{
  "version": 1,
  "topic": "DH11",
  "title": "Why a good rheology fit can fail outside the measured range",
  "createdAt": "2026-10-24",
  "case": {
    "title": "A two-point high-shear fit extrapolated",
    "premise": "Bingham line fitted only at 400 and 900 s⁻¹ from τ = 10 + 0.2√γ. Compare its low-shear extrapolation with the generating curve.",
    "formula": "τBingham = 12.4 + 0.004γ; τHB = 10 + 0.2√γ",
    "variables": "Stress in Pa, shear rate in s⁻¹. The Bingham and power-law objectives differ; no common-objective accuracy ranking is implied.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Shear rate (1/s) = 100",
        "explanation": "Bingham line fitted only at 400 and 900 s⁻¹ from τ = 10 + 0.2√γ. Compare its low-shear extrapolation with the generating curve."
      },
      {
        "label": "Select the relation",
        "calculation": "τBingham = 12.4 + 0.004γ; τHB = 10 + 0.2√γ",
        "explanation": "Stress in Pa, shear rate in s⁻¹. The Bingham and power-law objectives differ; no common-objective accuracy ranking is implied."
      },
      {
        "label": "Substitute the values",
        "calculation": "At γ = 100 s⁻¹: τHB = 10 + 0.2√100 = 12 Pa; Bingham uses the recorded fitted intercept and slope; power law uses exp(intercept) × 100^slope",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "HB generating stress (Pa) = 12; Bingham predicted stress (Pa) = 12.8; Power-law predicted stress (Pa) = 12.8416811",
        "explanation": "An exact two-point fit at high shear does not establish low-shear behavior."
      }
    ],
    "result": "HB generating stress (Pa) = 12; Bingham predicted stress (Pa) = 12.8; Power-law predicted stress (Pa) = 12.8416811",
    "interpretation": "An exact two-point fit at high shear does not establish low-shear behavior.",
    "control": "Shear rate (1/s)",
    "columns": [
      "Shear rate (1/s)",
      "HB generating stress (Pa)",
      "Bingham predicted stress (Pa)",
      "Power-law predicted stress (Pa)"
    ],
    "rows": [
      [
        1,
        10.2,
        12.404,
        9.583832069412978
      ],
      [
        4,
        10.4,
        12.416,
        10.466341363678211
      ],
      [
        25,
        11,
        12.5,
        11.7588859995263
      ],
      [
        100,
        12,
        12.8,
        12.84168107665487
      ],
      [
        400,
        14,
        14,
        14.024183318144175
      ],
      [
        900,
        16,
        16,
        14.765757442983329
      ]
    ],
    "baseline": 3,
    "rowNotes": [
      "Shear rate (1/s) = 1. HB generating stress (Pa): 10.2; Bingham predicted stress (Pa): 12.404; Power-law predicted stress (Pa): 9.58383207.",
      "Shear rate (1/s) = 4. HB generating stress (Pa): 10.4; Bingham predicted stress (Pa): 12.416; Power-law predicted stress (Pa): 10.4663414.",
      "Shear rate (1/s) = 25. HB generating stress (Pa): 11; Bingham predicted stress (Pa): 12.5; Power-law predicted stress (Pa): 11.758886.",
      "Shear rate (1/s) = 100. HB generating stress (Pa): 12; Bingham predicted stress (Pa): 12.8; Power-law predicted stress (Pa): 12.8416811.",
      "Shear rate (1/s) = 400. HB generating stress (Pa): 14; Bingham predicted stress (Pa): 14; Power-law predicted stress (Pa): 14.0241833.",
      "Shear rate (1/s) = 900. HB generating stress (Pa): 16; Bingham predicted stress (Pa): 16; Power-law predicted stress (Pa): 14.7657574."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline HB generating stress (Pa) without rounding intermediate values.",
        "answer": "At γ = 100 s⁻¹: τHB = 10 + 0.2√100 = 12 Pa; Bingham uses the recorded fitted intercept and slope; power law uses exp(intercept) × 100^slope. HB generating stress (Pa) = 12. An exact two-point fit at high shear does not establish low-shear behavior."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Shear rate (1/s) changes from 1 to 900. Other controls remain as stated in the premise. HB generating stress (Pa) changes from 10.2 to 16."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. Stress in Pa, shear rate in s⁻¹. The Bingham and power-law objectives differ; no common-objective accuracy ranking is implied. An exact two-point fit at high shear does not establish low-shear behavior."
      }
    ]
  },
  "recorded": null
}