Jan Stebel

Weak solutions to the barotropic Navier–Stokes system with slip boundary conditions in time dependent domains

Abstract We consider the compressible (barotropic) Navier–Stokes system on time dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.

STABILITY WITH RESPECT TO DOMAIN OF THE LOW MACH NUMBER LIMIT OF COMPRESSIBLE VISCOUS FLUIDS

We study the asymptotic limit of solutions to the barotropic Navier–Stokes system, when the Mach number is proportional to a small parameter e → 0 and the fluid is confined to an exterior spatial domain Ωe that may vary with e. As e → 0, it is shown that the fluid density becomes constant while the velocity converges to a solenoidal vector field satisfying the incompressible Navier–Stokes equations on a limit domain. The velocities approach the limit strongly (a.a.) on any compact set, uniformly with respect to a certain class of domains. The proof is based on spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.

INCOMPRESSIBLE LIMITS OF FLUIDS EXCITED BY MOVING BOUNDARIES

We consider the motion of a viscous compressible fluid confined to a physical space with a time dependent kinematic boundary. We suppose that the characteristic speed of the fluid is dominated by the speed of sound and perform the low Mach number limit in the framework of weak solutions. The standard incompressible Navier--Stokes system is identified as the target problem.

On Topological Derivatives for Contact Problems in Elasticity

In this article, a general method for shape-topology sensitivity analysis of contact problems is proposed. The method uses domain decomposition combined with specific properties of minimizers for the energy functional. The method is applied to the static problem of an elastic body in frictionless contact with a rigid foundation. The contact model allows a small interpenetration of the bodies in the contact region. This interpenetration is modeled by means of a scalar function that depends on the normal component of the displacement field on the potential contact zone. We present the asymptotic behavior of the energy shape functional when a spheroidal void is introduced at an arbitrary point of the elastic body. For the asymptotic analysis, we use a nonoverlapping domain decomposition technique and the associated Steklov–Poincaré pseudodifferential operator. The differentiability of the energy with respect to the nonsmooth perturbation is established, and the topological derivative is presented in the closed form.

Shape Sensitivity Analysis of Incompressible Non-Newtonian Fluids

We study the shape differentiability of a cost function for the steady flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design.

Original article: On a mathematical model of journal bearing lubrication

Abstract: We consider the isothermal steady motion of an incompressible fluid whose viscosity depends on the pressure and the shear rate. The system is completed by suitable boundary conditions involving non-homogeneous Dirichlet, Naviers slip and inflow/outflow parts. We prove the existence of weak solutions and show that the resulting level of the pressure is fixed by the boundary conditions. The paper is motivated by the journal bearing lubrication problem and extends the earlier results for homogeneous boundary conditions.

Topology Design of Elastic Structures for a Contact Model

Contact problems are very important in the engineering design and the correct interpretation of the physical phenomena, and its influence in this process, is of paramount importance for the engineers. In this paper we employ the topological derivative concept for optimum design problems in contact solid mechanics. A nonlinear contact model governed by a variational inequality is considered. Beside the theoretical developments, some computational examples are included. The influence of the parameters of the contact model in the optimal results for the structures is studied. The numerical results show that the proposed method of optimum design can be applied to a broad class of engineering problems.