# 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.[1][2]

## Definition

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

${\displaystyle \tau =\tau _{0}+k{\dot {\gamma }}^{n}}$

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.

As a generalized Newtonian fluid model, the effective viscosity is given as [3]

${\displaystyle \mu _{\operatorname {eff} }={\begin{cases}\mu _{0},&|{\dot {\gamma }}|\leq {\dot {\gamma }}_{0}\\k|{\dot {\gamma }}|^{n-1}+\tau _{0}|{\dot {\gamma }}|^{-1},&|{\dot {\gamma }}|\geq {\dot {\gamma }}_{0}\end{cases}}}$

The limiting viscosity ${\displaystyle \mu _{0}}$ is chosen such that ${\displaystyle \mu _{0}=k{\dot {\gamma }}_{0}^{n-1}+\tau _{0}{\dot {\gamma }}_{0}^{-1}}$. 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

${\displaystyle \tau _{ij}=2\mu _{\operatorname {eff} }(|{\dot {\gamma }}|)E_{ij}=\mu _{\operatorname {eff} }(|{\dot {\gamma }}|)\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right),}$

where the magnitude of the shear rate is given by

${\displaystyle |{\dot {\gamma }}|={\sqrt {2E_{ij}E^{ij}}}}$.

The magnitude of the shear rate is an isotropic approximation, and it is coupled with the second invariant of the rate-of-strain tensor

${\displaystyle II_{E}=tr(E_{ij}E^{jk})=E_{ij}E^{ij}}$.

## Channel flow

A schematic diagram pressure-driven horizontal flow. The flow is uni-directional in the direction of the pressure gradient.

A frequently-encountered situation in experiments is pressure-driven channel flow [4] (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. Then, the Navier-Stokes equations, together with the rheological model, reduce to a single equation:

Velocity profile of the Herschel–Bulkley fluid for various flow indices n. In each case, the non-dimensional pressure is ${\displaystyle \pi _{0}=-10}$. The continuous curve is for an ordinary Newtonian fluid (Poiseuille flow), the broken-line curve is for a shear-thickening fluid, while the dotted-line curve is for a shear-thinning fluid.
${\displaystyle {\frac {\partial p}{\partial x}}={\frac {\partial }{\partial z}}\left(\mu {\frac {\partial u}{\partial z}}\right)\,\,\,={\begin{cases}\mu _{0}{\frac {\partial ^{2}u}{\partial {z}^{2}}},&\left|{\frac {\partial u}{\partial z}}\right|<\gamma _{0}\\\\{\frac {\partial }{\partial z}}\left[\left(k\left|{\frac {\partial u}{\partial z}}\right|^{n-1}+\tau _{0}\left|{\frac {\partial u}{\partial z}}\right|^{-1}\right){\frac {\partial u}{\partial z}}\right],&\left|{\frac {\partial u}{\partial z}}\right|\geq \gamma _{0}\end{cases}}}$

To solve this equation it is necessary to non-dimensionalize the quantities involved. The channel depth H is chosen as a length scale, the mean velocity V is taken as a velocity scale, and the pressure scale is taken to be ${\displaystyle P_{0}=k\left(V/H\right)^{n}}$. This analysis introduces the non-dimensional pressure gradient

${\displaystyle \pi _{0}={\frac {H}{P_{0}}}{\frac {\partial p}{\partial x}},}$

which is negative for flow from left to right, and the Bingham number:

${\displaystyle Bn={\frac {\tau _{0}}{k}}\left({\frac {H}{V}}\right)^{n}.}$

Next, the domain of the solution is broken up into three parts, valid for a negative pressure gradient:

• A region close to the bottom wall where ${\displaystyle \partial u/\partial z>\gamma _{0}}$;
• A region in the fluid core where ${\displaystyle |\partial u/\partial z|<\gamma _{0}}$;
• A region close to the top wall where ${\displaystyle \partial u/\partial z<-\gamma _{0}}$,

Solving this equation gives the velocity profile:

${\displaystyle u\left(z\right)={\begin{cases}{\frac {n}{n+1}}{\frac {1}{\pi _{0}}}\left[\left(\pi _{0}\left(z-z_{1}\right)+\gamma _{0}^{n}\right)^{1+\left(1/n\right)}-\left(-\pi _{0}z_{1}+\gamma _{0}^{n}\right)^{1+\left(1/n\right)}\right],&z\in \left[0,z_{1}\right]\\{\frac {\pi _{0}}{2\mu _{0}}}\left(z^{2}-z\right)+k,&z\in \left[z_{1},z_{2}\right],\\{\frac {n}{n+1}}{\frac {1}{\pi _{0}}}\left[\left(-\pi _{0}\left(z-z_{2}\right)+\gamma _{0}^{n}\right)^{1+\left(1/n\right)}-\left(-\pi _{0}\left(1-z_{2}\right)+\gamma _{0}^{n}\right)^{1+\left(1/n\right)}\right],&z\in \left[z_{2},1\right]\\\end{cases}}}$

Here k is a matching constant such that ${\displaystyle u\left(z_{1}\right)}$ is continuous. The profile respects the no-slip conditions at the channel boundaries,

${\displaystyle u(0)=u(1)=0,}$

Using the same continuity arguments, it is shown that ${\displaystyle z_{1,2}={\tfrac {1}{2}}\pm \delta }$, where

${\displaystyle \delta ={\frac {\gamma _{0}\mu _{0}}{|\pi _{0}|}}\leq {\tfrac {1}{2}}.}$

Since ${\displaystyle \mu _{0}=\gamma _{0}^{n-1}+Bn/\gamma _{0}}$, for a given ${\displaystyle \left(\gamma _{0},Bn\right)}$ pair, there is a critical pressure gradient

${\displaystyle |\pi _{0,\mathrm {c} }|=2\left(\gamma _{0}+Bn\right).}$

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. The flow associated with a shear-thickening fluid is retarded relative to that associated with a shear-thinning fluid.

## Pipe flow

For laminar flow Chilton and Stainsby [5] 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.

${\displaystyle {\frac {\Delta P}{L}}={\frac {4K}{D}}\left({\frac {8V}{D}}\right)^{n}\left({\frac {3n+1}{4n}}\right)^{n}{\frac {1}{1-X}}\left({\frac {1}{1-aX-bX^{2}-cX^{3}}}\right)^{n}}$
${\displaystyle X={\frac {4L\tau _{y}}{D\Delta P}}}$
${\displaystyle a={\frac {1}{2n+1}}}$
${\displaystyle b={\frac {2n}{\left(n+1\right)\left(2n+1\right)}}}$
${\displaystyle c={\frac {2n^{2}}{\left(n+1\right)\left(2n+1\right)}}}$
For turbulent flow the authors propose a method that requires knowledge of the wall shear stress, but do not provide a method to calculate the wall shear stress. Their procedure is expanded in Hathoot [6]
${\displaystyle R={\frac {4n\rho VD\left(1-aX-bX^{2}-cX^{3}\right)}{\mu _{Wall}\left(3n+1\right)}}}$
${\displaystyle \mu _{Wall}=\tau _{Wall}^{1-1/n}\left({\frac {K}{1-X}}\right)^{1/n}}$
${\displaystyle \tau _{Wall}={\frac {D\Delta P}{4L}}}$
All units are SI
${\displaystyle \Delta P}$ Pressure drop, Pa.
${\displaystyle L}$ Pipe length, m
${\displaystyle D}$ Pipe diameter, m
${\displaystyle V}$ Fluid velocity, ${\displaystyle m/s}$
Chilton and Stainsby state that defining the Reynolds number as
${\displaystyle Re={\frac {R}{n^{2}\left(1-X\right)^{4}}}}$

allows standard Newtonian friction factor correlations to be used.