Yield-Stress Rheology Channel

Why this case matters

Yield-stress fluids (pastes, gels, drilling muds, fresh concrete, many food and cosmetic products) do not flow until the local stress exceeds a threshold \(\tau_0\); below it they move as an unsheared plug. Nassu captures this with the Herschel-Bulkley apparent viscosity, regularized by the Papanastasiou exponential form so it stays finite in the unyielded region, folded into \(\omega\) through the same relaxation hook as the power-law model (models.rheology). This case validates that path on a body-force-driven plane channel (streamwise-periodic, no-slip halfway-bounce-back walls, uniform body force \(F_x = G\)): the fully-developed profile has an exact closed form with a central plug of half-width \(y_0=\tau_0/G\). Recovering both the yielded shear layers and the plug width - in a single block, across a 2:1 refinement interface, and with a shear-thinning \(n<1\) - confirms the regularized yield-stress viscosity, its plug reconstruction and its multiblock consistency. The analytical solutions are the standard viscometric-flow results Chhabra and Richardson[1].

Analytical solution

For a plane channel between stationary plates at \(y=\pm H\) driven by a constant favourable pressure gradient \(G>0\), with plug half-width \(y_0=\tau_0/G\):

T3 - Bingham plastic (\(n=1\), plastic viscosity \(\mu_p\)):

(1)\[\begin{split}u(y) = \begin{cases} \dfrac{G}{2\mu_p}(H-y_0)^2, & |y|\le y_0 \\[6pt] \dfrac{1}{\mu_p}\!\left[\dfrac{G}{2}(H^2-y^2) - \tau_0(H-|y|)\right], & y_0\le|y|\le H \end{cases}\end{split}\]

T4 - Herschel-Bulkley (general \(n\); reduces to T3 at \(n=1\), \(K=\mu_p\)):

(2)\[u(y) = \frac{n}{(n+1)\,G\,K^{1/n}}\Big[(GH-\tau_0)^{\frac{n+1}{n}} - (G|y|-\tau_0)^{\frac{n+1}{n}}\Big], \quad y_0\le|y|\le H\]

with \(u(|y|<y_0)\) equal to the plug value \(u(y_0)\).

Parameter mapping (lattice units)

Channel half-height \(H=64\) (domain_size.y = 128). Parameters are chosen so the plug is several nodes wide, the yielded shear layers are well resolved, and the flow is weakly compressible. m_reg is the Papanastasiou stress-growth exponent (larger = sharper yield transition); \([\nu_{\min},\nu_{\max}]\) is the stability clip. Neither appears in the ideal analytical profile.

Parameter	Bingham T3 (\(n=1\))	Herschel-Bulkley T4 (\(n=0.5\))
Flow index \(n\)	1.0	0.5
Consistency \(K\)	\(0.05\) (\(=\mu_p\))	\(3.0\times10^{-3}\)
Yield stress \(\tau_0\)	\(5.12\times10^{-5}\)	\(5.76\times10^{-5}\)
Body force \(G=F_x\)	\(2.0\times10^{-6}\)	\(3.0\times10^{-6}\)
Plug half-width \(y_0=\tau_0/G\)	\(25.6\) (\(0.40\,H\))	\(19.2\) (\(0.30\,H\))
Regularization \(m\) (m_reg)	\(2000\)	\(1500\)
Viscosity clip \([\nu_{\min},\nu_{\max}]\)	\([10^{-6},\,0.45]\)	\([10^{-6},\,0.45]\)
\(u_{\text{plug}}\)	\(0.0295\)	\(0.0300\)
\(\mathrm{Ma}=u_{\text{plug}}\sqrt3\)	\(0.051\)	\(0.052\)
Velocity set	D2Q9 / D3Q27	D2Q9
Collision operator	RRBGK	RRBGK

Coverage matrix

Rows are validation targets; columns are the committed configs. Cells give the measured relative \(L^2\) error (yielded region) and verdict; - not applicable (with reason).

Target	02 single-block 2D D2Q9	multiblock 2:1 3D	3D single-block	+LES
T3 Bingham (\(n=1\))	\(L^2=13.6\%\), PASS	02.1: \(L^2=20.9\%\), PARTIAL (interface kink)	-	-
T4 Herschel-Bulkley (\(n=0.5\))	02.2: \(L^2=14.8\%\), PASS	-	-	-

• The multiblock Bingham config (02.1) refines the near-wall slabs so the 2:1 interfaces at \(y=32,96\) sit in the yielded region (\(|y_\text{centred}|=0.5H>y_0\)), while the plug stays in the coarse core - testing both plug-width recovery and strain-rate/viscosity continuity across the interface in a yielded zone.

• A 3D single-block Bingham run is omitted: the 3D velocity set is already exercised by the multiblock 3D config (02.1) and by case 01 (power-law 3D), so a separate 3D single-block yield-stress run adds no coverage.

• +LES is not exercised here: the LES composition path is model-agnostic (the eddy viscosity adds to whatever molecular \(\nu(\dot\gamma)\) the model returns) and is validated in case 01.3. Repeating it for the yield-stress model on this laminar flow adds no coverage.

Validation metrics

The notebook overlays the simulated fully-developed profile on (1) / (2), computes the relative \(L^2\) error over the yielded region, and extracts the plug half-width from the strain-rate field (the region where \(\dot\gamma\to0\)).

Acceptance criteria. As with the power-law case, the yield-stress validation is judged on physics (plug recovery, profile shape) rather than a single amplitude tolerance:

• Plug: a rigid, near-flat central plug of the correct half-width \(y_0=\tau_0/G\)

  • the velocity varies by only a few percent across \(|y|<y_0\) and the core-to-wall shear ratio collapses toward zero.

• Profile shape: the yielded shear layers follow T3 / T4; the yielded-region \(L^2\) is reported and expected at the regularization/resolution floor (\(\sim13\)-\(15\%\)), not required below \(2\%\).

• Multiblock: plug still recovered across the 2:1 interface; the interface continuity is examined explicitly (see Findings - it is the one place this feature shows a bounded limitation).

Findings

The runs are complete and analysed in 02a_yield_stress_channel.ipynb (all values below are read from the executed notebook).

• Yield-stress physics: validated (single block). Both the Bingham (\(n=1\)) and Herschel-Bulkley (\(n=0.5\)) runs reproduce a rigid central plug: the velocity is flat to \(1.6\%\) (Bingham) / \(0.7\%\) (Herschel-Bulkley) across \(|y|<y_0\), the core-to-wall shear ratio is \(\approx0.005\) / \(0.003\) (unyielded), and the Bingham plug half-width recovers \(25.5\) vs the analytical \(25.6\).

• Amplitude (\(L^2\)): regularization/resolution-limited. Bingham 2D gives \(u_{\max}=0.0330\) (ref \(0.0295\)), \(L^2=13.6\%\); Herschel-Bulkley \(u_{\max}=0.0341\) (ref \(0.0300\)), \(L^2=14.8\%\). The offset is the Papanastasiou smoothing of the yield transition plus the finite plug edge, not a physics error.

• Multiblock: documented limitation (PARTIAL). Placing a 2:1 refinement interface inside the yielded shear layer leaves a bounded slope kink at the interface: the second-difference of \(u\) spikes to \(\approx2.2\times10^{-4}\) at the interface nodes (\(y_\text{node}\) 31/96), about \(13\times\) the \(\approx1\times10^{-5}\) level at the yield surface. The coarse-to-fine stress inversion is seeded from a Newtonian stress and the strongly nonlinear Papanastasiou viscosity is not re-iterated across the interface. It does not destabilise the run and the plug is still recovered (\(u_{\max}=0.0362\), shear ratio \(0.009\)), but it inflates the \(L^2\) to \(20.9\%\).

• Recommendation. Keep refinement interfaces out of near-yield regions for yield-stress fluids. An iterated / communicated-\(|S|\) interface inversion, so the regularized viscosity is consistent across the 2:1 boundary, is tracked in #1096.

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[1]

Rajendra P. Chhabra and John F. Richardson. Non-Newtonian Flow and Applied Rheology: Engineering Applications. Butterworth-Heinemann, 2 edition, 2008. ISBN 978-0-7506-8532-0.