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 τ 0 {\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.
Definition In one dimension, the constitutive equation of the Herschel-Bulkley model after the yield stress has been reached can be written in the form:
γ ˙ = 0 , i f τ < τ 0 {\displaystyle {\dot {\gamma }}=0,\qquad \qquad \mathrm {if} \ \tau <\tau _{0}}
τ = τ 0 + k γ ˙ n , i f τ ≥ τ 0 {\displaystyle \tau =\tau _{0}+k{\dot {\gamma }}^{n},\qquad \mathrm {if} \ \tau \geq \tau _{0}}
where τ {\displaystyle \tau } is the shear stress [Pa], τ 0 {\displaystyle \tau _{0}} the yield stress [Pa], k {\displaystyle k} the consistency index [Pa ⋅ {\displaystyle \cdot } s n {\displaystyle ^{n}} ], γ ˙ {\displaystyle {\dot {\gamma }}} the shear rate [s − 1 {\displaystyle ^{-1}} ], and n {\displaystyle n} the flow index [dimensionless]. If τ < τ 0 {\displaystyle \tau <\tau _{0}} the Herschel-Bulkley fluid behaves as a rigid (non-deformable) solid, otherwise it behaves as a fluid. For n < 1 {\displaystyle n<1} the fluid is shear-thinning, whereas for n > 1 {\displaystyle n>1} the fluid is shear-thickening. If n = 1 {\displaystyle n=1} and τ 0 = 0 {\displaystyle \tau _{0}=0} , this model reduces to that of a Newtonian fluid. Reformulated as a tensor, we can instead write:
γ ˙ _ _ = 0 , i f | τ _ _ | < τ 0 {\displaystyle {\underline {\underline {\dot {\gamma }}}}=0,\qquad \qquad \qquad \qquad \mathrm {if} \ |{\underline {\underline {\tau }}}|<\tau _{0}}
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