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The '''Herschel–Bulkley fluid''' is a generalized model of a [[non-Newtonian fluid]], in which the [[strain (materials science)|strain]] experienced by the fluid is related to the [[Stress (physics)|stress]]  in a complicated, non-linear way.  Three parameters characterize this relationship: the consistency ''k'', the flow index ''n'', and the yield shear stress <math>\tau_0</math>.  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 [[Non-newtonian fluid#Oobleck|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 Herschel and Bulkley in 1926.<ref>{{citation
 |  doi = 10.1007/BF01432034
 |  first1 = W.H. | last1 = Herschel 
 | first2 = R. | last2 = Bulkley
 | title= Konsistenzmessungen von Gummi-Benzollösungen
 | journal = Kolloid Zeitschrift
 | volume = 39
 | year = 1926
 | pages = 291–300
}}</ref>
<ref name=Tang2004>{{citation
 | last1 = Tang | first1 =  Hansong S.
 | last2 = Kalyon | first2 = Dilhan M.
 | year = 2004
 | title = Estimation of the parameters of Herschel–Bulkley fluid under wall slip using a combination of capillary and squeeze flow viscometers
 | journal = Rheologica Acta
 | volume = 43
 | issue = 1
 | pages = 80–88
 | doi = 10.1007/s00397-003-0322-y
}}</ref>

==Definition==

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

:<math>\tau_{ij}=2\mu E_{ij}=\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right),</math>

where in contrast to the Newtonian fluid, the viscosity is itself a function of the strain tensor.  This is constituted through the formula <ref>K. C. Sahu, P. Valluri, P. D. M. Spelt, and O. K. Matar (2007) 'Linear instability of pressure-driven channel flow of a Newtonian and a Herschel–Bulkley fluid' Phys. Fluids 19, 122101</ref>

:<math>\mu=\begin{cases}\mu_0,&\Pi\leq\Pi_0\\k\Pi^{n-1}+\tau_0\Pi^{-1},&\Pi\geq\Pi_0\end{cases},</math>

where <math>\Pi</math> is the second invariant of the rate-of-strain tensor:

:<math>\Pi=\sqrt{2E_{ij}E^{ij}}</math>.

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

:<math>\tau = \tau_{0} + K \gamma ^ {n}</math>

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

==Channel flow==

[[Image:Os schematic.png|thumb|right|300px|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 <ref>D. J. Acheson 'Elementary Fluid Mechanics' (1990), Oxford, p. 51</ref> (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 [[rheology|rheological]] model, reduce to a single equation:

[[Image:Channel one phase.png|thumb|right|300px|Velocity profile of the Herschel–Bulkley fluid for various flow indices ''n''.  In each case, the non-dimensional pressure is <math>\pi_0=-10</math>.  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.]]

:<math>\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}
</math>

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

<math>\pi_0=\frac{H}{P_0}\frac{\partial p}{\partial x},</math>

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

:<math>Bn=\frac{\tau_0}{k}\left(\frac{H}{V}\right)^n.</math>

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

Solving this equation gives the velocity profile:

<math>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}</math>

Here ''k'' is a matching constant such that <math>u\left(z_1\right)</math> is continuous.  The profile respects the [[no-slip condition|no-slip]] conditions at the channel boundaries,

:<math>u(0)=u(1)=0,</math>

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

<math>\delta=\frac{\gamma_0\mu_0}{|\pi_0|}\leq \tfrac{1}{2}.</math>

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

<math>|\pi_{0,\mathrm{c}}|=2\left(\gamma_0+Bn\right).</math>

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|laminar]] flow Chilton and Stainsby <ref>Chilton, RA and R Stainsby, 1998, "Pressure loss equations for laminar and turbulent non-Newtonian pipe flow", ''Journal of Hydraulic Engineering'' '''124'''(5) pp. 522 ff.</ref> 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.

:<math>
\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
</math>

:<math>
X = \frac {4 L \tau_ y} {D \Delta P}
</math>
:<math> a = \frac {1} {2 n + 1} </math>
:<math> b = \frac {2 n}     { \left( n + 1 \right) \left( 2 n + 1 \right) } </math>
:<math> c = \frac {2 n ^ 2} { \left( n + 1 \right) \left( 2 n + 1 \right) } </math>

: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 <ref>Hathoot, HM, 2004, "Minimum-cost design of pipelines transporting non-Newtonian fluids", ''Alexandrian Engineering Journal'', '''43'''(3) 375 - 382</ref>

:<math> R = \frac {4 n \rho V D \left( 1 - a X - b X^2 - cX^3 \right)} {\mu_{Wall} \left(3 n + 1 \right) } </math>
:
:<math> \mu_{Wall} = \tau_{Wall}^{1 - 1/n} \left( \frac {K} {1 - X} \right) ^ {1/n} </math>
:
:<math> \tau_{Wall} = \frac {D \Delta P} {4 L} </math>

:All units are SI
:<math>\Delta P</math> Pressure drop, Pa.
:<math>L</math> Pipe length, m
:<math>D</math> Pipe diameter, m
:<math>V</math> Fluid velocity, <math>m/s</math>

:Chilton and Stainsby state that defining the Reynolds number as

:<math>Re = \frac {R} {n ^ 2 \left( 1 - X \right) ^ 4} </math>

allows standard Newtonian [[fanning friction factor|friction factor]] correlations to be used.

==See also==

[[Viscosity]]

==References==

<references/>

==External links==
* [http://www.glossary.oilfield.slb.com/Display.cfm?Term=Herschel-Bulkley%20fluid Description of Herschel–Bulkley fluid; graphical comparison between rheological models]

{{DEFAULTSORT:Herschel-Bulkley fluid}}
[[Category:Fluid dynamics]]
[[Category:Non-Newtonian fluids]]
[[Category:Equations of fluid dynamics]]