Definitions

# Herschel-Bulkley fluid

The Herschel-Bulkley fluid is a generalized model of a Non-Newtonian fluid, in which the stress experienced by the fluid is related to the strain 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.

## Definition

The viscous stress tensor is given, in the usual way, as a viscosity, multiplied by the rate-of-strain tensor:

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

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

$mu=begin\left\{cases\right\}mu_0,&PileqPi_0kPi^\left\{n-1\right\}+tau_0Pi^\left\{-1\right\},&PigeqPi_0end\left\{cases\right\},$

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

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

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 $mu_0=kPi_0^\left\{n-1\right\}+tau_0Pi_0^\left\{-1\right\}$. 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.

## Channel flow

A frequently-encountered situation in experiments is pressure-driven channel flow (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:

$frac\left\{partial p\right\}\left\{partial x\right\}=frac\left\{partial\right\}\left\{partial z\right\}left\left(mufrac\left\{partial u\right\}\left\{partial z\right\}right\right),,,$
=begin{cases}mu_0frac{partial^2 u}{partial{z}^2},&left|frac{partial u}{partial z}right|

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

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

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

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

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
• A region close to the top wall where $partial u/partial z<-gamma_0$,

Solving this equation gives the velocity profile:

$uleft\left(zright\right)=begin\left\{cases\right\} frac\left\{n\right\}\left\{n+1\right\}frac\left\{1\right\}\left\{pi_0\right\}left\left[left\left(pi_0left\left(z-z_1right\right)+gamma_0^nright\right)^\left\{1+left\left(1/nright\right)\right\}-left\left(-pi_0z_1+gamma_0^nright\right)^\left\{1+left\left(1/nright\right)\right\}right\right],&zinleft\left[0,z_1right\right] frac\left\{pi_0\right\}\left\{2mu_0\right\}left\left(z^2-zright\right)+k,&zinleft\left[z_1,z_2right\right], frac\left\{n\right\}\left\{n+1\right\}frac\left\{1\right\}\left\{pi_0\right\}left\left[left\left(-pi_0left\left(z-z_2right\right)+gamma_0^nright\right)^\left\{1+left\left(1/nright\right)\right\}-left\left(-pi_0left\left(1-z_2right\right)+gamma_0^nright\right)^\left\{1+left\left(1/nright\right)\right\}right\right],&zinleft\left[z_2,1right\right] end\left\{cases\right\}$

Here k is a matching constant such that $uleft\left(z_1right\right)$ is continuous. The profile respects the no-slip conditions at the channel boundaries,

$u\left(0\right)=u\left(1\right)=0,$

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

$delta=frac\left\{gamma_0mu_0\right\}$

>leq tfrac{1}{2}.

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

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

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.