Abstract

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b ( x ) , and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P 1 -discontinuous, P 2 -continuous and P 1 -continuous finite elements respectively.

Highlights

  • In nature, most fluids are non-Newtonian, and many researchers have investigated the behaviour of non-Newtonian fluid extensively

  • + C12 (1 − α)−1 H 3/2, which completes the proof of the error analysis of the two-level method for Oseen viscoelastic fluid flow

  • In 4:1 contraction channel flow, we demonstrate the graphical representation of streamlines, pressure oscillation, and reentrant corner flow behaviour of the steady-state viscoelastic fluid flow model and Oseen viscoelastic fluid flow model for one-level and two-level algorithms

Read more

Summary

Introduction

Most fluids are non-Newtonian, and many researchers have investigated the behaviour of non-Newtonian fluid extensively. In the viscoelastic fluid flow model, the non-linearity occurs only in the constitutive equation [2]. To solve Oseen viscoelastic fluid flow, Lee et al used domain decomposition method in [12], the defect correction process at high Weissenberg number in [13], two-level stabilized mixed finite element method in [14], stabilized Lagrange-Galerkin method for the nonlinear scheme in [15]. We consider the two-level method to investigate the Oseen viscoelastic fluid flow for error estimation. Since in the Oseen viscoelastic fluid flow model, the constitutive equation is already linear where the non-linearity vanishes because of creeping flow. This new feature allows us to consider a two-level approach differently.

Model Equations
The Weak Derivative and Finite Element Discretization
Two-Level Method for Steady State Viscoelastic Fluid Flow Model
Two-Level Method for Oseen Viscoelastic Fluid Flow
Existence and Uniqueness of the Finite Element Solution
C2 Mdh
Error Analysis
Numerical Tests
Analytic Solution Test
Conclusions
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call