Herschel–Bulkley fluid

The Herschel–Bulkley fluid is a generalized model of a non-Newtonian fluid, in which the strain experienced by the fluid is related to the stress in a complicated, non-linear way. Three parameters characterize this relationship: the consistency k, the flow index n, and the yield shear stress ${\displaystyle \tau _{0}}$. The consistency is a simple constant of proportionality, while the flow index measures the degree to which the fluid is shear-thinning or shear-thickening. Ordinary paint is one example of a shear-thinning fluid, while oobleck provides one realization of a shear-thickening fluid. Finally, the yield stress quantifies the amount of stress that the fluid may experience before it yields and begins to flow.

This non-Newtonian fluid model was introduced by Winslow Herschel and Ronald Bulkley in 1926.

The constitutive equation of the Herschel-Bulkley model is commonly written as

where ${\displaystyle \tau }$ is the shear stress, ${\displaystyle {\dot {\gamma }}}$ the shear rate, ${\displaystyle \tau _{0}}$ the yield stress, ${\displaystyle k}$ the consistency index, and ${\displaystyle n}$ the flow index. If ${\displaystyle \tau <\tau _{0}}$ the Herschel-Bulkley fluid behaves as a solid, otherwise it behaves as a fluid. For ${\displaystyle n<1}$ the fluid is shear-thinning, whereas for ${\displaystyle n>1}$ the fluid is shear-thickening. If ${\displaystyle n=1}$ and ${\displaystyle \tau _{0}=0}$, this model reduces to the Newtonian fluid.

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