Stokes flow in logarithmic-viscosity space: velocity renormalization and a reference-operator formulation

Alessandro Forte , Shayan Kamali Lima

Effective viscosity in crystalline solids commonly depends exponentially on temperature, pressure, composition and other state variables, producing spatial variations over many orders of magnitude and making variable-viscosity Stokes flow numerically demanding. We develop an exact, coordinate-independent reformulation using logarithmic viscosity
\(\Lambda=\ln(\eta/\eta_0)\) and renormalized velocity \(\mathbf{v}=(\eta/\eta_0)\mathbf{u}\). The transformation identifies \(\boldsymbol{\xi}=\nabla\Lambda\) as the local measure of viscosity heterogeneity: its magnitude and direction determine how spatial viscosity variations modify the flow, while the absolute viscosity scale controls conversion back to the physical velocity. Multiplicative viscosity contrasts thereby become additive variations in \(\Lambda\),
and a fixed-reference-viscosity Stokes operator can be retained. Viscosity heterogeneity enters through coupling terms constructed from \(\mathbf{v}\) and \(\boldsymbol{\xi}\) together with a corresponding transformed mass-conservation source. We verify the formulation with an exact manufactured solution, a two-region Cartesian benchmark and a three-dimensional spherical-shell calculation with lateral viscosity contrast \(10^4\). In the spherical test, viscosity heterogeneity redistributes a simple buoyancy-driven poloidal response into broad-spectrum poloidal and toroidal flow contributions whose harmonic content agrees with theoretically predicted selection rules. The degeneracy related to the degree-one toroidal mode is resolved by imposing zero angular momentum on the reconstructed physical flow, while preserving dynamically forced differential rotation. Therefore, compared to an isoviscous reference solution, the variable-viscosity case develops substantial shorter-wavelength surface divergence and radial vorticity. The strong-viscosity-contrast calculation converges in 49 fixed-point updates, and spectral, resolution and physical-field reconstruction tests confirm that theoretically-predicted flow couplings are resolved.