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 :

$\tau _{ij}=2\mu E_{ij}=\mu \left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right),$

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

$\mu =k\Pi ^{n-1}+\tau _{0}\Pi ^{-1},$

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.

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 (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:

${\frac {\partial p}{\partial x}}={\frac {\partial }{\partial z}}\left[k\left(\Pi ^{n-1}+{\frac {\tau _{0}}{\Pi }}\right){\frac {\partial u}{\partial z}}\right].$

Solving this equation gives the velocity profile:

$u(z)=c_{1}+{\frac {k}{\partial p/\partial x}}{\frac {n}{n+1}}\left|{\frac {1}{k}}{\frac {\partial p}{\partial x}}z+c_{2}-{\frac {\tau _{0}}{k}}\right|^{1+\left(1/n\right)}$.

Here $c_{1}$ and $c_{2}$ are constants of integration to be determined. These are fixed by the no-slip condition

$u(0)=u(H)=0,$

where H is the channel depth. Next, the velocity is non-dimensionalized on a characteristic velocity V and the channel depth H. Finally then, the velocity profile is obtained:

$u(z)={\frac {1}{2}}{\frac {n}{n+1}}\left({\frac {|\pi _{0}|}{2}}\right)^{1/n}-{\frac {1}{|\pi _{0}|}}{\frac {n}{n+1}}\left|-|\pi _{0}|z+{\frac {|\pi _{0}|}{2}}\right|^{1+\left(1/n\right)},$

where

$\pi _{0}={\frac {H}{P_{0}}}{\frac {\partial p}{\partial x}},\,\,P_{0}=k\left({\frac {V}{H}}\right)^{n},$

is the non-dimensional pressure gradient, which is negative for flow from left to right. The Bingham number is also defined through this analysis:

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

External links

• Description of Herschel-Bulkley fluid; graphical comparison between rheological models