Complementary variational principles for poiseuille flow of an Oldroyd fluid

The complementary variational principles are given for the poiseuille flow of an Oldroyd fluid by taking the pressure gradient to be exponentially increasing with time and the bounds on the flux are obtained.     

On the theory of periodic layers in incompressible fluid

A conceptual error in the formally closed but physically incomplete Kim-Moin-Moser form of equations for viscous incompressible fluid in a horizontal periodic layer is corrected. This form, which has lately become popular, assumes that the vertical projections of the rotor and the second rotor of the field of accelerations vanish. This assumption considerably simplifies the calculations; however, it is insufficient for the equations of motion. In this paper, the fulfillment of these assumptions is ensured by the additional condition that the vector of the horizontal projection averaged over the period of the acceleration vorticity vanishes, which opens new possibilities. The resulting complete form of the equations with the rotors of three orders admits a reduction to two scalar equations (of the fourth and the sixth orders), which, however, are not less complicated than the equivalent Navier-Stokes equations.     

Oscillation of membranes in an incompressible fluid

This article considers the problem of natural oscillations of membranes in an unbounded incompressible fluid. The article contains a qualitative analysis of the equations used and upper and lower bounds are obtained for the natural frequencies of the membrane. Translated fromDinamicheskic Sistemy. Vol. 12. pp. 23–29, 1993.     

Regularity for minimizers of some variational problems in plasticity theory

A variational problem for a functional depending on the symmetric part of the gradient of the unknown vectorvalued function is considered. We assume that the integrand of the problem has power growth with exponent less than two. We prove the existence of summable second derivatives near a flat piece of the boundary. In the two-dimensional case, Hölder continuity up to the boundary of the strain and stress tensors is established. Bibliography: 6 titles.     

Positive solutions of some eigenvalue problems in the theory of variational inequalities

We prove the existence of positive solutions of some eigenvalue problems relative to variational inequalities. The operators considered here belong to a class of differential nonlinear elliptic operators in divergence form.     

Problem of the infusion of a fluid in an elastic deformable medium

The problem of the infusion of a fluid in an elastic coal seam via a hole drilled from the daylight surface is solved. Analytical expressions that make it possible to investigate the effect of the stress—strain state, physicomechanical and collecting properties of the coal, and the rate of injection on the process of fluid infusion under prescribed conditions, are obtained.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 112–116, 1190.     

The uniqueness of solutions of some variational problems of the theory of phase equilibrium in solid bodies

A variational problem in the theory of phase equilibrium is considered in a geometrically linear statement. A solid body is assumed to have two different phases with the same elasticity tensor. In the case of the Dirichlet condition, it is proved that under some restrictions on the elasticity tensor a solution of the relaxational variational problem is unique provided that the difference of the strain tensors relative to the zero stress state of the corresponding phase is anisotropic. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 220–232.     

Gibb's variational principles for the equilibrium of continuous systems with an interface

A mathematical formulation of the Gibbs variational principle is supplied for the equilibrium of a system of to fluids with an interface adsorbing or not and permeable or not. These properties of the interface are represented by suitable constraints for the admissible variations.

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Sunto Viene fornita una rigorosa formulazione del principio variazionale di Gibbs per l'equilibrio di un sistema di due fluidi con un'interfaccia assorbente o non ed impermeabile o non. Dette proprietà dell'interfaccia sono tradotte da opportuni vincoli per le variazioni ammissibili.

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Work partially supported by G.N.F.M. of the Italian C.N.R.     

Determining nodes of the global attractor for an incompressible non-Newtonian fluid

This paper estimates the finite number of the determining nodes to the equations for an incompressible non-Newtonian fluid with space-periodic or no-slip boundary conditions. The authors prove that, whenever the second order derivatives of two different solutions within the global attractor have the same time-asymptotic behavior at finite number of points in the physical space, then the two solutions possess the same time-asymptotic behavior at almost everywhere points of the physical space.     

On the phase transition temperature in a variational problem of elasticity theory for two-phase media

We obtain sufficient conditions for the noncoincidence of the phase transition temperatures and illustrate this result by examples of problems in two-phase elastic media. We also indicate some cases where equilibrium states exist or not depending on the values of the temperature lying between the lower and upper phase transition temperatures. Bibliography: 10 titles. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 37–48.     

Smooth solutions for the motion of a ball in an incompressible perfect fluid

In this paper we investigate the motion of a rigid ball surrounded by an incompressible perfect fluid occupying RN. We prove the existence, uniqueness, and persistence of the regularity for the solutions of this fluid-structure interaction problem.     

On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications

Dedicated to Professor Shmuel Agmon     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Canonical variables for the two-dimensional hydrodynamics of an incompressible fluid with vorticity

All-Union Scientific-Research Institute of Physicotechnical and Radio Engineering Measurements. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 1, pp. 136–139, January, 1989.