Unsteady homothermal motions of fluids and isotropic solids

Motions of a certain type are shown to be possible in every compressible or incompressible, homogeneous, isotropic simple material, in the absence of body force. These motions have a specific spatial dependence and the nature of the time dependence, for a particular material, is determined by the material response function or functional for the shear stress in simple shear. It is shown further that motions of this type, and spatially homogeneous, time dependent temperatures, can be supported without application of body force or radiative supply of heat.     

Unsteady motions of degrading or aging linearized elastic solids

Many structures are made of concrete and steel. The response characteristics of such structures that are exposed to the environment change due to its impact such as oxidation due to moisture. Usually the structural element “degrades” in that the load bearing capacity decreases; it also decreases due to “aging”. On the other hand, chemicals are infused into biological systems in order to enhance their strength. Thus far, not much attention has been paid to studying such problems. In this investigation, we study the response of a linearized elastic solid whose material properties change due to the diffusion of a chemical.     

Unsteady flows of a viscoplastic medium in channels

We numerically study the nonstationary Poiseuille problem for a Bingham-Il’yushin viscoplastic medium in ducts of various cross-sections. The medium acceleration and deceleration problems are solved by using the Duvaut-Lions variational setting and the finite-difference scheme proposed by the authors. The dependence of the stopping time on internal parameters such as density, viscosity, yield stress, and the cross-section geometry is studied. The obtained results are in good agreement with the well-known theoretical estimates of the stopping time. The numerical solution revealed a peculiar characteristic of the stagnant zone location, which is specific to unsteady flows. In the annulus, disk, and square, the stagnant zones arising shortly before the flow cessation surround the entire boundary contour; but for other domains, the stagnant zones go outside the critical curves surrounding the stagnant zones in the steady flow. The steady and unsteady flows are studied in some domains of complicated shape.     

Unsteady wave motions of a fluid above an inclined floor

The propagation of unsteady waves above a flat inclined floor within the framework of a linear dispersion model was first studied in [1]. This paper shows how to solve the three-dimensional problem for the case = /4, where is the angle of inclination of the floor plane to the free surface. The two-dimensional problem was studied in [2–4]. In articles [2, 3] asymptotic solutions were found for the Cauchy-Poisson problem for certain values of . In [4], a method is proposed for solving the problem of the wave motion of a fluid due to the displacement of a section of the floor of the basin. However, the complicated structure of the expression obtained by reducing the problem to an inhomogeneous functional equation makes it impossible to study the solution. The aim of the present work is to obtain some exact solutions for the two- and three-dimensional problems of unsteady waves above an inclined floor, which are suitable for calculations and asymptotic estimates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 65–70, November–December, 1984.     

Formation fronts of a nonlinear elastic medium from a medium without shear stresses

The fronts of phase transition of a medium without shear stresses to a nonlinear incompressible anisotropic elastic medium are considered. The mass flux through unit area of a front is assumed to be known. The variation of the tangential components of the medium’s velocity and the variation of the arising shear stresses are studied. An explicit form of boundary conditions is found using the existence condition of a discontinuity front structure. The Kelvin–Voight viscoelastic model is adopted for this structure.     

Unsteady outflow of a warm dense Van der Vaals medium from a plane layer,a cylinder,and a sphere

The unsteady outflow of a warm dense boiling compressiblemedium, initially at rest, from a plane layer, a cylinder, and a sphere into a vacuumis investigated in the approximation of an inviscid and non-heat-conducting two-parameter “gas-liquid”, whose thermodynamic properties are determined by the Van der Vaals equation of state. The expansion, the boiling, and the two-phase medium generation are assumed to be thermodynamically equilibrium, while the transition is instantaneous. The speed of sound suffers a discontinuity across the phase transition line of the second kind (binodal), whereas the pressure, the temperature, the density, the entropy, and the enthalpy remain continuous. The main issue in the thermodynamic calculations, which are the same for all the problems, is the construction of the binodal and, at the same time, an isentrope in an equilibrium two-phase mixture, reduced to the numerical integration of two ordinary differential equations. The one-dimensional problems of unsteady outflow are solved by means of the method of characteristics using the isentropes obtained in the thermodynamic calculations. As distinct from the plane problem, in the cylindrically and spherically symmetric problems there are no regions of homogeneous boiling liquid, which would be finite in space and time.     

Unsteady flow through a porous medium in the presence of chemical reactions

The underground leaching of uranium ores and nonferrous and precious metals under natural conditions is one of the latest methods of mineral extraction [1]. It consists of pumping into isolated formations through reaction wells an acid solution which upon reacting with the rock yields a readily soluble salt that can be brought to the surface with water through extraction wells. Together with the acid solution, it is also possible to pump in other reactants to participate in the chemical reaction, for example, gases such as oxygen. Moreover, secondary gases may be formed as a result of the chemical reaction. Thus, the chemical reaction proceeds in the presence of a one or two-phase flow in the porous medium. The mathematical modeling of these processes is usually based on the approximation of one-phase flow without allowance for the changes in the porosity and permeability of the medium as a result of the reaction [2]. In this paper we solve the problem of unsteady flow in the presence of a chemical reaction for a two-phase system taking into account the changes in the flow parameters of the porous medium. The condition of stability of the plane reaction front is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 82–87, January–February, 1987.The author is grateful to R. I. Nigmatulin for his useful comments and interest in the work.     

Interaction of evaporation fronts with a formation interface in a porous medium

The evolutionarity and structure of water-vapor phase discontinuities formed in a geothermal reservoir on the interface between permeable formations with different properties are considered. In the short-wave approximation a graphic method is proposed for solving the problem of breakdown of an arbitrary discontinuity in a geothermal reservoir consisting of two formations with different properties.     

Geometrical disorder of the fronts of a tunnel-crack propagating in shear in some heterogeneous medium

A statistical analysis of the deformation of the fronts of a tensile tunnel-crack propagating in fatigue in some medium with spatially varying Paris constant was recently performed, with special emphasis on the evolution of the power spectra and correlation functions of the fluctuations of the fronts around reference straight lines. This study is extended here to coplanar propagation (along a weak plane) of a tunnel-crack loaded in mode 2+3. The results are rather similar to those previously obtained for mode 1. In particular, just like for tensile loadings, there is an effect of gradual selection in time of Fourier components of the fluctuations of the fronts of large wavelength. One novelty, however, is that for shear loadings, the fronts no longer tend to become symmetrical in time, so that correlations between crack front fluctuations at two points are higher for points located on the same front than for points located on distinct ones.     

Convective motions in a porous medium heated from below

Convective motions in a porous medium filling a horizontal cylinder with a cross section of arbitrary shape and heated from below are studied. The small-parameter method is used to obtain infinitely many stationary motions forming a one-parameter family. For small values of the parameter, all of these motions are stable with respect to small perturbations. The article also discusses the case of heating which is not strictly vertical. It is found that in this case only one stationary motion is stable.     

Unsteady dissipative structures in non-Newtonian fluid flow through a porous medium

The nonuniform space-time pressure and velocity distributions in an initially nonempty stratum with constant initial pressure created by pumping a non-Newtonian fluid through the boundary of the stratum are investigated. The injected fluid and the fluid present in the stratum before injection have identical physical properties. The conditions of formation of traveling fronts and localized structures are analyzed as functions of the nonlinearity of the rheological law of the fluid and the injection regime.Baku. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–112, November–December, 1994.     

Unsteady viscometric circular flows of a viscoplastic medium in an annular gap

The time of initiation and cessation of a circular flow of a viscoplastic medium in an annular gap is estimated analytically and is verified numerically.     

Similarity in stationary motions of gas and two-phase medium with incompressible component

In this paper we suggest the transformation between the equations for a perfect gas and the equations describing in one-velocity approach the two-phase medium with any volume occupied by the incompressible phase. It is proved that the motion of a two-phase medium in the transformed coordinate system is similar with certain accuracy to that of a perfect gas. It means that the solutions obtained for perfect gas can be used to solve wave problems for media with incompressible component. There is no necessity directly to solve the problem for medium with incompressible component, and it is only sufficient to transform the known solution of the similar problem for a homogeneous medium. Thus, the solutions of many hydrodynamic problems for multi-component media with incompressible phase can be obtained without solving the original set of equations. The scope for the suggested transformation is demonstrated by reference to the strong explosion in a two-phase medium.     

Far fields of internal gravitywaves generated by a perturbation source in a stratified rotating medium

The problem of constructing uniform asymptotics for the far fields of internal gravity waves generated by a moving source of perturbations in flow of a finite-depth stratified rotating medium is considered. The solutions obtained describe the wave perturbations both inside and outside the wave fronts and can be expressed in terms of the Airy function and its derivatives. Numerically calculated wave patterns of the excited wave fields are presented.