Non-Newtonian Fluid
Non-Newtonian Fluid: Theoretical Foundations
What is a Non-Newtonian Fluid?
Professor, my image of a non-Newtonian fluid is just "a fluid with weird viscosity." Could you explain it a bit more properly?
For Newtonian fluids, shear stress $\tau$ and shear rate $\dot{\gamma}$ have a linear relationship $\tau = \mu \dot{\gamma}$. Fluids for which this linear relationship does not hold are non-Newtonian fluids. There are many around us. Blood, paint, ketchup, shampoo, polymer solutions, cement slurry, etc.
Non-Newtonian fluids are broadly classified as follows.
| Type | Characteristics | Examples |
|---|---|---|
| Shear-thinning (Pseudoplastic) | Shear rate ↑ → Viscosity ↓ | Paint, blood, polymer solutions |
| Shear-thickening (Dilatant) | Shear rate ↑ → Viscosity ↑ | Cornstarch-water mixture |
| Bingham Plastic | Does not deform below yield stress | Toothpaste, chocolate |
| Viscoelastic Fluid | Possesses both viscosity and elasticity | Polymer melt, DNA solution |
| Thixotropy | Viscosity decreases over time | Paint, yogurt |
So ketchup being hard to get out is also non-Newtonian! Shaking it lowers the viscosity, making it easier to pour.
Constitutive Equation (Viscosity Model)
The core of non-Newtonian fluids lies in the formulation of the viscosity function $\eta(\dot{\gamma})$, which depends on shear rate. Let me introduce the main models.
1. Power-law Model
It is the simplest model.
$K$ is the consistency index, $n$ is the power-law index. $n < 1$ for shear-thinning, $n > 1$ for shear-thickening. It reduces to a Newtonian fluid when $n = 1$.
2. Carreau Model
A model that improves upon the shortcomings of the Power-law (unrealistic behavior at low/high shear rates).
$\eta_0$ is the zero-shear viscosity, $\eta_\infty$ is the infinite-shear viscosity, $\lambda$ is the relaxation time.
3. Herschel-Bulkley Model
For fluids with yield stress (a generalization of Bingham plastic).
$\tau_y$ is the yield stress. For $\tau < \tau_y$, there is no flow (rigid behavior). It reduces to the Bingham model when $n = 1$.
Fluids with yield stress seem numerically tricky to handle.
Exactly. Since the location of the yield surface is unknown, regularization techniques are often used. In the Papanastasiou model:
Increasing the parameter $m$ brings it closer to ideal Bingham behavior, but numerically increases stiffness. $m = 100\text{--}1000\,\text{s}$ is a typical range.
The Foundation of Non-Newtonian Fluid Theory—Bingham and Ostwald-de Waele (1906–1929)
Systematic research on non-Newtonian fluids began in the early 20th century. Eugene Bingham (1916) measured the "Yield Stress" of mud slurries and pastes and proposed the "Bingham Plastic" model, which behaves linearly after yielding. Meanwhile, Ostwald (1925) and de Waele (1923) independently formulated the "Power Law" fluid (τ=K·γ̇ⁿ), providing a unified description for shear-thinning (n<1: blood, polymer solutions) and shear-thickening (n>1: cornstarch-water mixture). These century-old models are still implemented as the foundation of CFD material model libraries and served as the starting point for more refined Cross, Casson, and Herschel-Bulkley models.
Computational Methods for Non-Newtonian Fluid
Numerical Methods for Non-Newtonian Fluids
When solving non-Newtonian fluids with CFD, what's different from Newtonian fluids?
The fundamental difference is that viscosity depends on the velocity field. The viscous term in the Navier-Stokes equations becomes nonlinear.
$\mathbf{D} = \frac{1}{2}(\nabla\mathbf{u} + \nabla\mathbf{u}^T)$ is the strain rate tensor.
As a solution method, Picard iteration (successive substitution method) becomes the basis.
1. Calculate $\dot{\gamma}$ from the previous step's velocity field
2. Update $\eta(\dot{\gamma})$
3. Solve the N-S equations with the updated viscosity to obtain a new velocity field
4. Repeat steps 1-3 until convergence
So there's one more iteration loop compared to Newtonian fluids.
Correct. The difficulty of non-Newtonian fluid computation lies in this outer iteration loop being hard to converge. Special care is needed when shear-thinning is strong (small $n$) or when yield stress is present.
Generalized Reynolds Number
For non-Newtonian fluids, the definition of Reynolds number also changes. For pipe flow of Power-law fluids, the Metzner-Reed generalized Reynolds number is used.
Laminar-turbulent transition occurs around $\text{Re}_{\text{MR}} \approx 2100$ (almost the same critical value as Newtonian fluids). However, post-transition turbulent behavior differs significantly from Newtonian fluids.
The generalized Re number formula is quite complex.
Discretization in the Finite Volume Method
Let me explain the discretization points to note when solving non-Newtonian fluids with CFD.
Face value interpolation of viscosity becomes important. When interpolating viscosity calculated at cell centers to faces:
- Harmonic Mean: Recommended for interfaces where viscosity changes sharply. $\eta_f = \frac{2\eta_L \eta_R}{\eta_L + \eta_R}$
- Arithmetic Mean: For cases where viscosity variation is gradual. $\eta_f = \frac{\eta_L + \eta_R}{2}$
For shear-thinning fluids, viscosity can differ by orders of magnitude between near-wall regions and the channel center. For example, in injection molding of polymer melts, $\eta$ can vary in the range of $10^1 \sim 10^4\,\text{Pa}\cdot\text{s}$. To handle this sharp change, the mesh near walls needs to be sufficiently refined.
| Power-law $n$ | Viscosity Ratio $\eta_{\text{center}}/\eta_{\text{wall}}$ | Mesh Requirement |
|---|---|---|
| 0.8 | ~3 | Mild refinement sufficient |
| 0.5 | ~30 | 10+ layers near wall recommended |
| 0.2 | ~1000 | Very fine mesh required |
The smaller $n$ is, the stricter the mesh requirement becomes. Convergence probably gets harder too.
Exactly. A practical technique is to lower the relaxation factor to 0.3-0.5 and also apply relaxation to the viscosity update.
Numerical Stability of Non-Newtonian Fluid CFD—Singularity Handling for Yield-Stress Fluids (Bingham Fluids)
In CFD for Bingham fluids (yield stress τ_y>0), the discontinuity in material properties—"solid below yield stress, liquid above"—causes numerical instability. In the limit of shear rate γ̇→0, the viscosity of the Bingham model diverges to infinity, causing Newton's method to fail to converge. A practical solution is Papanastasiou's (1987) regularization approximation: τ = (τ_y/γ̇)(1-exp(-mγ̇)) + ηγ̇, where parameter m (typical values 10³–10⁵) balances convergence and accuracy. If m is too large, numerical stiffness causes divergence; if too small, the yield stress effect is underestimated. An appropriate m value is around 100 times τ_y/η (inverse of viscous time scale), and it is recommended to verify this along with mesh refinement.
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