Modelling of Advection-Dominated Transport in a Porous Column with a Time-Decaying Flow Field

This paper examines the problem of the advective-dispersive movement of a non-decaying, inert chemical dye solution through the pore space of a fluid saturated porous column. The objective of the paper is to present a complete study of the one-dimensional advective-dispersive transport problem by considering certain analytical solutions, experimental results and their comparisons with specific computational simulations. Dye concentrations obtained by means of an image processing method are used in conjunction with an analytical solution to identify the hydrodynamic dispersion coefficient that governs the advective-dispersive transport problem. The experimental results and identified parameters are also used to assess the computational estimates derived from several stabilized computational schemes available in the literature, for examining advection-dominated transport processes in porous media.     

Symmetric low-frequency motion in incompressible transversely isotropic elastic plates

The problem of long-wave low-frequency extensional (symmetric) motion in a layer composed of incompressible, transversely isotropic elastic material is investigated. Motivated by appropriate approximations of the dispersion relation, a hierarchy of asymptotically approximate boundary value problems is set up and solved. A leading order system of equations is obtained for the governing extensions, together with a refined system for their second order counterparts. A one-dimensional model problem, involving impact edge loading, is set up and solved in order to illustrate the derived theory.     

Asymptotic form of the admixture transport equation for a channel flow with an anisotropic diffusion tensor

The problem of the asymptotically correct reduction of a 3-D mass (heat) transfer equation to a 1-D equation in a flow with anisotropic diffusion properties is considered. The convective mass (heat) transfer domain is a cylindrical channel of arbitrary cross section. The diffusion coefficient matrix is assumed to be independent of the spatial coordinates. In the equivalent diffusion equation constructed, a certain effective diffusion (dispersion [1]) coefficient is introduced. Formulas for this coefficient are obtained. A relation between the effective diffusion coefficient calculations and the problem of minimization of a certain functional is established, i. e. the possibility of calculations based on variational methods is noted. An example of an exact calculation of the effective diffusion coefficient is considered. The possibility of a generalization of the problem, in which the effective diffusion (heat conduction) equation is essentially a nonlinear equation of general form for the one-dimensional case, is indicated. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 110–123, March–April, 2000.     

The transient buoyancy driven motion of bubbles across a two-dimensional quiescent domain

The transient buoyancy driven motion of two-dimensional bubbles across a domain bounded by two horizontal walls is studied by direct numerical simulations. The bubbles are initially released next to the lower wall and as they rise, they disperse. Eventually all the bubbles collect at the top wall. The goal of the study is to examine how a simple one-dimensional model for the averaged void fraction captures the unsteady bubble motion. By using void fraction dependent velocities, where the exact dependency is obtained from simulations of homogeneous bubbly flows, the overall dispersion of the bubbles is predicted. Significant differences remain, however. We suggest that bubble dispersion by the bubble induced liquid velocity must be included, and by using a simple model for the bubble dispersion we show improved agreement.     

Plane one-dimensional stationary flow of an ideal charged gas in its own electric field

The plane one-dimensional flow of an incompressible gas consisting of a neutral and a charged component in its own electric field has been investigated by Stuetzer [1]. Stuetzer's results are valid when the electrostatic pressure is small compared with the hydraulic pressure. In the present paper an analogous problem is considered for a compressible gas under the more general assumption that the pressures are comparable. Three cases are analyzed: a) the velocity of the relative motion of the charged and the neutral particles is equal to zero; b) it is nonvanishing but the flow can be assumed to be approximately isentropic; c) a nonisentropic flow, i.e., one cannot ignore irreversible losses due to the relative motion of the charged and neutral particles. In the first two cases, closed solutions are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 24–32, January–February, 1971.I should like to thank I. V. Bespalov and Yu. M. Trushin for their interest and helpful comments.     

Solute dispersion in a pulsating flow

The one-dimensional model proposed by Taylor [1] of the dispersion of soluble matter describes approximately the distribution of the solute concentration averaged over the tube section in Poiseuille flow. Aris [2] obtained more accurately the effective diffusion coefficient in Taylor's model and solved the problem for the general case of steady flow in a channel of arbitrary section. Many papers have been published in the meanwhile devoted to particular applications of this theory (for example, [3–5]). Various dispersion models have been constructed [6–8] that make the Taylor—Aris model more accurate at small times and agree with it at large times. The acceleration of the mixing of the solute considered in these models in the presence of the simultaneous influence of molecular diffusion and convective transport also operates in unsteady flows. In particular, the presence of velocity pulsations influences the growth of the dispersion even if the mean flow velocity is equal to zero at every point of the flow. In the present paper, the Taylor—Aris theory is extended to the case of laminar flows with periodically varying flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 24–30, September–October, 1982.     

Wave propagation and dispersion in microstructured wool felt

On the basis of the experimental data of the piano hammers study the one-dimensional constitutive equation of the wool felt material is proposed. This relation enables deriving a nonlinear partial differential equation of motion with third order terms, which takes into account the elastic and hereditary properties of a microstructured felt. This equation of motion is used to study pulse evolution and propagation in the one-dimensional case. Thorough analysis both of the linear and nonlinear problems is presented. The physical dimensionless parameters are established and their importance in describing the dispersion effects is discussed. It is shown that both normal and anomalous dispersion types exist in wool felt material. The dispersion analysis shows also that for the certain ranges of physical parameters negative group velocity will appear. The initial value problem is considered and the analysis of the numerical solution describing the strain wave evolution is provided. The influence of the material parameters on the form of a propagating pulse is demonstrated and explained.     

The speed of sound in a reacting gas mixture

The one-dimensional problem of the propagation of sound in a two-component mixture is solved. An expression is obtained for the speed of sound under conditions of chemical equilibrium. The results for the dissociation of hydrogen are compared with similar results in [1].     

Two-dimensional non-linear evolution equations: The derivation and the transient wave solution

The general theory of two-dimensional evolution equations describing transient wave propagation in non-linear continuous media is presented. The ray method is used and the two-dimensional evolution equations for plane and cylindrical wave-beams are obtained. The transient wave solutions are discussed briefly. A transformation of variables is proposed that permits the transformation of the two-dimensional evolution equation into a one-dimensional evolution equation with coordinate-dependent coefficients. A breakdown time analysis is carried out for this case. The dispersion relations for plane and cylindrical wave-beams are presented. The non-linear dispersion relation is obtained by making use of a series representation.     

The Motion Behavior of a Group of Charged Droplets in Liquid-Liquid Electrostatic Spray

The droplet sizes and electrical charges under different applied electrical voltages are experimentally measured for a liquid-liquid electrostatic spray system. Considering droplet size and charge distributions, the two-dimensional motion for a group of charged droplets in a liquid-liquid electrostatic atomization system is simulated. From measured droplet size and charge distributions, the simulation can obtain the velocities and positions in a two-dimensional domain for all simulated droplets at different times. The various forces acting on droplet as well as their effects on droplet velocity and trajectory are analyzed and the liquid-liquid electrostatic atomization characteristics are revealed. In addition, for one-dimensional motion trajectory of larger droplet, the comparison between simulation and experiment is also conducted and a general agreement can be obtained.     

Self-similar problems of propagation of an extended hydrofracture in a permeable medium

The propagation of an extended hydrofracture in a permeable elastic medium under the influence of an injected viscous fluid is considered within the framework of the model proposed in [1, 2]. It is assumed that the motion of the fluid in the fracture is turbulent. The flow of the fluid in the porous medium is described by the filtration equation. In the quasisteady approximation and for locally one-dimensional leakage [3] new self-similarity solutions of the problem of the hydraulic fracture of a permeable reservoir with an exponential self-similar variable are obtained for plane and axial symmetry. The solution of this two-dimensional evolution problem is reduced to the integration of a one-dimensional integral equation. The asymptotic behavior of the solution near the well and the tip of the fracture is analyzed. The difficulties of using the quasisteady approximation for solving problems of the hydraulic fracture of permeable reservoirs are discussed. Other similarity solutions of the problem of the propagation of plane hydrofractures in the locally one-dimensional leakage approximation were considered in [3, 4] and for leakage constant along the surface of the fracture in [5–7].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 91–101, March–April, 1992.     

One-dimensional simulation of viscous fluid displacement from a porous medium with consideration of lateral inflow

A one-dimensional model of fluid displacement in a porous medium is discussed with consideration of lateral inflow. The time period required for the complete displacement of an initially injected fluid from a region is studied. Some numerical results obtained for two formulations of the problem are given; these results are in good agreement with the estimates considered in this paper. The problem under study is of interest in practice for enhancing the oil recovery from oil fields.     

Mathematical modeling of the motion of a portion of helium under pulsed injection over a fixed bed of cenospheres

A mathematical model is constructed and an analytical solution is obtained for the problem of a one-dimensional steady flow of a mixture of different gases with hollow permeable particles. The case of a one-dimensional unsteady flow of such a mixture is analyzed numerically. The numerical solutions are compared with experimental data on the motion of the peak concentration of helium in a fixed bed filled with cenospheres (solid hollow permeable spherical particles). The permeability of cenosphere walls and the drag coeficient of cenospheres in the gas flow are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 92–102, May–June, 2007.     

Exact solution of an automodel (self-sustaining) problem regarding the motion of a nonviscous heat-conducting gas bounded by a vacuum

The one-dimensional transient motion of a nonviscous, compressible, heat-conducting, gas bordered on one side by free space and on the other by a solid wall is considered. The boundary conditions at the wall are taken in a form which ensures an automodel (self-sustaining) state. An exact solution is obtained for the problem, and this is then compared with a numerical calculation.     

On a Discrete Fourier Series Solution to Static Problem for Conical Shells of Circumferentially Varying Thickness

An approach is developed to solve the two-dimensional boundary-value problems of the stress-strain state of conical shells with circumferentially varying thickness. The approach employs discrete Fourier series to separate variables and make the problem one-dimensional. The one-dimensional boundary-value problem is solved by the stable discrete-orthogonalization method. The results obtained are presented as plots and tables __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 26–37, September 2005.     

ADVECTION–DIFFUSION PROCESSES AND RESIDENCE TIMES IN SEMIENCLOSED MARINE BASINS

This paper addresses the problem of estimating the residence times in a marine basin of a passive constituent released in the sea. The dispersion process is described by an advection–diffusion model and the hydrodynamics is assumed to be known. We have performed the analysis of two different scenarios: (i) basins with unidirectional flows, in three space dimensions and under the rigid lid approximation, and (ii) basins with flows forced by the tide, under the shallow water approximation. Let the random variable τ be defined as the time spent in the basin by a particle released at a given point. The probability distribution of τ is obtained from the solution of the advection–diffusion problem and the residence time of a particle is defined as the mean value of τ. Two different numerical approximations have been used to solve the continuous problem: the finite volume and Monte Carlo methods. For both continuous and discrete formulations it is proved that if all the particles eventually leave the basin, then the residence time has a finite value. We present here the results obtained for two study cases: a two- dimensional basin with a steady flow and a one-dimensional channel with flow induced by the tide. The results obtained by the finite volume and Monte Carlo methods are in very good agreement for both scenarios.     

Numerical solution of the two-dimensional unsteady-state problem of the motion of a shell under the action of the products of an axial detonation

A numerical solution is obtained to the problem of the motion of an incompressible cylindrical shell with a charge of explosive, with excitation of the detonation simultaneously along the whole axis of the charge. The strength of the shell is not taken into consideration. A three-term equation of state is adopted for the products of the detonation. In [1] a numerical solution is obtained to the problem of the one-dimensional motion of a shell with the axial detonation of a charge of explosive.     

Solution of some problems of particle motion in a gas stream

The problem of designing the contour of an optimum nozzle for particle acceleration is considered in the one-dimensional formulation. In [1] a similar problem was solved in the general formulation using a numerical method. Here, in contrast to [1], the solution is obtained in analytic form for the particular case of low particle concentration. The problem of the motion of a particle in a uniform stream is solved in the same form. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 86–90, September–October, 1988.     

An Approximate Roe-Type Riemann Solver for a Class of Realizable Second Order Closures

A realizable, objective second-moment turbulence closure, allowing for an entropy characterisation, is analyzed with respect to its convective subset. The distinct characteristic wave system of these equations in non-conservation form is exposed. An approximate solution to Ihe associated one-dimensional Riemann problem is constructed making use of approximate jump conditions obtained by assuming a linear path across shock waves. A numerical integration method based on a new approximate Riemann solver (flux-difference-splitting) is proposed for use in conjunction with either unstructured or structured grids. Test calculations of quasi one-dimensional flow cases demonstrate the feasibility of the current technique even where Euler-based approaches fail.