Ruxandra Stavre

ASYMPTOTIC METHODS FOR MICROPOLAR FLUIDS IN A TUBE STRUCTURE

The steady motion of a micropolar fluid through a wavy tube with the dimensions depending on a small parameter is studied. An asymptotic expansion is proposed and error estimates are proved by using a boundary layer method. We apply the method of partial asymptotic decomposition of domain and we prove that the solution of the partially decomposed problem represents a good approximation for the solution of the considered problem.

Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall

In this paper we continue the study of a fluid-structure interaction problem with the non periodic case. We consider the non stationary flow of a viscous fluid in a thin rectangle with an elastic membrane as the upper part of the boundary. The physical problem which corresponds to non homogeneous boundary conditions is stated. By using a boundary layer method, an asymptotic solution is proposed. The properties of the boundary layer functions are established and an error estimate is obtained.

Algorithms and convergence results for an inverse problem in heat propagation

The aim of this paper is to study from the theoretical and numerical points of view an optimal control problem in heat propagation theory. An iteration algorithm is defined and its convergence is proved. For the discretization of this problem an internal approximation in the space variables and a backward Euler scheme in the time variable are used. Stability and convergence results are also established. Some numerical examples are given to illustrate the accuracy of the proposed algorithms in the cases of traction and torsion tests.

Optimization and Numerical Approximation for Micropolar Fluids

Abstract In the theory of micropolar fluids, a special case appears when the microrotation is equal to the vorticity of the fluid. The aim of this article is to determine an external field which realises this case. An existence result for the proposed control problem is obtained and the necessary conditions of optimality are derived. For solving the optimality system, an iterative algorithm is proposed and its convergence is obtained. The discretization of the approximation is studied; stability and convergence theorems are proved.

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiters equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiters equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.

Well Posedness and Asymptotic Expansion of Solution of Stokes Equation Set in a Thin Cylindrical Elastic Tube

We extend the previous results for an interaction problem betweena viscous fluid and an elastic structure to a three-dimensional case. We consider anonstationary, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is describedby the Stokes equations, and for the wall displacement we consider the Koiter equation. The well posedness of the problem is proved bymeans of its variational formulation. We perform an asymptotic analysis ofthe problem with respect to two small parameters, for the periodic case. Thesmall error between theexact solution and the asymptotic one justifies our asymptotic expansions.

Optimal control of a non isothermal Navier-Stokes flow

Abstract This paper deals with a boundary optimal control problem associated with the stationary Navier-Stokes equations coupled with the heat equation. The most general type of boundary condition for the temperature is considered. The existence of a solution of this problem and, for some values of the viscosity coefficient, the uniqueness are proved. The control problem consists in finding a temperature of the surrounding medium which leads to a desired configuration of the temperature of the fluid. The existence of an optimal control is proved and necessary conditions of optimality are derived by introducing a family of regularized optimal problems.

Viscous fluid-thin cylindrical elastic layer interaction: asymptotic analysis

This article represents a generalization of our previous work. We consider a periodic, non-steady, axially symmetric, creeping flow of a viscous incompressible fluid that fills a cylindrical elastic hollow tube. We study the interaction problem “viscous fluid-thin cylindrical elastic layer” when the thickness of the tube wall, , tends to zero, while the density and the Young’s modulus of the elastic material are of order and , respectively. We construct a complete asymptotic expansion when tends to zero. The error between the exact solution and the asymptotic one is evaluated in order to justify the asymptotic construction.

Asymptotic analysis of a thin rigid stratified elastic plate – viscous fluid interaction problem

A three-dimensional model for the interaction of a thin stratified rigid plate and a viscous fluid layer is considered. This problem depends on a small parameter which is the ratio of the thickness of the plate and that of the fluid layer. The right-hand side functions are 1-periodic with respect to the tangential variables of the plate. The plate’s Young’s modulus is of order , i.e. it is great, while its density is of order 1. At the solid–fluid interface, the velocity and the normal stress are continuous. The variational analysis of this model (including the existence, uniqueness of the solution and its regularity) is provided. An asymptotic expansion of the solution is constructed and justified. The error estimate is proved for the partial sums of the asymptotic expansion. The limit problem contains a non-standard boundary condition for the Stokes equations. The existence, uniqueness, and regularity of its solution are proved. The asymptotic analysis is applied to the partial asymptotic dimension reduction of the solid phase and the derivation of the asymptotically exact junction conditions between two-dimensional and three-dimensional models of the plate.

A Viscous Fluid Flow through a Thin Channel with Mixed Rigid-Elastic Boundary: Variational and Asymptotic Analysis

We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and some a priori estimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of the a priori estimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.