to L.A. Galin and N. N. Gvozdkov for assistance with the study.     

Dispersion of a filtration flow

In this paper we shall consider the transport of a dynamically neutral impurity in a porous medium containing random inhomogeneities. The original versions of the equations for the mean impurity concentration [1, 2] were based on the hyphothesis that the random motions obeyed the Markov principle, use being made of the diffusion equations of A. N. Kolmogorov. Later [3, 4] the method of perturbations was used to study the complete system of equations for the impurity concentration and random filtration velocity in the case of a constant, nonrandom porosity; after an averaging process this yields a generalized equation for the average concentration. In the limiting cases of small- and large-scale inhomogeneities in the permeability of the medium, the basic integrodifferential equation may be, respectively, reduced to parabolic and hyperbolic equations of the second order. In the present analysis we shall use the perturbation method to study the transport of an impurity by a flow when the filtration velocity of the latter fluctuates around inhomogeneities in the permeability field, the porosity of the medium in which the flow is taking place also constituting a random field, correlating with the field of permeability. We shall derive equations for the average concentration and should formulate the corresponding boundary-value problems for these equations; we shall also calculate the components of the dispersion tensor and shall consider the equilibrium sorption of an impurity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–69, July–August, 1976.The author is grateful to A. I. Shnirel'man for useful discussions.     

Bifurcations at Combination Resonance and Quasiperiodic Vibrations of Flexible Beams

The nonlinear dynamic behavior of flexible beams is described by nonlinear partial differential equations. The beam model accounts for the tension of the neutral axis under vibrations. The Bubnov–Galerkin method is used to derive a system of ordinary differential equations. The system is solved by the multiple-scale method. A system of modulation equations is analyzed     

Boltzmann equation and moment equations in curvilinear coordinates

Various forms of writing the Boltzmann equation in an arbitrary orthogonal curvilinear coordinate system are discussed. The derivation is presented of a general transport equation and moment equations containing moments of the distribution function no higher than the fourth. For a gas of Maxwellian molecules it is shown that the system of moment equations for flows which differ little from equilibrium flows transforms into the system of hydrodynamic equations. The resulting equations may be useful in solving problems on motions of a rarefied gas by the moment methods. The results are valid for both the Boltzmann equation and model kinetic equations.The author wishes to thank A. A. Nikol'skii for discussions and helpful comments.     

On the parametrization of three-point nonlinear boundary-value problems

A three-point boundary-value problem for a system of nonlinear differential equations is reduced to a family of two-point problems, whose solutions are investigated by using the numerical-analytic method.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 395–413, July–September, 2004.     

Conditional averaging of the equations of flow in random composite porous media

The problem of conditional averaging of the transport equations is solved for a neutral impurity in a composite medium with random porosity and impurity diffusion tensor. An unclosed system of conditionally averaged equations is constructed and closed using the globally averaged equations. The average impurity concentration fields for the individual phases of the composite medium and the phase-continuum interaction characteristics are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 1, pp. 75–81, January–February, 1987.     

Thermoelastoplastic Nonaxisymmetric Stress-Strain Analysis of Laminated Shells of Revolution

A technique for nonaxisymmetric thermoelastoplastic stress–strain analysis of laminated shells of revolution is developed. It is assumed that there is no slippage and the layers are not separated. The problem is solved using the geometrically linear theory of shells based on the Kirchhoff–Love hypotheses. The thermoplastic relations are written down in the form of the method of elastic solutions. The order of the system of partial differential equations obtained is reduced by means of trigonometric series in the circumferential coordinate. The systems of ordinary differential equations thus obtained are solved by Godunov's discrete-orthogonalization method. The nonaxisymmetric thermoelastoplastic stress–strain state of a two-layered shell is analyzed as an example     

Transport coefficients of a dissociating gas

The calculation of the transport coefficients of a dissociating gas involves fundamental difficulties which arise when the internal degrees of freedom of the molecules are taken strictly into account. In practical calculations extensive use is made of the approximation proposed in [1], in the context of which the dependence of the diffusion velocity of the molecule on its internal state is totally neglected. In this case the expressions for the stress tensor and the diffusion velocities coincide with the corresponding expressions for a mixture of structureless particles; in the expression for the heat flux the diffusion transport of internal energy is taken only approximately into account. Here, analytic expressions for the diffusion velocities, heat flux and stress tensor are obtained without introducing simplifying assumptions. The calculation method is based on the results of [2], in which an approximate method of calculating the transport coefficients of a multicomponent mixture of structureless particles was proposed, and [3], in which the transport coefficients of a rotationally excited gas were calculated. The relations obtained are analyzed and compared with the existing results; their accuracy is estimated. A closed system of equations of gas dynamics is presented for a number of cases of practical importance.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–165, January–February, 1987.     

Homogeneous condensation in turbulent submerged isobaric jets

Turbulent isobaric vapor-air jet flows with homogeneous condensation are examined. A general system of equations, including the gas dynamic and kinetic equations, the thermodynamic relations and the equations for the turbulence model, is formulated. The moment kinetic equations valid for the free-molecular regime of drop growth in the surrounding medium are extended to other drop mass transfer regimes. The structure of the condensation shock, which includes the nucleation zone and the zone of drop growth on pre-existing nuclei, is investigated on the basis of a general asymptotic approach. Additional conditions at the nucleation and condensation shocks, the need for which follows from the requirement that the shocks be evolutionary, are obtained. Certain problems of averaging of the source terms in the moment equations are discussed, and with reference to the simple example of averaging of the frozen nucleation rate it is shown that the latter is nonzero for a mean supersaturation less than unity and that the condensation zone is displaced upstream. Condensation in a turbulent jet into which condensation-intensifying charged particles (corona discharge ions) are introduced is studied. A numerical method of analyzing homogeneous condensation in turbulent jets, which makes it possible to obtain the gas dynamic and disperse flow characteristics for various temperature conditions with allowance for the averaging of the source terms, is developed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 43–52, March–April, 1988.The authors wish to thank V. R. Kuznetsov for discussing various aspects of their work.