A. T. Il’ichev

Instabilities of uniform filtration flows with phase transition

New mechanisms of instability are described for vertical flows with phase transition through horizontally extended two-dimensional regions of a porous medium. A plane surface of phase transition becomes unstable at an infinitely large wavenumber and at zero wavenumber. In the latter case, the unstable flow undergoes reversible subcritical bifurcations leading to the development of secondary flows (which may not be horizontally uniform). The evolution of subcritical modes near the instability threshold is governed by the Kolmogorov-Petrovskii-Piskunov equation. Two examples of flow through a porous medium are considered. One is the unstable flow across a water-bearing layer above a layer that carries a vapor-air mixture under isothermal conditions in the presence of capillary forces at the phase transition interface. The other is the vertical flow with phase transition in a high-temperature geothermal reservoir consisting of two high-permeability regions separated by a low-permeability stratum.

Solitary waves in media with dispersion and dissipation (a review)

The latest results relating to the theory of nonlinear waves in dispersive and dissipative media are reviewed. Attention is concentrated on small-amplitude solitary waves and, in particular, on the classification of types of solitary waves, their conditions of existence, the evolution of local perturbations associated with the presence of solitary waves of various types, and problems of the existence of nonlinear waves localized with respect to a particular direction as the space dimension increases (spontaneous dimension breaking). As examples of dispersive and dissipative media admitting plane solitary waves of various types, we consider a cold collisionless plasma, an ideal incompressible fluid of finite depth beneath an elastic plate and with surface tension, and a fluid in a rapidly oscillating rectangular vessel (Faraday resonance). Examples of spontaneous dimension breaking are considered for the generalized Kadomtsev-Petviashvili equation.

Formation of a wave on an ice-sheet above the dipole, moving in a fluid

Theory of wave motions of a fluid with an ice-sheet was developed due to the necessity of solving of a number of problems of marine and land physics. The main attention in these investigations was focused on propagation and interaction of free waves, and also on appearance of waves under action of different loadings on the ice-sheet. From the other side, the problems dealing with waves on the fluid surface, free from the ice due to motion in the mass of the fluid of rigid bodies, has the known solutions. In this connection, it seems natural to disserminate the formulation and methods of such problems to the case of the fluid with the ice-sheet. In the present note we describe the character of formation of waves from the singularity, localized in the fluid of infinite depth beneath the ice-sheet. We use the example of the dipole, which models a cylinder in the infinite mass of the fluid. The character of the formation does not depend on the type of singularity. The ice-sheet is considered as a thin elastic plate of a constant width, floating on the water surface.

Dynamics of water evaporation fronts

The evolution and shapes of water evaporation fronts caused by long-wave instability of vertical flows with a phase transition in extended two-dimensional horizontal porous domains are analyzed numerically. The plane surface of the phase transition loses stability when the wave number becomes infinite or zero. In the latter case, the transition to instability is accompanied with reversible bifurcations in a subcritical neighborhood of the instability threshold and by the formation of secondary (not necessarily horizontal homogeneous) flows. An example of motion in a porous medium is considered concerning the instability of a water layer lying above a mixture of air and vapor filling a porous layer under isothermal conditions in the presence of capillary forces acting on the phase transition interface.

Envelope solitary waves and dark solitons at a water-ice interface

The article is devoted to the study of some self-focusing and defocusing features of monochromatic waves in basins with horizontal bottom under an ice cover. The form and propagation of waves in such basins are described by the full 2D Euler equations. The ice cover is modeled by an elastic Kirchhoff-Love plate and is assumed to be of considerable thickness so that the inertia of the plate is taken into account in the formulation of the model. The Euler equations involve the additional pressure from the plate that is freely floating at the surface of the fluid. Obviously, the self-focusing is closely connected with the existence of so-called envelope solitary waves, for which the envelope speed (group speed) is equal to the speed of filling (phase speed). In the case of defocusing, solitary envelope waves are replaced by socalled dark solitons. The indicated families of solitary waves are parametrized by the wave propagation speed and bifurcate from the quiescent state. The dependence of the existence of envelope solitary waves and dark solitons on the basin’s depth is investigated.

Stability of solitary waves in membrane tubes: A weakly nonlinear analysis

We study the problem of the stability of solitary waves propagating in fluid-filled membrane tubes. We consider only waves whose speeds are close to speeds satisfying a linear dispersion relation (it is well known that there can be four families of solitary waves with such speeds), i.e., the waves with small (but finite) amplitudes branching from the rest state of the system. In other words, we use a weakly nonlinear description of solitary waves and show that if the solitary wave speed is bounded from zero, then the solitary wave itself is orbitally stable independently of whether the fluid is in the rest state at the initial time.

Spectral stability theory of heteroclinic solutions to the Korteweg-de Vries-Burgers equation with an arbitrary potential

The analysis of stability of heteroclinic solutions to the Korteweg–de Vries–Burgers equation is generalized to the case of an arbitrary potential that gives rise to heteroclinic states. An example of a specific nonconvex potential is given for which there exists a wide set of heteroclinic solutions of different types. Stability of the corresponding solutions in the context of uniqueness of a solution to the problem of decay of an arbitrary discontinuity is discussed.

Solitary wave packets beneath a compressed ice cover

A family of plane solitary wave packets of a small (but finite) amplitude on the surface of an ideal incompressible fluid of finite depth beneath an ice cover is described. The solitary wave trains correspond to solutions of the two-dimensional system of Euler’s equations of an ideal incompressible fluid of the type of a traveling wave which decreases at infinity and has identical phase and group velocities. The ice cover is simulated by an elastic Kirchhoff-Love plate freely floating on the fluid surface in the compressed state.

Evolution of a long-wave train in loss of stability of a phase interface in geothermal systems

The transition to instability of phase interfaces in geothermal systems when a water stratum overlies a steam stratum and the most unstable mode corresponds to zero wavenumber is considered. The nonlinear Kolmogorov-Petrovskii-Piskunov equation describing the evolution of a narrow strip of weakly unstable modes is obtained. This equation is an analog of the well-known Ginzburg-Landau equation corresponding to the case of destabilization of modes with finite wavenumbers. It is shown that in the neighborhood of the critical points there exist two locations of the plane phase interface which coincide at the instant at which the instability threshold is reached and then disappear.

Rayleigh-Taylor instability of an interface in a nonwettable porous medium

The diffusion of vapor through the roof of an underground structure located beneath an aquifer is considered. In the process of evaporation, an interface between the upper water-saturated layer and the lower layer containing an air-vapor mixture is formed. A mathematical model of the evaporation process is proposed and a solution of the steady-state problem is found. It is shown that in the presence of capillary forces in the case of a nonwettable medium the solution is not unique. Using the normal mode method, it is shown that Rayleigh-Taylor instability of the interface can develop in the nonwettable porous medium. It is found that there are two scenarios of loss of stability corresponding to the occurrence of the most unstable wavenumber at zero and at infinity, respectively. It is shown that for zero wavenumber the stability limit is reached at the same time as the solution of the steady-state problem disappears.