Planar Poiseuille non-Newtonian flow

Overview

• Solver: pimpleFoam
• Experimental case described by Amoreira and Olivera 3
• Start-up planar Poiseuille flow of a non-Newtonian fluid
• Modelled using the Maxwell viscoelastic laminar stress model
• Initially at rest
• Constant pressure gradient applied from time zero
• planarPoiseuille

Mesh

• 1D structured mesh created using blockMesh

Results

The predictions are compared against the analytical solution for velocity given by Waters and King 87

$$u(y,t) = \frac{3}{2}(1 - y^2) + \sum\limits_{k=1}^\infty A_k(t)B_k(t),$$ $$A_k(t) = e^{-b_k t} \begin{cases} \frac{b_k-a_k^2}{c_k}\sinh c_k t + \cosh c_k t, & \text{if } b_k \geq a_k, \\ \frac{b_k-a_k^2}{c_k}\sin c_k t + \cos c_k t, & \text{if } b_k \lt a_k, \end{cases}$$ $$B_k(y) = \frac{48(-1)^k}{(2k -1)^3\pi^3} \cos \frac{2k - 1}{2}\pi y,$$

where

$$a_k = \frac{2k - 1}{2}\pi \sqrt{E},\; b_k = \frac{1+\beta a_k^2}{2},\; c_k = \sqrt{|b_k^2 - a_k^2|}.$$