- 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 thestrain in a complicated, non-linear way. Three parameters characterize this relationship: the consistency "k", the flow index "n", and the yield shear stress au_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, whileoobleck 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::au_{ij}=2mu E_{ij}=muleft(frac{partial u_i}{partial x_j}+frac{partial u_j}{partial x_i} ight),
where in contrast to the Newtonian fluid, the viscosity is itself a function of the strain tensor. This is constituted through the formula [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]
:mu=egin{cases}mu_0,&PileqPi_0\kPi^{n-1}+ au_0Pi^{-1},&PigeqPi_0end{cases},
where Pi is the second invariant of the rate-of-strain tensor:
:Pi=sqrt{2E_{ij}E^{ij.
If "n"=1 and au_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^{n-1}+ au_0Pi_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.
Channel flow
A frequently-encountered situation in experiments is
pressure -driven channel flow [D. J. Acheson 'Elementary Fluid Mechanics' (1990), Oxford, p. 51] (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, theNavier-Stokes equations , together with the rheological model, reduce to a single equation::frac{partial p}{partial x}=frac{partial}{partial z}left(mufrac{partial u}{partial z} ight),,,=egin{cases}mu_0frac{partial^2 u}{partial{z}^2},&left|frac{partial u}{partial z} ight|
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(V/H ight)^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{ au_0}{k}left(frac{H}{V} ight)^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|;
* A region close to the top wall where partial u/partial z<-gamma_0,Solving this equation gives the velocity profile:
uleft(z ight)=egin{cases}frac{n}{n+1}frac{1}{pi_0}left [left(pi_0left(z-z_1 ight)+gamma_0^n ight)^{1+left(1/n ight)}-left(-pi_0z_1+gamma_0^n ight)^{1+left(1/n ight)} ight] ,&zinleft [0,z_1 ight] \frac{pi_0}{2mu_0}left(z^2-z ight)+k,&zinleft [z_1,z_2 ight] ,\frac{n}{n+1}frac{1}{pi_0}left [left(-pi_0left(z-z_2 ight)+gamma_0^n ight)^{1+left(1/n ight)}-left(-pi_0left(1-z_2 ight)+gamma_0^n ight)^{1+left(1/n ight)} ight] ,&zinleft [z_2,1 ight] \end{cases}
Here "k" is a matching constant such that uleft(z_1 ight) 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}= frac{1}{2}pmdelta, where
delta=frac{gamma_0mu_0}leq frac{1}{2}.
Since mu_0=gamma_0^{n-1}+Bn/gamma_0, for a given left(gamma_0,Bn ight) pair, there is a critical pressure gradient
pi_{0,mathrm{c|=2left(gamma_0+Bn ight).
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.
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]
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