Herschel–Bulkley formulation for non-Newtonian flows

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TL;DR

The Herschel–Bulkley model unifies Newtonian, Bingham, and power-law fluids via a yield stress and a strain-rate-dependent viscosity. An -regularization ensures stable computations and recovers simpler models (Newtonian, Bingham) by tuning model parameters. Dimensionless groups (e.g., the plasto-capillary number  and the effective Ohnesorge) capture the interplay of fluid rheology, capillarity, and flow scales. Implementation details are provided, along with references, open-source code, and demonstrations of bubble-burst simulations in viscoplastic media.

Features:

• Yield stress
• Power law dependance on the strain rate
  • Shear thinning for .
  • Shear thickening for .
• Bingham model for .
• Newtonian fluid for and .

-formulation

Normalizing stresses with , length with , and velocity with …

Here, the effective Ohnesorge is

The plasto-capillary number is

One can easily see that putting recovers the Bingham model with . Additionally, with & = 0, the model will give a Newtonian response.

More details on the implementation

Calculate the norm of the deformation tensor :

The second invariant is (this is the Frobenius norm)

Note: .

We use the formulation as given in Balmforth et al. (2013) [1], who use the strain rate tensor which and its norm . Of course, given .

Calculate the equivalent viscosity

Factorizing with to obtain an equivalent viscosity

In this formulation, is a small number to ensure numerical stability. The term is equivalent to the of the previous (v1.0, see: GitHub) formulation [2].

Note: The fluid flows always, it is not a solid, but a very viscous fluid.

Reproduced from: P.-Y. Lagrée’s Sandbox. Here, we use a face implementation of the regularisation method, described here.

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Further exploration:

Video showcasing a typical simulation of bubble bursting in a Herschel–Bulkley fluid medium

Open on YouTube

More resources

GitHub	Demo	License	Latest Changes	pdf

References

[1] N. J. Balmforth, I. A. Frigaard, and G. Ovarlez, “Yielding to Stress: Recent Developments in Viscoplastic Fluid Mechanics,” Annu. Rev. Fluid Mech., vol. 46, pp. 121–146, Jan. 2014, doi: 10.1146/annurev-fluid-010313-141424.

[2] V. Sanjay, D. Lohse, and M. Jalaal, “Bursting bubble in a viscoplastic medium,” J. Fluid Mech., vol. 922, p. A2, 2021.

Metadata

Author:: Vatsal Sanjay Date published:: Dec 31, 2024
Date modified:: Jan 26, 2025 at 11:50 CET

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