it means that the fluid in question exhibits pseudoplastic behavior but does not follow the power law model. It suggests that the fluid's viscosity decreases as the shear rate increases, but the relationship between viscosity and shear rate does not follow a simple power law equation.

Thanks, Milad Sharifipour ! Good answer.

The statement: “The fluid is pseudoplastic, but not power law.” means that the fluid is a type of non-Newtonian fluid that has a lower apparent viscosity at higher shear rates, but does not follow the simple mathematical relationship of the power law model

The power law model is a two-parameter equation that relates the shear stress and the shear rate of a fluid as follows:

τ=Kγ^n

where τ is the shear stress, K is the flow consistency index, γ˙​ is the shear rate, and n is the flow behavior index

A pseudoplastic fluid has a flow behavior index of less than one, which means that its apparent viscosity decreases as the shear rate increases, However, this model does not account for the minimum and maximum effective viscosity that a real fluid has, and it only applies to a limited range of shear rates.

Therefore, a pseudoplastic fluid may not fit the power law model well, and may require a more complex model to describe its entire flow behavior.

Yield – pseudo plastic fluids in laminar flow in pipes

The Herschel–Bulkley model describes the behavior of pseudo plastic fluids that exhibit a yield stress. The flowrate of these fluids in laminar flow in a round pipe can be calculate by substituting the model equation 5.8 into equation 5.18

(5.57)V¯=D2kH1/nτw3∫τhτwτ2(τ−τH)1/ndτ

A change of variable to

(5.58)u=τ−τH

leads to a simple integral

(5.59)V¯=D2kH1/nτw3∫0τw−τH(u2+1/n+2τHu1+1/n+τH2u1/n)du=D2kH1/nτw3(4n3n+1(1−τHτw)3n+1n+8n2n+1τhτw(1−τHτw)2n+1n)+4nn+1(τHTw)2(1−τHτw)n+1

Shear-Thinning Fluids

Shear rate in the vessel varies with distance from the impeller; it depends on velocity gradients and is lowest near the vessel walls and at the fluid surface. Thus, the apparent viscosity is lowest near the impeller and highest at the walls.Shear thinning, or pseudoplastic, fluids are characterized by consistency and flow behavior indices, K and n. These values are determined experimentally by measuring the fluid viscosity at different shear rates, satisfying the power law relationship:(16-9)μA=Kγ˙n−1Where:μA = apparent viscosityK = consistency indexγ˙ = shear rate, s−1n = flow behavior index

Knowing how to characterize and accurately analyze non-Newtonian fluids is a vital skill when it comes to the development and distribution of these fluids. Non-Newtonian fluids are manufactured and sold in multiple industries including the chemical, oil and gas, biopharmaceutical, food and beverage, inks, paints and coatings and cosmetics industries. When characterizing non-Newtonian fluids we produce and sell, the goal is to extract important information about the fluids quickly and simply.

Scientists have been studying, analyzing, and characterizing non-Newtonian fluid behavior for decades and have developed numerous analytical models to facilitate their analysis. These mathematical model outputs can be used to interpolate data, rank, and categorize fluids, and even help make calculations for fluid dynamics simulations. One simple model used to characterize non-Newtonian fluids is the power law model (Macosko, 1994):

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η is the viscosity (mPa-s) n is the power law constant (unitless) K is the flow consistency index, (mPa-s)

The power law model, also known as the Ostwald de Waele relationship, is used to fit non-Newtonian data across shear rates where there is no evidence of a Newtonian plateau region. Fitting to the power law model is appropriate where the measured data is entirely within the shear-thinning regime across all the shear rates tested. The power law model only has two fitting constants making it much simpler than other models that fit non-Newtonian fluids such as the Cross model or Carreau-Yasuda model which contain four and five fitting constants, respectively.

Mathematically modeling your data improves your data analysis and fluid characterization. The constants generated by these fits have multiple uses from ranking fluid performance to performing calculations that are relevant for your samples. Parameters useful for fluid characterization like flow consistency index and flow behavior can be determined using this model.

Download our application note for a detailed walk through of how to perform power law fitting on non-Newtonian fluids using Microsoft Excel.

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