# 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 $\tau_0$. 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 viscous stress tensor is given, in the usual way, as a viscosity, multiplied by the rate-of-strain tensor:

$\tau_{ij}=2\mu E_{ij}=\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right),$

where in contrast to the Newtonian fluid, the viscosity is itself a function of the strain tensor. This is constituted through the formula [3]

$\mu=\begin{cases}\mu_0,&\Pi\leq\Pi_0\\k\Pi^{n-1}+\tau_0\Pi^{-1},&\Pi\geq\Pi_0\end{cases},$

where $\Pi$ is the second invariant of the rate-of-strain tensor:

$\Pi=\sqrt{2E_{ij}E^{ij}}$.

If n=1 and $\tau_0=0$, this model reduces to the Newtonian fluid. If $n<1$ the fluid is shear-thinning, while $n>1$ produces a shear-thickening fluid. The limiting viscosity $\mu_0$ is chosen such that $\mu_0=k\Pi_0^{n-1}+\tau_0\Pi_0^{-1}$. 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.

This equation is also commonly written as

$\tau = \tau_{0} + K \gamma ^ {n}$

where $\tau$ is the shear stress, $\gamma$ the shear rate, $\tau_{0}$ the yield stress, and K and n are regarded as model factors.

## 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 $\pi_0=-10$. 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.
$\frac{\partial p}{\partial x}=\frac{\partial}{\partial z}\left(\mu\frac{\partial u}{\partial z}\right)\,\,\, =\begin{cases}\mu_0\frac{\partial^2 u}{\partial{z}^2},&\left|\frac{\partial u}{\partial z}\right|<\gamma_0\\ \\\frac{\partial}{\partial z}\left[\left(k\left|\frac{\partial u}{\partial z}\right|^{n-1}+\tau_0\left|\frac{\partial u}{\partial z}\right|^{-1}\right)\frac{\partial u}{\partial z}\right],&\left|\frac{\partial u}{\partial z}\right|\geq\gamma_0\end{cases}$

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 $P_0=k\left(V/H\right)^n$. This analysis introduces the non-dimensional pressure gradient

$\pi_0=\frac{H}{P_0}\frac{\partial p}{\partial x},$

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

$Bn=\frac{\tau_0}{k}\left(\frac{H}{V}\right)^n.$

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 $\partial u/\partial z>\gamma_0$;
• A region in the fluid core where $|\partial u/\partial z|<\gamma_0$;
• A region close to the top wall where $\partial u/\partial z<-\gamma_0$,

Solving this equation gives the velocity profile:

$u\left(z\right)=\begin{cases} \frac{n}{n+1}\frac{1}{\pi_0}\left[\left(\pi_0\left(z-z_1\right)+\gamma_0^n\right)^{1+\left(1/n\right)}-\left(-\pi_0z_1+\gamma_0^n\right)^{1+\left(1/n\right)}\right],&z\in\left[0,z_1\right]\\ \frac{\pi_0}{2\mu_0}\left(z^2-z\right)+k,&z\in\left[z_1,z_2\right],\\ \frac{n}{n+1}\frac{1}{\pi_0}\left[\left(-\pi_0\left(z-z_2\right)+\gamma_0^n\right)^{1+\left(1/n\right)}-\left(-\pi_0\left(1-z_2\right)+\gamma_0^n\right)^{1+\left(1/n\right)}\right],&z\in\left[z_2,1\right]\\ \end{cases}$

Here k is a matching constant such that $u\left(z_1\right)$ is continuous. The profile respects the no-slip conditions at the channel boundaries,

$u(0)=u(1)=0,$

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

$\delta=\frac{\gamma_0\mu_0}{|\pi_0|}\leq \tfrac{1}{2}.$

Since $\mu_0=\gamma_0^{n-1}+Bn/\gamma_0$, for a given $\left(\gamma_0,Bn\right)$ pair, there is a critical pressure gradient

$|\pi_{0,\mathrm{c}}|=2\left(\gamma_0+Bn\right).$

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.

$\frac {\Delta P} {L} = \frac {4 K} {D} \left( \frac {8 V} {D} \right) ^ n \left( \frac {3 n + 1} {4 n} \right) ^ n \frac {1} {1 - X} \left( \frac {1} {1 - a X - b X^2 - cX^3} \right) ^ n$
$X = \frac {4 L \tau_ y} {D \Delta P}$
$a = \frac {1} {2 n + 1}$
$b = \frac {2 n} { \left( n + 1 \right) \left( 2 n + 1 \right) }$
$c = \frac {2 n ^ 2} { \left( n + 1 \right) \left( 2 n + 1 \right) }$
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]
$R = \frac {4 n \rho V D \left( 1 - a X - b X^2 - cX^3 \right)} {\mu_{Wall} \left(3 n + 1 \right) }$
$\mu_{Wall} = \tau_{Wall}^{1 - 1/n} \left( \frac {K} {1 - X} \right) ^ {1/n}$
$\tau_{Wall} = \frac {D \Delta P} {4 L}$
All units are SI
$\Delta P$ Pressure drop, Pa.
$L$ Pipe length, m
$D$ Pipe diameter, m
$V$ Fluid velocity, $m/s$
Chilton and Stainsby state that defining the Reynolds number as
$Re = \frac {R} {n ^ 2 \left( 1 - X \right) ^ 4}$

allows standard Newtonian friction factor correlations to be used.