(vc_yield_stress_channel)=

# 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
{footcite:t}`chhabra2008nonnewtonian`.

## 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 $\mu_p$):

```{math}
---
label: vc_yield_stress_channel_t3
---
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}
```

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

```{math}
---
label: vc_yield_stress_channel_t4
---
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
{eq}`vc_yield_stress_channel_t3` / {eq}`vc_yield_stress_channel_t4`, 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](https://github.com/AeroSim-CFD/nassu/issues/1096).

```{toctree}
---
hidden:
---
02a_yield_stress_channel.ipynb
```

```{footbibliography}
```