{
  "version": 1,
  "topic": "DH08",
  "title": "Explore yield stress and the unyielded annular region",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Herschel-Bulkley Annular Pressure Loss Calculator",
    "premise": "Fixed controls: Outer bore inside diameter = 0.2 m; Inner pipe outside diameter = 0.1 m; Actual annular volume flow rate = 0.001 m3/s; Measured consistency factor = 0.2 Pa.s^n; Measured flow exponent = 0.5 dimensionless; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m.",
    "formula": "stress magnitude = Gp/2 * abs(r1*r2/r-r); r2-r1 = 2*tau_y/Gp; onset = 2*tau_y/(Ro-Ri); shear rate = (max(0,stress-tau_y)/K)^(1/n).; Inner and outer yielded-layer shear-rate integrals give the same plug velocity; Q = integral(2*pi*r*u(r), Ri..Ro); solve Gp for the requested Q; friction loss = Gp*L.",
    "variables": "Measured dynamic yield stress is varied within the supported input range. Incompressible, time-independent HB liquid; fully developed laminar axial flow in a stationary concentric circular annulus with no wall slip. No eccentricity, pipe rotation, turbulence, entrance losses, thixotropy, viscoelasticity, cuttings transport, gas or thermal/property variation is modeled.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Measured dynamic yield stress (Pa) = 5",
        "explanation": "Fixed controls: Outer bore inside diameter = 0.2 m; Inner pipe outside diameter = 0.1 m; Actual annular volume flow rate = 0.001 m3/s; Measured consistency factor = 0.2 Pa.s^n; Measured flow exponent = 0.5 dimensionless; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m."
      },
      {
        "label": "Select the relation",
        "calculation": "stress magnitude = Gp/2 * abs(r1*r2/r-r); r2-r1 = 2*tau_y/Gp; onset = 2*tau_y/(Ro-Ri); shear rate = (max(0,stress-tau_y)/K)^(1/n).; Inner and outer yielded-layer shear-rate integrals give the same plug velocity; Q = integral(2*pi*r*u(r), Ri..Ro); solve Gp for the requested Q; friction loss = Gp*L.",
        "explanation": "Measured dynamic yield stress is varied within the supported input range. Incompressible, time-independent HB liquid; fully developed laminar axial flow in a stationary concentric circular annulus with no wall slip. No eccentricity, pipe rotation, turbulence, entrance losses, thixotropy, viscoelasticity, cuttings transport, gas or thermal/property variation is modeled."
      },
      {
        "label": "Substitute the values",
        "calculation": "tau_y = 5 Pa; with the fixed controls above, the workflow evaluates Annular friction pressure loss = 243142.551 Pa; Inner plug boundary radius = 0.053535408 m; Outer plug boundary radius = 0.0946635444 m",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Annular friction pressure loss (Pa) = 243142.551; Inner plug boundary radius (m) = 0.053535408; Outer plug boundary radius (m) = 0.0946635444",
        "explanation": "Software envelopes: 0.05 <= d/D <= 0.95, 0.3 <= n <= 1, nonnegative yield stress, gradient <= 2,000,000 Pa/m and inertia screening number <= 1,000. These are conditioning/conservative implementation limits, not experimentally validated field ranges. A yield-stress/reference-stress ratio above 10,000 is rejected as unresolved near onset. The output excludes hydrostatic head and all local losses. It is not total standpipe pressure, an ECD design or a hole-cleaning assessment. At tau_y = 0 and n = 1 it reduces to the Newtonian concentric solution; n < 1 and tau_y = 0 gives power-law flow. Density changes the screen, not the viscous gradient."
      }
    ],
    "result": "Annular friction pressure loss (Pa) = 243142.551; Inner plug boundary radius (m) = 0.053535408; Outer plug boundary radius (m) = 0.0946635444",
    "interpretation": "Software envelopes: 0.05 <= d/D <= 0.95, 0.3 <= n <= 1, nonnegative yield stress, gradient <= 2,000,000 Pa/m and inertia screening number <= 1,000. These are conditioning/conservative implementation limits, not experimentally validated field ranges. A yield-stress/reference-stress ratio above 10,000 is rejected as unresolved near onset. The output excludes hydrostatic head and all local losses. It is not total standpipe pressure, an ECD design or a hole-cleaning assessment. At tau_y = 0 and n = 1 it reduces to the Newtonian concentric solution; n < 1 and tau_y = 0 gives power-law flow. Density changes the screen, not the viscous gradient.",
    "control": "Measured dynamic yield stress (Pa)",
    "columns": [
      "Measured dynamic yield stress (Pa)",
      "Annular friction pressure loss (Pa)",
      "Inner plug boundary radius (m)",
      "Outer plug boundary radius (m)"
    ],
    "rows": [
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        0,
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        1,
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        2,
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      [
        10,
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    "baseline": 3,
    "rowNotes": [
      "Measured dynamic yield stress (Pa) = 0. Annular friction pressure loss (Pa): 20694.7344; Inner plug boundary radius (m): 0.0728289495; Outer plug boundary radius (m): 0.0728289495.",
      "Measured dynamic yield stress (Pa) = 1. Annular friction pressure loss (Pa): 69055.0864; Inner plug boundary radius (m): 0.0587379667; Outer plug boundary radius (m): 0.0877003517.",
      "Measured dynamic yield stress (Pa) = 2. Annular friction pressure loss (Pa): 113907.47; Inner plug boundary radius (m): 0.056054346; Outer plug boundary radius (m): 0.091170568.",
      "Measured dynamic yield stress (Pa) = 5. Annular friction pressure loss (Pa): 243142.551; Inner plug boundary radius (m): 0.053535408; Outer plug boundary radius (m): 0.0946635444.",
      "Measured dynamic yield stress (Pa) = 10. Annular friction pressure loss (Pa): 452762.896; Inner plug boundary radius (m): 0.0522977601; Outer plug boundary radius (m): 0.0964709911."
    ],
    "provenance": "Original recorded synthetic cases solved with the site's herschel-bulkley-annular-pressure-loss engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.",
    "questions": [
      {
        "question": "Reproduce the baseline Annular friction pressure loss (Pa) without rounding intermediate values.",
        "answer": "tau_y = 5 Pa; with the fixed controls above, the workflow evaluates Annular friction pressure loss = 243142.551 Pa; Inner plug boundary radius = 0.053535408 m; Outer plug boundary radius = 0.0946635444 m. Annular friction pressure loss (Pa) = 243142.551. Software envelopes: 0.05 <= d/D <= 0.95, 0.3 <= n <= 1, nonnegative yield stress, gradient <= 2,000,000 Pa/m and inertia screening number <= 1,000. These are conditioning/conservative implementation limits, not experimentally validated field ranges. A yield-stress/reference-stress ratio above 10,000 is rejected as unresolved near onset. The output excludes hydrostatic head and all local losses. It is not total standpipe pressure, an ECD design or a hole-cleaning assessment. At tau_y = 0 and n = 1 it reduces to the Newtonian concentric solution; n < 1 and tau_y = 0 gives power-law flow. Density changes the screen, not the viscous gradient."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Measured dynamic yield stress (Pa) changes from 0 to 10. Other controls remain as stated in the premise. Annular friction pressure loss (Pa) changes from 20694.7344 to 452762.896."
      },
      {
        "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. Measured dynamic yield stress is varied within the supported input range. Incompressible, time-independent HB liquid; fully developed laminar axial flow in a stationary concentric circular annulus with no wall slip. No eccentricity, pipe rotation, turbulence, entrance losses, thixotropy, viscoelasticity, cuttings transport, gas or thermal/property variation is modeled. Software envelopes: 0.05 <= d/D <= 0.95, 0.3 <= n <= 1, nonnegative yield stress, gradient <= 2,000,000 Pa/m and inertia screening number <= 1,000. These are conditioning/conservative implementation limits, not experimentally validated field ranges. A yield-stress/reference-stress ratio above 10,000 is rejected as unresolved near onset. The output excludes hydrostatic head and all local losses. It is not total standpipe pressure, an ECD design or a hole-cleaning assessment. At tau_y = 0 and n = 1 it reduces to the Newtonian concentric solution; n < 1 and tau_y = 0 gives power-law flow. Density changes the screen, not the viscous gradient."
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    "workflow": "herschel-bulkley-annular-pressure-loss",
    "parameter": "tau_y",
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          "D": 0.2,
          "d": 0.1,
          "Q": 0.001,
          "K": 0.2,
          "n": 0.5,
          "tau_y": 0,
          "rho": 1200,
          "L": 1000
        },
        "result": {
          "outputs": [
            {
              "label": "Annular friction pressure loss",
              "value": 20694.734372656,
              "unit": "Pa"
            },
            {
              "label": "Friction pressure gradient",
              "value": 20.694734372656,
              "unit": "Pa/m"
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            {
              "label": "Ideal yield-onset gradient",
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              "unit": "Pa/m"
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              "label": "Inner plug boundary radius",
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            {
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              "label": "Mean axial velocity",
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            {
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            {
              "label": "Outer-wall shear rate",
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            {
              "label": "Wall-viscosity inertia screening number",
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              "unit": "dimensionless"
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            {
              "label": "Relative flow-rate residual",
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            {
              "label": "Relative two-wall velocity residual",
              "value": 5.862276576605476e-12,
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          "columns": [
            "Radius (m)",
            "Axial velocity (m/s)",
            "Shear-rate magnitude (1/s)",
            "Shear-stress magnitude (Pa)"
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          "chart": {
            "xLabel": "Radius from common axis (m)",
            "yLabel": "Axial velocity (m/s)",
            "series": [
              {
                "name": "Concentric HB profile",
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          "warnings": [
            "Laminar stationary concentric HB solution only. The inertia screen is not a transition prediction; confirm laminar conditions independently. No hydrostatic, eccentricity, rotation, turbulence or cuttings losses are included."
          ]
        }
      }
    ]
  }
}