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14.2 Rheological Models

2 min readLast Updated on July 19, 2024

Non-Newtonian fluids don't follow simple rules like water. They can get thicker or thinner when you stir them. Scientists use math to figure out how these weird fluids behave.

These math models help engineers design stuff like pipes and pumps for tricky fluids. They look at things like how much force it takes to get the fluid moving and how it changes when you mix it faster.

Rheological Models for Non-Newtonian Fluids

Purpose of rheological models

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  • Mathematical equations describe relationship between shear stress and shear rate in fluids
    • Newtonian fluids have linear relationship with constant viscosity (water, honey)
    • Non-Newtonian fluids have nonlinear relationship with varying viscosity (ketchup, shampoo)
  • Accurately predict and characterize flow behavior of complex fluids
  • Aid in design and optimization of processes involving non-Newtonian fluids (extrusion, pumping)
  • Enable selection of appropriate equipment and operating conditions (pipe diameter, pump type)

Power-law model for viscosity

  • Power-law model (Ostwald-de Waele model) described by equation: τ=Kγ˙n\tau = K \dot{\gamma}^n
    • τ\tau = shear stress
    • KK = consistency index, measures fluid's viscosity at shear rate of 1 s1^{-1}
    • γ˙\dot{\gamma} = shear rate
    • nn = flow behavior index, characterizes degree of non-Newtonian behavior
  • Shear-thinning fluids (pseudoplastic) have n<1n < 1, viscosity decreases with increasing shear rate (blood, paint, polymer solutions)
  • Shear-thickening fluids (dilatant) have n>1n > 1, viscosity increases with increasing shear rate (cornstarch suspensions, some colloidal dispersions)

Bingham plastic model for yield stress

  • Bingham plastic model used for fluids exhibiting yield stress, minimum stress required to initiate flow
  • Model described by equation: τ=τ0+μpγ˙\tau = \tau_0 + \mu_p \dot{\gamma}
    • τ\tau = shear stress
    • τ0\tau_0 = yield stress
    • μp\mu_p = plastic viscosity, slope of shear stress-shear rate curve above yield stress
    • γ˙\dot{\gamma} = shear rate
  • Fluids following Bingham plastic model do not flow until applied stress exceeds yield stress (toothpaste, mayonnaise, some drilling fluids)

Herschel-Bulkley vs other models

  • Herschel-Bulkley model combines Power-law and Bingham plastic models, incorporating yield stress and shear-thinning/thickening behavior
  • Model described by equation: τ=τ0+Kγ˙n\tau = \tau_0 + K \dot{\gamma}^n
    • τ\tau = shear stress
    • τ0\tau_0 = yield stress
    • KK = consistency index
    • γ˙\dot{\gamma} = shear rate
    • nn = flow behavior index
  • Compared to Power-law model:
    1. Herschel-Bulkley includes yield stress term (τ0\tau_0), Power-law does not
    2. When τ0=0\tau_0 = 0, Herschel-Bulkley reduces to Power-law model
  • Compared to Bingham plastic model:
    1. Herschel-Bulkley incorporates shear-thinning/thickening behavior through flow behavior index (nn)
    2. Bingham plastic assumes constant plastic viscosity
    3. When n=1n = 1, Herschel-Bulkley reduces to Bingham plastic model
  • Herschel-Bulkley model more versatile, describes behavior of wider range of non-Newtonian fluids (yield-stress fluids with shear-thinning or shear-thickening properties)
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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