Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

Abstract

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity-pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier-Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.

Comparison of computational times and accuracy for finite element solution of Navier–Stokes equations and Ladyzhenskaya equations

“…Argyis triangle, Clough-Tocher triangle, Bogner-Fox-Schmidt rectangle, and Bicubic spline rectangle [8] are examples of finite-element spaces for the streamfunction formulation of the LE. In [12], we established the error bound given in the following theorem. This theorem and its proof can also be found in [12].…”

“…In [12], we established the error bound given in the following theorem. This theorem and its proof can also be found in [12]. Table I shows the error estimates.…”

“…This uniqueness property has been proved in [12]. In this paper, we focus our attention on finite element approximations of the model problems [WSL1] and [WSL2] described above.…”

“…Existence and uniqueness and finite element error analysis of the weak form [WSL1] were carried out in [12].…”

On the well-posedness of generalized Darcy–Forchheimer equation

“…To see this, we multiply equation (2) by ϕ ∈ X and integrate over . We then apply the conditions in equation (3) to the Green's formula (14):…”