# ⚛  L1 Principle — Oldroyd-B Viscoelastic Constitutive Model

**ID:** `L1-477` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Polymer Physics — *Viscoelastic flow*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** stress tensor field
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555300

---

## 🧠 1. Introduction

**Oldroyd-B Viscoelastic Constitutive Model** is a **nonlinear inverse problem** whose unknown lives in **stress tensor field** space, within the **Viscoelastic flow** sub-domain of **Polymer Physics**.

Measurements consist of none via a **rheometry plus velocimetry** sensing mechanism.

The forward operator applies, in order: E · momentum operator; E · UCM stress operator; a fixed-point or gradient iteration on the unknown; O · stress field operator.

Observations are corrupted by additive Gaussian noise. High-Weissenberg number problem (HWNP); stress singularities near geometric corners; log-conformation reformulation needed for Wi>10.

## ⚙ 2. Forward Model

Physical chain: **x** → E · momentum → E · UCM stress → O · stress field → **y** (detector).

```
y = `O.stress_field` `E.UCM_stress` `E.momentum` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.momentum` | E · momentum operator |
| `E.UCM_stress` | E · ucm stress operator |
| `O.stress_field` | O · stress field operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Polymer Physics |
| Sub domain | Viscoelastic flow |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | stress_tensor_field |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 5 |
| L dag | 3.6 |

## 📡 4. Measurement Model

High-Weissenberg number problem (HWNP); stress singularities near geometric corners; log-conformation reformulation needed for Wi>10.

| Metric | Value |
|---|---|
| Metric | N1_N2_relative_error |
| Secondary | velocity_profile_RMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `viscoelastic_parameter_inversion` · **Forward operator:** `oldroyd_b_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Re | — | 0.1 |
| Wi | — | 1 |
| Mesh n | N | 32 |
| Geometry | — | 4to1_contraction |
| Beta eta s over eta total | — | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Re | — | 0.0 – 100.0 |
| Wi | — | 0.1 – 20.0 |
| Mesh n | N | 16 – 256 |
| Beta eta s over eta total | — | 0.05 – 0.95 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** N1 relative error <= 0.05

| Metric | Range |
|---|---|
| N1 n2 relative error | 0.02 – 0.4 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **N1_N2_relative_error**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xa50db07913dec22c16ef6efaa491371a451618d9267f4a9e00c628a14bc3ad0d`
- **Chain tx hash:** `0xdbbaee78ff2caf4205f3c7c4624ea1cf07552e059453b6278e1f223e76984686`
- **Chain block:** `41555300`

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## File Mapping

This bundle consists of: `L1-477.md`, `L1-477.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-477.md` | Source of truth — edit this | Human or LLM |
| `L1-477.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-477`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._