Non-isothermal non-Newtonian fluids: the stationary case

TL;DR

This work analyzes stationary flows of incompressible non-Newtonian fluids with temperature-dependent viscosity under Dirichlet conditions, focusing on shear-thinning regimes with $p\in(1,2)$. It develops a rigorous theory for the coupled Navier–Stokes and heat equations, proving existence of weak solutions and conditional regularity; an approximating regularized problem is introduced to enable compactness arguments. The authors then formulate a Carreau-law finite element method, derive a priori error estimates, and validate the theory with 2D numerical experiments, including an analysis of the impact of the regularization parameter $\sigma$. The results provide solid theoretical guarantees and practical FE strategies for non-isothermal non-Newtonian flows, with implications for polymer processing and related applications.

Abstract

The stationary Navier-Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suit-able power law depending on $p \in (1,2)$ (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier-Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.

Figures (7)

• Figure 1: Test $1$. Convergence test for the Carreau model with $\eta_\infty=0.5, \eta_0=2, \lambda = 1$ and different values of $p=2, 1.6, 1.2$ (from top to bottom) using $\mathcal{P}_2/\mathcal{P}_1/\mathcal{P}_2$ (left) and $\mathcal{P}_3/\mathcal{P}_2/\mathcal{P}_3$ (right) finite elements
• Figure 2: Test $2$. Convergence test for the Carreau model with $\eta_\infty=0, \eta_0=2. \lambda = 1, p=2$ and different values of the regularization parameter $\sigma$ using $\mathcal{P}_2/\mathcal{P}_1/\mathcal{P}_2$ finite elements.
• Figure 3: Test $2$. Convergence test for the Carreau model with $\eta_\infty=0, \eta_0=2, \lambda = 1, p=1.6$ and different values of the regularization parameter $\sigma$ using $\mathcal{P}_2/\mathcal{P}_1/\mathcal{P}_2$ finite elements.
• Figure 4: Test $2$. Convergence test for the Carreau model with $\eta_\infty=0, \eta_0=2, \lambda = 1, p=1.2$ and different values of the regularization parameter $\sigma$ using $\mathcal{P}_2/\mathcal{P}_1/\mathcal{P}_2$ finite elements.
• Figure 5: Test $2$. Convergence test for the Carreau model with $\eta_\infty=0, \eta_0=2, \lambda = 1, p=2$ and different values of the regularization parameter $\sigma$ using $\mathcal{P}_3/\mathcal{P}_2/\mathcal{P}_3$ finite elements.

Theorems & Definitions (34)

• Definition 2.1
• Remark 2.1
• Theorem 3.1
• Remark 3.1
• Remark 3.2
• Theorem 3.2
• Remark 3.3
• Theorem 3.3
• Remark 3.4
• Theorem 3.4