Analysis of partial differential equations with time-periodic forcing, applications to Navier-Stokes equations

Abstract

Flows with time-periodic forcing can be found in various applications, such as the circulatory and respiratory systems, or industrial mixers. In this thesis, we address few questions in relation with the time-periodic forcing of flows and related partial differential equations (PDE), including the linear Advection-Diffusion equation. In Chapter 2, we first study linear PDE's with non-symmetric operators subject to time-periodic forcing. We prove that they have a unique time-periodic solution which is stable and attracts any initial solution if the bilinear form associated to the operator is coercive, and we obtain an error estimate for finite element method with a backward Euler time-stepping scheme. That general theory is applied to the Advection-Diffusion equation and the Stokes problem. The first equation has a non-symmetric operator, while the second has a symmetric operator but two unknowns, the velocity and pressure. To apply the general theory, we prove an error estimate for a Riesz projection operator, using a special Aubin-Nistche argument for the Advection-Diffusion equation with a tune-dependent advective velocity. A spectral analysis for the 1-D Advection-Diffusion equation, relevant parameters that control the speed of convergence of any initial solution to the time-periodic solution are identified. In Chapter 3, we extend a theorem of J.L. Lions about the existence of time-periodic solutions of Navier-Stokes equations under periodic distributed forcing with homogeneous Dirichlet boundary conditions to the case of non-homogeneous time-periodic Dirichlet boundary conditions. Our theorem predicts the existence of a time-periodic solution for Navier-Stokes equations subject to time-periodic forcing but the stability of these time-periodic solutions is not known. In Chapter 4, we investigate the stability of these time-periodic solutions, through numerical simulations with test cases in a 2-D time-periodic lid driven cavity and a 2-D constricted channel with a time-periodic inflow. From our numerical simulations, it seems that a bifurcation occurs in the range 3000--8000 in the periodically driven cavity, and the range 400--1200 in the periodically driven channel.