[ADDITION] Added a variant of the Herschel Bulkley rheology

[mthielma]

Whats the purpose of this PR?

• Bug fix
• New feature
• Documentation update
• Other, please explain

Describe it in more detail below:
Based on Albert's earlier work, I have added a variant of Hershel-Bulkley rheology that is a bit more general than the one in Massmeyer et al. (2013). The rheology is given by:

$$ \eta( \dot{\varepsilon}_{II},T ) = \left( 1.0 - e^{-\frac{2\eta_0\dot{\varepsilon}_{II}}{\tau_0}} \right) \left( \frac{\tau_0}{2\dot{\varepsilon}_{II}} + η_T \left(\frac{\dot{\varepsilon}_{II}}{\dot{\varepsilon}_{r}}\right)^{\frac{1}{n} - 1} \right) $$

where $\eta_T$ is a temperature-dependent reference viscosity at a reference temperature $T_r$ and a reference strain rate $\dot\varepsilon_{r}$:

$$ \eta_T (T) = \eta_r e^{Q\left(T^{-1} - T_r^{-1}\right)} $$

and

$$ \dot{\varepsilon}_{r} = \frac{\tau_0}{2\eta_0} $$

is the reference strain rate at the Bingham yield stress.
This formulation is loosely based on the paper by Souza Mendes and Dutra (2004) and Massmeyer et al. (2013). I have adapted it so that it resembles more the formulations that are used in geodynamics.

This rheology may be useful for lower mantle rheology, but it is also often used to model magma rheology (when the magma contains crystals or bubbles). However, the current implementation does not include the effects of bubbles/crystals etc. on e.g. the yield stress or the power law exponent.

Before going on with this, I would like to get some feedback on the implementation and the formulation. I have not added any tests/documentation yet, but I will do so once everybody is happy with the implementation.

Mendes, P. R. S., & Dutra, E. S. S. (2004). Viscosity Function for Yield-Stress Liquids. Applied Rheology, 14(6), 296–302. https://doi.org/10.1515/arh-2004-0016

Massmeyer, A., Giuseppe, E. D., Davaille, A., Rolf, T., & Tackley, P. J. (2013). Numerical simulation of thermal plumes in a Herschel–Bulkley fluid. Journal of Non-Newtonian Fluid Mechanics, 195, 32–45. https://doi.org/10.1016/j.jnnfm.2012.12.004

Checklist

• The PR title is descriptive and starts with the appropriate tag: [BUGFIX], [ADDITION], [DOC], etc.
• New tests were added, or old tests were updated
• The new feature was added in a way that does not break public API
• New documentation related to the new feature was added
• The new code follows the Runic Style

[codecov]

Codecov Report

❌ Patch coverage is 89.13043% with 10 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/CreepLaw/HerschelBulkley.jl	91.11%	8 Missing ⚠️
src/Viscosity/Viscosity.jl	0.00%	2 Missing ⚠️

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[albert-de-montserrat]

Thanks Marcel! Looks mostly good to me!

[aelligp]

@albert-de-montserrat I think its either fastpow() or @pow but the macro @fastpow doesn't exist in GP that's why the tests fail now

[albert-de-montserrat]

Ah yes my bad, @fastpow is from a different pkg :D