Herschel–Bulkley fluid

Three parameters characterize this relationship: the consistency k, the flow index n, and the yield shear stress

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.

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.

[1][2] In one dimension, the constitutive equation of the Herschel-Bulkley model after the yield stress has been reached can be written in the form:[3][4] where

, this model reduces to that of a Newtonian fluid.

denotes the second invariant of the strain rate tensor

Note that the double underlines indicate a tensor quantity.

The viscosity associated with the Herschel-Bulkley stress diverges to infinity as the strain rate approaches zero.

This divergence makes the model difficult to implement in numerical simulations, so it is common to implement regularized models with an upper limiting viscosity.

For instance, the Herschel-Bulkley fluid can be approximated as a generalized Newtonian fluid model with an effective (or apparent) viscosity being given as [5] Here, the limiting viscosity

replaces the divergence at low strain rates.

to ensure the viscosity is a continuous function of strain rate.

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.

This is because a finite effective viscosity will always lead to a small degree of yielding under the influence of external forces (e.g. gravity).

The characteristic timescale of the phenomenon being studied is thus an important consideration when choosing a regularisation threshold.

In an incompressible flow, the viscous stress tensor is given as a viscosity, multiplied by the rate-of-strain tensor (Note that

indicates that the effective viscosity is a function of the shear rate.)

Furthermore, the magnitude of the shear rate is given by The magnitude of the shear rate is an isotropic approximation, and it is coupled with the second invariant of the rate-of-strain tensor A frequently-encountered situation in experiments is pressure-driven channel flow [6] (see diagram).

This situation exhibits an equilibrium in which there is flow only in the horizontal direction (along the pressure-gradient direction), and the pressure gradient and viscous effects are in balance.

This analysis introduces the non-dimensional pressure gradient

which is negative for flow from left to right, and the Bingham number: Next, the domain of the solution is broken up into three parts, valid for a negative pressure gradient: Solving this equation gives the velocity profile:

The profile respects the no-slip conditions at the channel boundaries, Using the same continuity arguments, it is shown that

Apply any pressure gradient smaller in magnitude than this critical value, and the fluid will not flow; its Bingham nature is thus apparent.

Any pressure gradient greater in magnitude than this critical value will result in flow.

For laminar flow Chilton and Stainsby [7] provide the following equation to calculate the pressure drop.

The equation requires an iterative solution to extract the pressure drop, as it is present on both sides of the equation.

allows standard Newtonian friction factor correlations to be used.

The pressure drop can then be calculated, given a suitable friction factor correlation.