Table 4 Governing equations, fitting parameters, and key features of the Power-law, Bingham Plastic, and Herschel–Bulkley models.

Power-law model

\(\:\tau\:=k.{\dot{\gamma\:}}^{n}\)

τ: shear stress (Pa);

k: consistency index (Pa·sⁿ);

\(\:\dot{\varvec{\gamma\:}}\)​: shear rate (s^− 1);

n: flow behavior index (-)

Simple, easy to apply to non-Newtonian fluids; classifies fluids as shear-thinning (n < 1), Newtonian (n = 1), or shear-thickening (n > 1)⁴¹

Does not predict yield stress; limited accuracy outside tested shear rate range⁴¹

Bingham Plastic model

\(\:\tau\:={\tau\:}_{^\circ\:}+k\dot{\gamma\:}\)

τ₀​ ​: yield stress (Pa);

k​: plastic viscosity (Pa·s);

\(\:\dot{\varvec{\gamma\:}}\): shear rate (s^− 1)

Accounts for yield stress; suitable for materials that behave as rigid bodies below τ₀⁴²

Cannot describe shear-thinning or shear-thickening; oversimplifies complex fluids⁴²

Herschel–Bulkley model

\(\:\tau\:={\tau\:}_{^\circ\:}+k\dot{{\gamma\:}^{n}}\)

τ₀​: yield stress (Pa);

k: consistency index (Pa·sⁿ);

n: flow behavior index (-);

\(\:\dot{\varvec{\gamma\:}}\) ​: shear rate (s^− 1)

Combines yield stress and non-Newtonian flow; flexible model applicable to many complex fluids⁴³

Requires more parameters; fitting may be complex; overfitting risk with limited data^43,44