Computational analysis of a contraction rheometer for the grade-two fluid model

TL;DR

This work develops a forward-modeling framework for the grade-two fluid in a contraction-rheometer geometry, introducing a transformed-equations formulation and an iterative solution scheme augmented by filtered Anderson acceleration to handle general parameters and inflow conditions. The authors implement high-order finite element discretizations and localized mesh refinement to robustly solve the resulting system, enabling computation of the contraction-region force $F$ as a function of flow rate $U$ and parameters $\nu,\alpha_1,\alpha_2$. They analyze force-based identifiability of the grade-two parameters, revealing a regime in which $\alpha_1,\alpha_2$ may be identifiable from rheometer data and highlighting limitations due to degeneracies and angular ambiguity in the parameter vector. The results demonstrate that viscosity can be inferred from small-$U$ asymptotics of $F/U$ and provide practical guidance on experimental design and mesh strategies to extract parameters in contraction-rheometer experiments.

Abstract

We explore the possibility of simulating the grade-two fluid model in a geometry related to a contraction rheometer, and we provide details on several key aspects of the computation. We show how the results can be used to determine the viscosity $ν$ from experimental data. We also explore the identifiability of the grade-two parameters $α_1$ and $α_2$ from experimental data. In particular, as the flow rate varies, force data appears to be nearly the same for certain distinct pairs of values $α_1$ and $α_2$; however we determine a regime for $α_1$ and $α_2$ for which the parameters may be identifiable with a contraction rheometer.

Figures (11)

• Figure 1: Flow in a contracting duct for Stokes flow ${\mathbf u}_{\rm S}$ and Navier-Stokes flow ${\mathbf u}_{\rm N}$. (a) horizontal flow component of ${\mathbf u}_{\rm S}$, and (b) horizontal flow component of ${\mathbf u}_{\rm N}-{\mathbf u}_{\rm S}$, for $R=10$, both with mesh parameter 64. The computational domain ${\Omega}$ is as specified in \ref{['eqn:expandom']}, with $b_i=1$, $b_o=1$, $L=1$, $H=0.5$. Computed using \ref{['eqn:navstonlyo']}.
• Figure 2: Horizontal flow of the difference ${\mathbf u}_{\rm G}-{\mathbf u}_{\rm N}$ for ${\mathbf u}_{\rm G}$ being the solution of the grade-two model \ref{['eqn:simpgtoodeeo']} with (a) $R=10$, $\alpha_1=10$, $\alpha_2=-10$, and (b) $R=40$, $\alpha_1=1$, $\alpha_2=-1$, both with mesh parameter 64. The computational domain ${\Omega}$ is as specified in \ref{['eqn:expandom']}, with $b_i=1$, $L=1$, $H=0.5$, and with (a) $b_o=1$, (b) $b_o=2$. Computed using \ref{['eqn:simpgtoodeeo']}.
• Figure 3: Horizontal flow component of the difference ${\mathbf u}_{\rm G}-{\mathbf u}_{\rm S}$ for $\nu=1$, $U = 2^{-2}$ and (a) $\alpha_1=\alpha_2=0.02$, (b) $\alpha_1=\alpha_2=0.2$. The computational domain ${\Omega}$ is as specified in \ref{['eqn:expandom']}, with $b_o = 1$, $b_i=1$, $L=1$, and $H=0.5$. The computational mesh was generated by four uniform refinements of the left-most mesh of figure \ref{['fig:cduct-mesh']}.
• Figure 4: Vertical component of the vector-valued auxiliary variable ${\mathbf w}$ for $\nu=1$, $U = 2^{-2}$ and (a) $\alpha_1=\alpha_2=0.02$, (b) $\alpha_1=\alpha_2=0.2$. The computational domain ${\Omega}$ is as specified in \ref{['eqn:expandom']}, with $b_o = 1$, $b_i=1$, $L=1$, and $H=0.5$. The computational mesh was generated by four uniform refinements of the left-most mesh of figure \ref{['fig:cduct-mesh']}.
• Figure 5: Generated data $f(U,\nu,{\boldsymbol \alpha}) = F(U)/U$ with the same computational domain and mesh as in table \ref{['tab:aitkendenti']} for ${\boldsymbol \alpha} = \alpha (\cos( j \pi/8),\sin(j \pi/8))$ with $\nu = 1$ and varying $\alpha$ and $j$. Left: $U = 2^{-8}$; center: $U = 2^{-7}$; right: $U = 2^{-6}$.

Theorems & Definitions (2)

• Theorem 1.1
• Theorem 1.4