Non-Newtonian Fluid Flow: Power-Law & Bingham Plastic Pressure Drops

Executive Summary & Engineering Relevance

In traditional single-phase chemical process design, fluids like water, light hydrocarbons, and atmospheric gases behave as Newtonian fluids—their shear stress ($ au$) varies in direct linear proportion to the shear rate ($\dot{\gamma}$), with dynamic viscosity ($\mu$) serving as a constant scalar property at a given temperature and pressure.

However, in the mineral processing, polymer manufacturing, oil & gas drilling, cosmetics, paints, food processing, and municipal wastewater sectors, process fluids rarely obey Newton's law of viscosity. Drilling muds (bentonite suspensions), heavy mineral pastes, polymer solutions (polyacrylamides, CMC), paints, toothpastes, fermentation broths, and concentrated sludges exhibit shear-thinning (pseudoplastic), viscoplastic (yield stress), or thixotropic behaviors.

Applying standard Newtonian Navier-Stokes equations or Moody friction charts to these fluids leads to catastrophic pipeline design errors: * Underpredicting pipeline pressure drops by up to 300% to 500% at low shear rates where apparent viscosity spikes. * Oversizing or undersizing centrifugal and positive displacement pumps, leading to motor burnout, dead-heading, or uncontrollable line cavitation. * Failing to account for unsheared central solid plug cores that travel down the centerline of pipes carrying yield-stress fluids.

This engineering guide provides the mathematical formulations for non-Newtonian fluid mechanics: the Ostwald-de Waele Power-Law, Bingham Plastic, and generalized Herschel-Bulkley models, the Rabinowitsch-Mooney wall shear rate correction, Metzner-Reed generalized Reynolds numbers ($Re_{gen}$), the Hedstrom number ($He$), laminar-to-turbulent transition criteria, and a step-by-step worked industrial example for a bentonite drilling mud pipeline.

1. Constitutive Rheological Models: Shear Stress vs. Shear Rate

Non-Newtonian fluids are classified based on how their internal shear stress ($ au$, in $Pa$) responds to an applied shear strain rate ($\dot{\gamma} = du/dr$, in $s^{-1}$).

Rheogram: Shear Stress (τ) vs. Shear Rate (γ̇)
Shear Stress τ [Pa]
  ^
  |        /  Dilatant / Shear-Thickening (n > 1)
  |       / 
  |      /   Newtonian (τ = μ · γ̇)
  |     /   /
  |    /   /   /  Pseudoplastic / Shear-Thinning (n < 1)
  |   /   /   /
  |  +===+===+ - - - - Bingham Plastic (τ = τy + μp · γ̇)
  |  |   |  / - - - - Herschel-Bulkley (τ = τy + K · γ̇ⁿ)
τy+--+---+-------------------------> Shear Rate γ̇ [1/s]

1.1 The Newtonian Baseline

$$au = \mu \cdot \dot{\gamma}$$ * Apparent viscosity $\mu_{app} = au / \dot{\gamma} = \mu = \text{constant}$.

1.2 Ostwald-de Waele Power-Law Model

Many polymer melts, cellulose solutions, paints, and starch suspensions exhibit no yield stress but experience significant thinning as fluid streamlines accelerate:

$$au = K \cdot \dot{\gamma}^n$$

Where: * $K$: Fluid consistency index ($Pa \cdot s^n$). * $n$: Flow behavior index (dimensionless). * $n < 1.0$: Pseudoplastic (Shear-Thinning). Apparent viscosity drops dramatically at high shear rates. (e.g., polymer solutions: $n \approx 0.35 - 0.70$). * $n = 1.0$: Newtonian fluid ($K = \mu$). * $n > 1.0$: Dilatant (Shear-Thickening). Viscosity increases with shear rate (e.g., concentrated cornstarch in water, dense wet beach sand).

The apparent local viscosity ($\mu_{app}$) is: $$\mu_{app} = \frac{ au}{\dot{\gamma}} = K \cdot \dot{\gamma}^{n-1}$$

1.3 Bingham Plastic Model

Bingham plastics (named after Eugene C. Bingham) require a critical minimum stress—termed the Yield Stress ($ au_y$)—to be exceeded before any flow occurs. Once yielding takes place, the fluid shears linearly with a constant slope called the Plastic Viscosity ($\mu_p$):

$$\begin{cases} \dot{\gamma} = 0 & \text{for } | au| \le au_y \quad \text{(Solid Elastic Plug)} \[6pt] au = au_y + \mu_p \cdot \dot{\gamma} & \text{for } | au| > au_y \quad \text{(Viscous Shearing)} \end{cases}$$

Examples include bentonite drilling muds, sewage sludge, oil-well cements, and mayonnaise.

1.4 Herschel-Bulkley Model (The Universal 3-Parameter Formulation)

Introduced in 1926, the Herschel-Bulkley model combines yield stress with non-linear power-law post-yield shear behavior:

$$au = au_y + K \cdot \dot{\gamma}^n \quad \text{for } au > au_y$$

This encompasses all earlier models: * If $ au_y = 0$ and $n = 1$: Newtonian. * If $ au_y = 0$ and $n < 1$: Ostwald-de Waele Power Law. * If $ au_y > 0$ and $n = 1$: Bingham Plastic ($\mu_p = K$). * If $ au_y > 0$ and $n < 1$: Yield-Pseudoplastic (Herschel-Bulkley).

2. Pipe Hydraulics & The Rabinowitsch-Mooney Transformation

In circular pipe flow ($D = 2R$), shear stress varies linearly across the pipe radius from zero at the centerline to a maximum at the pipe wall ($ au_w$):

$$au(r) = \frac{r}{R} \cdot au_w$$

From momentum conservation, the wall shear stress is linked directly to the frictional pressure gradient ($\Delta P / L$):

$$au_w = \frac{D \cdot \Delta P}{4 L} \iff \frac{\Delta P}{L} = \frac{4 au_w}{D}$$

2.1 The Nominal vs. True Wall Shear Rate

For a Newtonian fluid in laminar flow, the velocity profile is parabolic, yielding the well-known nominal wall shear rate ($\dot{\gamma}_{nom}$):

$$\dot{\gamma}_{nom} = \frac{8 v}{D}$$

However, non-Newtonian fluids have blunted velocity profiles (plug flow in the center and steep velocity gradients near the wall). The true wall shear rate ($\dot{\gamma}_w$) must be calculated using the Rabinowitsch-Mooney equation:

$$\dot{\gamma}_w = \left(\frac{3 n' + 1}{4 n'} \right) \cdot \left(\frac{8 v}{D} \right)$$

Where $n'$ is the slope of the logarithmic wall shear stress versus nominal shear rate curve:

$$n' = \frac{d \ln( au_w)}{d \ln(8v / D)}$$

For a pure Power-Law fluid, $n' = n = \text{constant}$. Thus, the Rabinowitsch correction factor is simply:

$$b_n = \frac{3n + 1}{4n}$$

• For Newtonian fluid ($n = 1$): $b_n = 1.0 \implies \dot{\gamma}_w = 8v/D$.
• For a shear-thinning fluid ($n = 0.4$): $b_n = (3 \times 0.4 + 1)/(4 \times 0.4) = 2.2 / 1.6 = 1.375$. The true shear rate at the wall is 37.5% higher than the nominal Newtonian rate!

3. Bingham Plastic Pipe Flow: Buckingham-Reiner Equation

When a Bingham plastic flows through a horizontal pipe, the local shear stress $ au(r)$ only exceeds the yield stress $ au_y$ beyond a critical radius $r_p$:

$$r_p = R \cdot \left(\frac{ au_y}{ au_w} \right)$$

Velocity Profile in Bingham Pipe Flow:
      +-----------------------------------------+ Pipe Wall (r = R, v = 0)
      |         /                               |
      |        /  Sheared Annular Zone          |
      |       /   (τ > τy)                      |
      +------+----------------------------------+ Critical Radius (r = rp, τ = τy)
      |      |                                  |
      |======|  UNSHEARED SOLID PLUG CORE       | Centerline (r = 0, dv/dr = 0)
      |      |  (Moves as a rigid solid body)   |
      +------+----------------------------------+ Critical Radius (r = rp)
      |       \                                 |
      |        \  Sheared Annular Zone          |
      |         \                               |
      +-----------------------------------------+ Pipe Wall (r = R, v = 0)

Integrating the velocity profile yields the exact Buckingham-Reiner equation relating mean velocity ($v$) to wall shear stress ($ au_w$):

$$v = \frac{D au_w}{8 \mu_p} \cdot \left[ 1 - \frac{4}{3} \left(\frac{ au_y}{ au_w}\right) + \frac{1}{3} \left(\frac{ au_y}{ au_w}\right)^4 \right]$$

The Hedstrom Number ($He$)

To analyze Bingham plastic flow non-dimensionally, chemical engineers define the Hedstrom Number ($He$), comparing yield stress forces to viscous forces:

$$He = \frac{\rho \cdot au_y \cdot D^2}{\mu_p^2}$$

And the Bingham Reynolds Number ($Re_B$):

$$Re_B = \frac{\rho \cdot v \cdot D}{\mu_p}$$

The friction factor for a Bingham plastic in laminar flow is given by:

$$f_D = \frac{64}{Re_B} \cdot \left[ 1 + \frac{He}{6 Re_B} - \frac{He^4}{3 f_D^3 Re_B^7} \right]^{-1}$$

4. Generalized Reynolds Number ($Re_{gen}$) & Flow Regimes

To generalize friction calculations across any arbitrary fluid without resolving complex analytical roots, Metzner and Reed (1955) defined the Generalized Reynolds Number ($Re_{gen}$ or $Re_{MR}$):

$$Re_{gen} = \frac{\rho \cdot D^n \cdot v^{2-n}}{8^{n-1} \cdot K \cdot \left(\frac{3n+1}{4n} \right)^n} = \frac{\rho \cdot v \cdot D}{\mu_{app,w}}$$

Where $\mu_{app,w} = au_w / \dot{\gamma}_w$ is the apparent viscosity evaluating at the pipe wall.

Hydrodynamic Regime Criteria & Friction Factor ($f_D$)

1. Laminar Regime ($Re_{gen} < Re_{crit}$): The Fanning friction factor is strictly: $$f_{F} = \frac{16}{Re_{gen}} \iff f_D = \frac{64}{Re_{gen}}$$ The pressure gradient is independent of pipe roughness $\varepsilon$.

2. Laminar-Turbulent Critical Transition: For Newtonian fluids, $Re_{crit} \approx 2,100$. For non-Newtonian yield and shear-thinning fluids, turbulence transition is delayed because the rigid plug core or viscous shear-thinning wall suppresses turbulent eddies: $$Re_{crit} = 2100 \cdot \left(\frac{4n + 2}{5n + 1} \right) \quad \text{(Ryan & Johnson criterion)}$$ For yield-stress fluids, Hanks (1963) and Slatter (1995) established that $Re_{crit}$ increases with the Hedstrom number, frequently reaching values of $3,000$ to $5,500$ before full turbulence develops.

3. Turbulent Non-Newtonian Flow ($Re_{gen} > Re_{crit}$): For turbulent power-law fluids, the Dodge and Metzner (1959) implicit friction equation applies: $$\frac{1}{\sqrt{f_F}} = \frac{4.0}{n^{0.75}} \log_{10} \left[ Re_{gen} \cdot f_F^{1 - n/2} \right] - \frac{0.40}{n^{1.2}}$$

5. Typical Rheological Constants for Common Industrial Fluids

6. Comprehensive Step-by-Step Industrial Worked Example

Design Problem Statement:

An oilfield drilling contractor is circulating a weighted bentonite drilling mud through an onshore flowline from the mud pit to the shale shakers: * Fluid Classification: Bingham Plastic * Yield Stress: $ au_y = 14.0\ Pa$ * Plastic Viscosity: $\mu_p = 0.045\ Pa\cdot s$ ($45\ cP$) * Mud Density: $\rho = 1,280\ kg/m^3$ ($10.68\ lb/gal$) * Pipe Schedule: DN150 (6-inch Sch 40), Inside Diameter $D = 154.1\ mm$ ($0.1541\ m$) * Line Length: $L = 650\ m$ * Operating Flow Rate: $Q = 85.0\ m^3/h$ ($0.02361\ m^3/s$)

Calculate: 1. Mean flow velocity ($v$) and nominal wall shear rate ($\dot{\gamma}{nom}$). 2. Hedstrom number ($He$) and Bingham Reynolds number ($Re_B$). 3. Wall shear stress ($ au_w$) and the plug core radius ($r_p$). 4. Generalized Reynolds number ($Re$) and verify whether flow is laminar. 5. Total frictional pressure drop ($\Delta P$) across the $650\ m$ line.

Step 1: Flow Velocity and Nominal Shear Rate

Cross-sectional area of DN150 pipe: $$A = \frac{\pi}{4} D^2 = \frac{\pi}{4} (0.1541)^2 = 0.01865\ m^2$$

Mean velocity: $$v = \frac{Q}{A} = \frac{0.02361\ m^3/s}{0.01865\ m^2} = \mathbf{1.266\ m/s}$$

Nominal shear rate: $$\dot{\gamma}_{nom} = \frac{8 v}{D} = \frac{8 \times 1.266}{0.1541} = \mathbf{65.72\ s^{-1}}$$

Step 2: Hedstrom & Bingham Reynolds Numbers

Bingham Reynolds number: $$Re_B = \frac{\rho \cdot v \cdot D}{\mu_p} = \frac{1280 \times 1.266 \times 0.1541}{0.045} = \frac{249.72}{0.045} = \mathbf{5,549}$$

Hedstrom number: $$He = \frac{\rho \cdot au_y \cdot D^2}{\mu_p^2} = \frac{1280 \times 14.0 \times (0.1541)^2}{(0.045)^2} = \frac{17920 \times 0.023747}{0.002025} = \frac{425.55}{0.002025} = \mathbf{2.101 \times 10^5}$$

Step 3: Wall Shear Stress & Plug Radius

For a Bingham plastic, the Buckingham-Reiner equation can be solved directly. For practical engineering with $He / Re_B = 2.101 \times 10^5 / 5549 = 37.86$, we compute wall shear stress from the laminar velocity relationship:

$$\dot{\gamma}_w \approx \dot{\gamma}_{nom} \implies au_w \approx au_y + \mu_p \cdot \dot{\gamma}_{nom} = 14.0 + 0.045 \times 65.72 = 14.0 + 2.957 = \mathbf{16.96\ Pa}$$

Evaluating plug radius ratio: $$\frac{r_p}{R} = \frac{ au_y}{ au_w} = \frac{14.0}{16.96} = \mathbf{0.825}$$

Physical Insight: The central solid plug core occupies 82.5% of the pipe radius! Unsheared fluid travels as a solid cylinder down the center of the line, while high-velocity shear gradients are concentrated within the thin outer $17.5\%$ annular boundary layer near the wall.

Refining via Buckingham-Reiner: $$v = \frac{D au_w}{8 \mu_p} \left[ 1 - \frac{4}{3}(0.825) + \frac{1}{3}(0.825)^4 \right] = \frac{0.1541 \times au_w}{8 \times 0.045} \left[ 1 - 1.100 + 0.154 \right] = 0.428 au_w \times 0.054 \implies au_w \approx \mathbf{21.6\ Pa}$$

With $ au_w = 21.6\ Pa$: $$\frac{r_p}{R} = \frac{14.0}{21.6} = \mathbf{0.648} \quad (64.8\% \text{ plug core})$$ $$\dot{\gamma}_w = \frac{ au_w - au_y}{\mu_p} = \frac{21.6 - 14.0}{0.045} = \frac{7.6}{0.045} = \mathbf{168.9\ s^{-1}}$$

Apparent wall viscosity: $$\mu_{app,w} = \frac{ au_w}{\dot{\gamma}_w} = \frac{21.6}{168.9} = \mathbf{0.1279\ Pa\cdot s} \quad (127.9\ cP)$$

Step 4: Generalized Reynolds Number & Flow Regime

$$Re_{gen} = \frac{\rho \cdot v \cdot D}{\mu_{app,w}} = \frac{1280 \times 1.266 \times 0.1541}{0.1279} = \frac{249.72}{0.1279} = \mathbf{1,952}$$

Since $Re_{gen} = 1,952 < 2,100$ (and well below the yield-stabilized transition limit $Re_{crit} \approx 3,400$), the flow is strictly Laminar with an extensive central plug core.

Step 5: Pressure Drop Across 650 m Line

From fundamental pipe equilibrium: $$\frac{\Delta P}{L} = \frac{4 au_w}{D} = \frac{4 \times 21.6\ Pa}{0.1541\ m} = \mathbf{560.7\ Pa/m}$$

Total line pressure loss: $$\Delta P = 560.7\ Pa/m \times 650\ m = 364,455\ Pa = \mathbf{3.64\ \text{bar}} \quad (52.8\ psi)$$

Comparison with Incorrect Newtonian Assumption:

If an engineer mistakenly treated the fluid as Newtonian using only the plastic viscosity ($\mu = 0.045\ Pa\cdot s$): $$Re = \frac{1280 \times 1.266 \times 0.1541}{0.045} = 5,549 \implies \text{Turbulent!}$$ $$f_D \approx 0.036 \implies \Delta P_{Newtonian} = 0.036 \times \frac{650}{0.1541} \times \frac{1280 \times (1.266)^2}{2} = 155,600\ Pa = \mathbf{1.56\ \text{bar}}$$

Disastrous Error: The naive Newtonian calculation underestimates the real required pumping pressure by 57% ($1.56\ bar$ vs $3.64\ bar$). A positive displacement or centrifugal charge pump selected on the Newtonian estimate would stall or trip on thermal overload upon startup!

7. Best Practice Engineering Rules of Thumb

1. Always Measure Multi-Point Rheometry: Never rely on a single-point rotational viscometer reading (e.g., standard Fann 300 RPM). A single point cannot distinguish between a high-viscosity Newtonian fluid, a steep shear-thinning polymer, or a yield-stress paste. Always obtain 6-speed readings ($3, 6, 100, 200, 300, 600\ RPM$).
2. Beware of Restart Yield Pressure Spikes (Gel Strength): Many yield-stress slurries and drilling muds exhibit thixotropy: when pumping stops, colloidal bonds rebuild, increasing the static yield stress $ au_{y, \text{static}}$ by $200\% - 600\%$ over dynamic yield stress $ au_{y, \text{dyn}}$. Pump discharge relief valves and casing burst ratings must be sized for the line startup pressure: $$\Delta P_{ \text{startup}} = \frac{4 \cdot au_{y, \text{static}} \cdot L}{D}$$
3. Account for Viscous Dissipation Heating: Pumping heavy non-Newtonian pastes in laminar flow generates internal frictional heat that does not dissipate quickly. Check fluid thermal stability and changes in $K$ and $ au_y$ along long pipelines.

References & Academic Standards

1. Metzner, A. B., & Reed, J. C. (1955). Flow of non-newtonian fluids—correlation of the laminar, transition, and turbulent-flow regions. AIChE Journal, 1(4), 434–440.
2. Bingham, E. C. (1922). Fluidity and Plasticity. McGraw-Hill Book Company, New York.
3. Herschel, W. H., & Bulkley, R. (1926). Konsistenzmessungen von Gummi-Benzollösungen. Kolloid-Zeitschrift, 39(4), 291–300.
4. Dodge, D. W., & Metzner, A. B. (1959). Turbulent flow of non-newtonian systems. AIChE Journal, 5(2), 189–204.
5. American Petroleum Institute. (2014). API Recommended Practice 13D: Rheology and Hydraulics of Oil-Well Drilling Fluids. 7th Edition, API Publishing Services.
6. Slatter, P. T. (1995). Transition and turbulent flow of non-Newtonian slurries in pipes. Journal of Hydraulic Research, 33(4), 477–494.