Self-similar solutions of thermal two-phase filtration

This work is concerned with deriving generalized self-similar solutions for a thermal model of two-phase filtration in porous media. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 9–17, May–June, 1999.     

Elementary solutions of plane nonlinear filtration problems

In the solution of plane problems of filtration theory it is important to study the behavior of the solution near the singular points of the boundary of the flow region (corner points, points of boundary-condition change, and so on) and at infinity (see, for example, [1]). In the present study, this analysis is made for nonlinear filtration problems.Just as in the analogous problems of gasdynamics [2, 3] and nonlinear elasticity theory [4], to find the singular solutions we make the transformation to the filtration velocity hodograph plane. Examples relating basically to filtration with the limiting gradient are presented.The authors wish to thank I. I. Eremlna, T. N. Ericheva, and T. N. Ivanova for assistance in the calculations.     

Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation

Communicated by M. Gurtin     

ASYMPTOTIC NON-STABILITY AND BLOW-UP AT BOUNDARY FOR SOLUTIONS OF A FILTRATION EQUATION

For a class of nonlinear filtration equation with nonlinear second-third boundary value condition, it is shown that a priori boundary of the solution can be estimated and controlled by initial data and integral on the boundary of the region. The priori estimate of the solutions was established by iterative method. By using this estimate the solutions may blow-up on the boundary of the region and thus it may have asymptotic non-stability.     

Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation

In this paper, we consider a dissipative-dispersive nonlinear equation appliable to many physical phenomena. Using the geometric singular perturbation method based on the theory of dynamical systems, we investigate the existence of its traveling wave solutions with the dissipative terms having sufficiently small coefficients. The results show that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ordinary differential equations （ODEs）. Then, we use the Melnikov method to establish the existence of a homoclinic orbit in this manifold corresponding to a solitary wave solution of the equation. Furthermore, we present some numerical computations to show the approximations of such wave orbits.     

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.     

The similar solutions of nonlinear heat conduction equation

ＴＨＥＳＩＭＩＬＡＲＳＯＬＵＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＨＥＡＴＣＯＮＤＵＣＴＩＯＮＥＱＵＡＴＩＯＮＹｕａｎＹｉｗｕ（袁镒吾）（ＣｅｎｔｒａｌＳｏｕｔｈＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ,Ｃｈａｎｇｓｈａ４１００１２．Ｐ．Ｒ．Ｃｈｉｎａ...     

Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections

Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations （ODEs） are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times singularities, we also obtain solutions which Along with solutions with time-dependent do not exhibit time-dependent singularities.     

Propagation of nonlinear waves in a porous medium with two-phase saturation by a liquid and a gas

The propagation of nonlinear waves through a porous medium saturated with a viscous liquid and a gas is investigated with allowance for the capillary pressure. Numerical solutions of the traveling-wave type are constructed for the generalized Korteweg-de Vries-Burgers equation for the wave amplitudes. Three types of regimes of longitudinal wave propagation, including soliton-like regimes, are found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 86–95, July–August, 1996.     

Constitutive equation of a gas-filled two-phase medium

A set of equations governing the consolidation of a two-phase medium consisting of a porous elastic skeleton saturated with a highly compressible liquid (gas), is described. The homogenization method was utilized to deduce the equations. For the equivalent macroscopic medium, mass and momentum conservation equations and the flow equation of pore liquid are presented. Sample material constants were calculated using laboratory test results which were carried out at the Institute of Geotechnics, Technical University of Wroclaw.     

Reflection and collision of solitary wave solutions of the nonlinear Klein-Gordon equation

The reflection of solitary wave solutions of the cubically nonlinear Klein-Gordon equation from rigid and nonrigid boundaries, and their mutual interaction are studied using finite-difference numerical techniques. The instability of these solutions is shown to be removed by incorporating a small amount of a fifth-order saturating nonlinearity in the system.