A mathematical explanation of thetransition between laminar and turbulentflow in Newtonian fluids, using the Liegroups and finite element methods

摘要

In this scientific work, we use two effective methods : Lie groups theory and the finiteelement method, to explain why the transition from laminar flow to turbulence flowdepends on the variation of the Reynolds number. We restrict ourselves to the caseof incompressible viscous Newtonian fluid flows. Their governing equations, i.e. thecontinuity and Navier-Stokes equations are established and investigated. Their solutionsare expressed explicitly thanks to Lie's theory. The stability theory, which leads to aneigenvalue problem is used together with the finite element method, showing a way tocompute the critical Reynolds number, for which the transition to turbulence occurs.The stationary flow is also studied and a finite element method, the Newton method, isused to prove the stability of its convergence, which is guaranteed for small variations ofthe Reynolds number.