污染物在非饱和带内运移的流固耦合数学模型及其渐近解

污染物在非饱和带中运移过程是多组分多相渗流问题.在考虑气相的存在对水相影响的前提下,基于流固耦合力学理论,建立了污染物在非饱和带内运移的流固耦合数学模型.对该强非线性数学模型采用摄动法及积分变换法进行拟解析求解,得出了解析表达式.对非饱和带内的孔隙压力分布、孔隙水流速以及污染物的浓度在耦合与非耦合气相条件下的分布规律进行解析计算.对该渐近解与Faust模型的计算结果进行了对比分析,结果表明:该模型解与Faust解基本吻合,且气相作用以及介质的变形对溶质的输运过程产生较大的影响,从而验证了解析表达式的正确性和实用性.这为定量化预报预测污染物在非饱和带中迁移转化和实验室确定压力-饱和度-渗透率三者之间的关系提供了可靠的理论依据.     

精确计算Bingham流体圆管层流压降及速度分布规律的新方法

Bingham(宾汉)模型情况下,多采用通用公式进行圆管层流压降的解析计算,即将Bingham模型本构方程代入粘性流体圆管层流流动通用公式进行计算,仅能得到压降的解析解.新方法结合Bingham流体本构方程与运动方程,建立有关力学平衡方程,并运用代数方程的根式解理论对圆管层流流动时的非线性方程进行求解,可直接求得Bingham流体圆管层流压降及速度流核区半径的解析解,进一步可求得圆管层流速度解析解;Bingham流体圆管层流速度的直接影响因素为流量、塑性粘度和屈服值,研究发现速度流核宽度与屈服值成正比,与流量及塑性粘度成反比,且流核的宽度越大,流核区的速度越小.     

Some Self-Similar Two-Phase Flows of Non-Newtonian Fluids Through a Porous Medium

Pascal This paper addresses the question of the rheological effects of non-Newtonian fluids in a flow system, in which a two-phase flow zone is coupled to a single-phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non-Newtonian displacing fluid (a polymer solution) is used to displace a non-Newtonian heavy oil. The self-similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one-dimensional and plane radial flows. The effects associated with the presence of a two-phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non-Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer-solution floods.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

On front solutions of the saturation equation of two-phase flow in porous media

We investigate the existence of “front” solutions of the saturation equation of two-phase flow in porous media. By front solution we mean a monotonic solution connecting two different saturations. The Brooks–Corey and the van Genuchten models are used to describe the relative-permeability – and capillary pressure–saturation relationships. We show that two classes of front solutions exist: self-similar front solutions and travelling-wave front solutions. Self-similar front solutions exist only for horizontal displacements of fluids (without gravity). However, travelling-wave front solutions exist for both horizontal and vertical (including gravity) displacements. The stability of front solutions is confirmed numerically.     

Monotone iterative method of upper and lower solutions applied to a multilayer combustion model in porous media

This paper presents a new nonlinear reaction–diffusion–convection system coupled with a system of ordinary differential equations that models a combustion front in a multilayer porous medium. The model includes heat transfer between the layers and heat loss to the external environment. A few assumptions are made to simplify the model, such as incompressibility; then, the unknowns are determined to be the temperature and fuel concentration in each layer. When the fuel concentration in each layer is a known function, we prove the existence and uniqueness of a classical solution for the initial and boundary value problem for the corresponding system. The proof uses a new approach for combustion problems in porous media. We construct monotone iterations of upper and lower solutions and prove that these iterations converge to a unique solution for the problem, first locally and then, in time, globally.     

Flow in the tail interaction region of a spherically symmetrical and a uniform supersonic gas stream

The flow in the tail region of two interacting supersonic streams – spherically symmetrical and planeparallel – is simulated on a supercomputer. The numerical solution is obtained by Godunov’s method. Analysis of the solutions reveals the complex structure of the flow, which includes multiple interfering shock wave structures, a near-axial circulation zone, and a near-axial forward flow zone with a velocity deficit. The detection of such a structure is an unexpected result of the simulation procedure, but it is consistent with some computational and experimental studies, where structures have been observed in supersonic jets.     

Hybrid modeling of convective laminar flow in a permeable tube associated with the cross-flow process

The confined flows in tubes with permeable surfaces are associated to tangential filtration processes (microfiltration or ultrafiltration). The complexity of the phenomena do not allow for the development of exact analytical solutions, however, approximate solutions are of great interest for the calculation of the transmembrane outflow and estimate of the concentration polarization phenomenon. In the present work, the generalized integral transform technique (GITT) was employed in solving the laminar and permanent flow in permeable tubes of Newtonian and incompressible fluid. The mathematical formulation employed the parabolic differential equation of chemical species conservation (convective–diffusive equation). The velocity profiles for the entrance region flow, which are found in the connective terms of the equation, were assessed by solutions obtained from literature. The velocity at the permeable wall was considered uniform, with the concentration at the tube wall regarded as variable with an axial position. A computational methodology using global error control was applied to determine the concentration in the wall and concentration boundary layer thickness. The results obtained for the local transmembrane flux and the concentration boundary layer thickness were compared against others in literature.     

Numerical approaches for computation of fronts

Moving fronts and pulses appear in many engineering applications like flame propagation and a falling liquid film. Standard computation methods are inappropriate since the problem is defined over an infinite domain and a steady-state solution exists only for a certain front velocity. This work presents a transformation that converts the original problem into a boundary-value problem within a finite domain, in a way that preserves the behavior at the boundaries. Good low-order approximations can be obtained as demonstrated by two examples. In another approach, a central element of adjustable length is incorporated into a three-element structure where the edge-elements obey known asymptotic solutions. That yields multiplicity of travelling fronts in an infinite domain but it successfully approximates standing wave solutions in a finite domain. The approximate solutions are shown to obey the qualitative features known for the exact solutions, like asymptotic solutions or the bifurcation set–the boundary where a new solution emerges or disappears.     

Mathematical modeling of infiltration metasomatism in a porous medium

We construct a mathematical model describing the processes of dissolution and redeposition of minerals in a medium with a nonhomogeneous distribution of acidity. The dynamics of extraction of a mineral from a leaching solutions is investigated. We show that filtration of solutions through reduced acidity regions induces deposition, increasing the concentration of the target mineral in the solid phase; in high pH regions, on the other hand, the mineral dissolves. The stratum may retain certain reserves of the target mineral after leaching depending on the size of the reduced pH region and its proximity to the extraction borehole. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 5–17, 2007.     

On peculiarities of the application of Lagrangian variables in problems of time-dependent supersonic flow past bodies

The paper considers particularities in applying Lagrangian variables in problems of hypersonic flow past bodies. It is pointed out that, in problems with intense shock waves, it is advisable to introduce Lagrangian variables as the values of parameters that characterize a particle not on surface t = t ₀ (t ₀ = const) but on surface t = σ, where σ is the time instant at which the particle meets the surface of discontinuity. Considering examples of two-dimensional flow past two-dimensional and axisymmetric bodies moving with high time-dependent velocity, we show that the passage to Lagrangian variables enables us to obtain a system of equations describing the gas flow behind the front of an intense shock wave, which is suitable for the application of the thin-shock-layer flow method. A solution is constructed in the form of series in powers of a small parameter which characterizes the ratio of the gas densities at the front of the leading shock wave. We remark that all nonlinear effects of the problem are concentrated in the equation to determine the law of motion of a particle in zero-order approximation. The cases in which this equation can be integrated are pointed out. For the remaining unknowns, the solution is taken in quadratures. We study the rearrangement of the gas flow in the shock layer when the motion of the body is changed. The flow rearrangement zone is distinguished. Also, a condition to determine the life time of that region (the time of establishment of the new flow regime) is obtained. In a specific case of passing from a steady motion of a wedge to a uniformly accelerated motion, the time of establishment of the uniformly accelerated motion is determined from a quadratic equation.     

Mathematical theory of nonlinear filtration

For the stationary problem of the nonlinear filtration in a bounded domain, from variational principles one establishes a series of statements: for a fixed drop in pressure, an increase in the local filtration resistance or a symmetrization of the domain leads to a decrease of the modified flow rate; at a depression inside the domain of the entrance-surface, the flow and the velocity of the filtration do not decrease at any point of the outgoing surface, etc. When passing to an unbounded domain, it turnsout that at a filtration with a limit gradient, there may exist physically distinct solutions, which are confirmed by examples given In the paper.     

Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels

Transient electro-osmotic flow of viscoelastic fluids in rectangular micro-channels is investigated. The general twofold series solution for the velocity distribution of electro-osmotic flow of viscoelastic fluids with generalized fractional Oldroyd-B constitutive model is obtained by using finite Fourier and Laplace transforms. Under three limiting cases, the generalized Oldroyd-B model simplifies to Newtonian model, fractional Maxwell model and generalized second grade model, where all the explicit exact solutions for velocity distribution are found through the discrete Laplace transform of the sequential fractional derivatives. These exact solutions may be able to predict the flow behavior of viscoelastic biological fluids in BioMEMS and Lab-on-a-chip devices and thus could benefit the design of these devices.     

Jump Conditions for Hyperbolic Systems of Forced Conservation Laws with an Application to Gravity Currents

Weak solutions to systems of nonlinear hyperbolic conservation laws admit discontinuities that result from either an initial value or as part of the temporally developing solution itself. The propagation of such shocks or jumps is affected by forcing terms for the nonlinear system in a way that has not been investigated fully in standard references. Jump conditions for systems of conservation laws with discontinuous forcing terms are derived herein, following the method used to derive the Rankine–Hugoniot jump conditions, and the generalized results are illustrated for the one-dimensional inviscid Burger's equation with discontinuous forcing. The main application of this type of jump condition, and the primary motivation for its study, is its application to a shallow-water model of gravity currents previously described by the authors. Specifically, a new result relation between the front and height at a gravity current front is obtained by using the existing model. Front speeds for gravity currents resulting from instantaneous release are calculated numerically and used to determine the suitability of the jump conditions, which are then compared with existing theoretical expressions and experimental observations. New numerical results are portrayed for the gravity current model, suggesting that the standard method of modeling shallow-water gravity currents with a simple Froude number front condition may tend to suppress some of the finer details of the flow resolved by the numerical scheme used by the authors.     

The asymptotic form of the stationary separated circumfluence of a body at high reynolds numbers

An asymptotic theory of the stationary separated circumfluence of bodies at high Reynolds numbers, Re, is constructed. It is shown that the length and width of the separated zone (SZ) is proportional to Re and that the drag cofficient is proportional to Re⁻¹. A cyclic boundary layer is located around the separated zone with a constant vorticity. In the scale of the body, the flow tends towards a Kirchhoff flow with a velocity on a free line of flow of the order of Re^−1/2 which satisfies the Brillouin-Villat condition.

A review of the attempts which have been made to describe the two-dimensional separated circumfluence of a body at high Reynolds numbers is given in /1, 2/. Certain features of the asmyptotic structure of the solution based on qualitative arguments were pointed out in /3, 4/. The corresponding shape of the separated zone was calculated in /5/. However, no complete theory was constructed in these papers. The appearance of the numerical calculations in /6, 7/ stimulated further investigations and a model with a non-zero jump in the Bernoulli constant on the boundary of the separated zone was proposed in /8/. A number of hypotheses concerning the limiting structure of the flow were put forward in /9/.

In the solution obtained below the flow in the scale of the body is described as in /1, 2/ but the velocity is of the order of Re^−1/2. The flow characteristics in this zone are correspondingly renormalized. The flow in the scale of the separated zone corresponds to the assumptions made in /3, 4/. Unlike in /1–4/, the flow in the scale of the body is not directly combined with the flow in the scale of the separated zone. There are several embedded zones and the possibility of uniting these ensures the selfconsistency of the expansion. Moreover, the cyclic boundary layer on the boundary of the separated zone plays an important role.     

Piston Problems of Two-Dimensional Chaplygin Gas

In this paper, the authors study the piston problem for the unsteady two-dimensional Euler system for a Chaplygin gas. The angle of the piston is allowed to vary in a wide range. The piston can be pushed forward into the static gas, or pulled back from the gas. The global existence of solution to the piston problem with any initial speed is established, and the structures of the global solutions are clearly described. The authors find that for the proceeding piston problem the front shock can be detached, attached or even adhere to the surface of the piston depending on the parameters of the flow and the piston; while for the receding problem the front rarefaction wave is always detached and the concentration will never occur.     

Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source

We construct asymptotic solutions to the wave equation with velocity rapidly oscillating against a smoothly varying background and with localized initial perturbations. First, using adiabatic approximation in the operator form, we perform homogenization that leads to a linearized Boussinesq-type equation with smooth coefficients and weak “anomalous” dispersion. Then, asymptotic solutions to this and, as a consequence, to the original equations are constructed by means of a modified Maslov canonical operator; for initial perturbations of special form, these solutions are expressed in terms of combinations of products of the Airy functions of a complex argument. On the basis of explicit formulas obtained, we analyze the effect of fast oscillations of the velocity on the solution fronts and solution profiles near the front.     

Quenching of flames by fluid advection

We consider a simple scalar reaction‐advection‐diffusion equation with ignition‐type nonlinearity and discuss the following question: What kinds of velocity profiles are capable of quenching any given flame, provided the velocity's amplitude is adequately large? Even for shear flows, the answer turns out to be surprisingly subtle. If the velocity profile changes in space so that it is nowhere identically constant, (or if it is identically constant only in a region of small measure) then the flow can quench any initial data. But if the velocity profile is identically constant in a sizable region, then the ensuing flow is incapable of quenching large enough flames, no matter how much larger is the amplitude of this velocity. The constancy region must be wider across than a couple of laminar propagating front‐widths. The proof uses a linear PDE associated to the nonlinear problem and quenching follows when the PDE is hypoelliptic. The techniques used allow the derivation of new, nearly optimal bounds on the speed of traveling wave solutions. © 2000 John Wiley & Sons, Inc.     

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.