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 τ 0 {displaystyle au _{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. 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 au _{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 τ {displaystyle au } is the shear stress, γ ˙ {displaystyle {dot {gamma }}} the shear rate, τ 0 {displaystyle au _{0}} the yield stress, k {displaystyle k} the consistency index, and n {displaystyle n} the flow index. If τ < τ 0 {displaystyle au < au _{0}} the Herschel-Bulkley fluid behaves as a 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 au _{0}=0} , this model reduces to the Newtonian fluid. As a generalized Newtonian fluid model, the effective viscosity is given as The limiting viscosity μ 0 {displaystyle mu _{0}} is chosen such that μ 0 = k γ ˙ 0 n − 1 + τ 0 γ ˙ 0 − 1 {displaystyle mu _{0}=k{dot {gamma }}_{0}^{n-1}+ au _{0}{dot {gamma }}_{0}^{-1}} . A large limiting viscosity means that the fluid will only flow in response to a large applied force. This feature captures the Bingham-type behaviour of the fluid. The viscous stress tensor is given, in the usual way, as a viscosity, multiplied by the rate-of-strain tensor