{
  "version": 1,
  "topic": "RF09",
  "title": "Darcy versus Forchheimer when inertia changes the equation",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Darcy-Forchheimer Superficial Velocity",
    "premise": "Fixed controls: mu = 0.001 Pa.s; k = 1e-10 m^2; beta = 100000000 1/m; rho = 1000 kg/m3. G is varied as shown.",
    "formula": "v=\\frac{2G}{\\mu/k+\\sqrt{(\\mu/k)^2+4\\beta\\rho G}}",
    "variables": "G: Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m). mu: Dynamic Viscosity (Pa.s). k: Intrinsic Permeability (m^2). beta: Inertial Resistance Coefficient (Not Dimensionless c_F) (1/m). rho: Fluid Density (kg/m3).",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 100000",
        "explanation": "Fixed controls: mu = 0.001 Pa.s; k = 1e-10 m^2; beta = 100000000 1/m; rho = 1000 kg/m3. G is varied as shown."
      },
      {
        "label": "Select the relation",
        "calculation": "v=\\frac{2G}{\\mu/k+\\sqrt{(\\mu/k)^2+4\\beta\\rho G}}",
        "explanation": "G: Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m). mu: Dynamic Viscosity (Pa.s). k: Intrinsic Permeability (m^2). beta: Inertial Resistance Coefficient (Not Dimensionless c_F) (1/m). rho: Fluid Density (kg/m3)."
      },
      {
        "label": "Substitute the values",
        "calculation": "G = 100000; mu = 0.001; k = 1e-10; beta = 100000000; rho = 1000 → v = 0.00095124922 m/s",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "v (m/s) = 0.00095124922",
        "explanation": "The inertial term limits superficial velocity relative to the linear Darcy extrapolation."
      }
    ],
    "result": "v (m/s) = 0.00095124922",
    "interpretation": "The inertial term limits superficial velocity relative to the linear Darcy extrapolation.",
    "control": "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m)",
    "columns": [
      "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m)",
      "v (m/s)"
    ],
    "rows": [
      [
        1000,
        0.00006180339887498948
      ],
      [
        10000,
        0.00027015621187164243
      ],
      [
        100000,
        0.0009512492197250393
      ],
      [
        1000000,
        0.0031126729201736937
      ],
      [
        10000000,
        0.00995012499921876
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 1000. v (m/s): 0.0000618033989.",
      "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 10000. v (m/s): 0.000270156212.",
      "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 100000. v (m/s): 0.00095124922.",
      "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 1000000. v (m/s): 0.00311267292.",
      "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 10000000. v (m/s): 0.009950125."
    ],
    "provenance": "Replayed with the same equation record and calculation engine as /calculators/darcy-forchheimer-superficial-velocity. Original synthetic inputs; engine outputs are saved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline v (m/s) without rounding intermediate values.",
        "answer": "G = 100000; mu = 0.001; k = 1e-10; beta = 100000000; rho = 1000 → v = 0.00095124922 m/s. v (m/s) = 0.00095124922. The inertial term limits superficial velocity relative to the linear Darcy extrapolation."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) changes from 1000 to 10000000. Other controls remain as stated in the premise. v (m/s) changes from 0.0000618033989 to 0.009950125."
      },
      {
        "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. G: Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m). mu: Dynamic Viscosity (Pa.s). k: Intrinsic Permeability (m^2). beta: Inertial Resistance Coefficient (Not Dimensionless c_F) (1/m). rho: Fluid Density (kg/m3). The inertial term limits superficial velocity relative to the linear Darcy extrapolation."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "darcy-forchheimer-superficial-velocity",
    "parameter": "G",
    "inputs": {
      "G": "1000000",
      "mu": "0.001",
      "k": "1e-10",
      "beta": "100000000",
      "rho": "1000"
    },
    "values": [
      1000,
      10000,
      100000,
      1000000,
      10000000
    ],
    "outputs": [
      {
        "expression": "2 * G / (mu / k + sqrt((mu / k)^2 + 4 * beta * rho * G))",
        "intermediates": {},
        "inputs": {
          "G": 1000,
          "mu": 0.001,
          "k": 1e-10,
          "beta": 100000000,
          "rho": 1000
        },
        "target": "v",
        "unit": "m/s",
        "value": 0.00006180339887498948
      },
      {
        "expression": "2 * G / (mu / k + sqrt((mu / k)^2 + 4 * beta * rho * G))",
        "intermediates": {},
        "inputs": {
          "G": 10000,
          "mu": 0.001,
          "k": 1e-10,
          "beta": 100000000,
          "rho": 1000
        },
        "target": "v",
        "unit": "m/s",
        "value": 0.00027015621187164243
      },
      {
        "expression": "2 * G / (mu / k + sqrt((mu / k)^2 + 4 * beta * rho * G))",
        "intermediates": {},
        "inputs": {
          "G": 100000,
          "mu": 0.001,
          "k": 1e-10,
          "beta": 100000000,
          "rho": 1000
        },
        "target": "v",
        "unit": "m/s",
        "value": 0.0009512492197250393
      },
      {
        "expression": "2 * G / (mu / k + sqrt((mu / k)^2 + 4 * beta * rho * G))",
        "intermediates": {},
        "inputs": {
          "G": 1000000,
          "mu": 0.001,
          "k": 1e-10,
          "beta": 100000000,
          "rho": 1000
        },
        "target": "v",
        "unit": "m/s",
        "value": 0.0031126729201736937
      },
      {
        "expression": "2 * G / (mu / k + sqrt((mu / k)^2 + 4 * beta * rho * G))",
        "intermediates": {},
        "inputs": {
          "G": 10000000,
          "mu": 0.001,
          "k": 1e-10,
          "beta": 100000000,
          "rho": 1000
        },
        "target": "v",
        "unit": "m/s",
        "value": 0.00995012499921876
      }
    ]
  }
}