Lattice Boltzmann Approach to Viscous Flows Between Parallel Plates

Four different kinds of laminar flows between two parallel plates are investigated using the Lattice Boltzmann Method (LBM). The LBM accuracy is estimated in two cases using numerical fits of the parabolic velocity profiles and the kinetic energy decay curves, respectively. The error relative to the analytical kinematic viscosity values was found to be less than one percent in both cases. The LBM results for the unsteady development of the flow when one plate is brought suddenly at a constant velocity, are found in excellent agreement with the analytical solution. Because the classical Schlichting's approximate solution for the entrance--region flow is not valid for small Reynolds numbers, a Finite Element Method solution was used in order to check the accuracy of the LBM results

A lattice Boltzmann study of phase separation in liquid-vapor systems with gravity

Phase separation of a two-dimensional van der Waals fluid subject to a gravitational force is studied by numerical simulations based on lattice Boltzmann methods (LBM) implemented with a finite difference scheme. A growth exponent $\alpha=1$ is measured in the direction of the external force.Comment: To appear in Communications in Computational Physics (CiCP

Time-dependent variational inequalities for viscoelastic contact problems

AbstractWe consider a class of abstract evolutionary variational inequalities arising in the study of contact problems for viscoelastic materials. We prove an existence and uniqueness result, using standard arguments of time-dependent elliptic variational inequalities and Banach's fixed point theorem. We then consider numerical approximations of the problem. We use the finite element method to discretize the spatial domain and we introduce spatially semi-discrete and fully discrete schemes. For both schemes, we show the existence of a unique solution, and derive error estimates. Finally, we apply the abstract results to the analysis and numerical approximations of a viscoelastic contact problem with normal compliance and friction

Analysis and control of a nonlinear boundary value problem

We consider a nonlinear two-dimensional boundary value problem which models the frictional contact of a bar with a rigid obstacle. The weak formulation of the problem is in the form of an elliptic variational inequality of the second kind. We establish the existence of a unique weak solution to the problem, then we introduce a regularized version of the variational inequality for which we prove existence, uniqueness and convergence results. We proceed with an optimal control problem for which we prove the existence of an optimal pair. Finally, we consider the corresponding optimal control problem associated to the regularized variational inequality and prove a convergence result