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The Herschel–Bulkley fluid is a generalized model of a
non-Newtonian fluid A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for exa ...
, in which the
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
experienced by the fluid is related to the
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
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.


Definition

In one dimension, the
constitutive equation In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
of the Herschel-Bulkley model after the yield stress has been reached can be written in the form: :\dot = 0, \qquad\qquad \mathrm\ \tau < \tau_ :\tau = \tau_ + k \dot ^ , \qquad\mathrm\ \tau \geq \tau_ where \tau is the
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
a \tau_ the yield stress a k the consistency index a\cdots^ \dot the
shear rate In physics, shear rate is the rate at which a progressive shearing deformation is applied to some material. Simple shear The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary ...
^ and n the flow index imensionless If \tau < \tau_ the Herschel-Bulkley fluid behaves as a rigid (non-deformable) solid, otherwise it behaves as a fluid. For n<1 the fluid is shear-thinning, whereas for n>1 the fluid is shear-thickening. If n=1 and \tau_0=0, this model reduces to that of a
Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
. Reformulated as a tensor, we can instead write: :\underline = 0, \qquad\qquad\qquad\qquad \mathrm\ , \underline, < \tau_ :\underline =\left(\frac + k , \underline, ^ \right)\underline, \qquad\qquad \mathrm\ , \underline, \geq \tau_ Note that the double underlines indicate a tensor quantity.


Modelling Herschel-Bulkley fluids using regularization

The viscosity associated with the Herschel-Bulkley stress diverges to infinity as the strain rate approaches zero. This divergence makes the model difficult to implement in numerical simulations, so it is common to implement ''regularized'' models with an upper limiting viscosity. For instance, the Herschel-Bulkley fluid can be approximated as a
generalized Newtonian fluid A generalized Newtonian fluid is an idealized fluid for which the shear stress is a function of shear rate at the particular time, but not dependent upon the history of deformation. Although this type of fluid is non-Newtonian (i.e. non-linear) in ...
model with an effective (or apparent) viscosity being given as : \mu_ = \begin \mu_0, & , \dot, \leq \dot_0 \\ \tau_0 , \dot, ^ + k , \dot, ^, & , \dot, \geq \dot_0 \end Here, the limiting viscosity \mu_0 replaces the divergence at low strain rates. Its value is chosen such that \mu_0=k \dot_0^+\tau_0 \dot_0^ to ensure the viscosity is a continuous function of strain rate. 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. It is not entirely possible to capture ''rigid'' behavior described by the constitutive equation of the Herschel-Bulkley model using a regularised model. This is because a finite effective viscosity will always lead to a small degree of yielding under the influence of external forces (e.g. gravity). The characteristic timescale of the phenomenon being studied is thus an important consideration when choosing a regularisation threshold. In an incompressible flow, the
viscous stress tensor The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress ...
is given as a viscosity, multiplied by the rate-of-strain tensor :\tau_ = 2 \mu_(, \dot, ) E_ = \mu_(, \dot, ) \left(\frac+\frac\right), (Note that \mu_(, \dot, ) indicates that the effective viscosity is a function of the shear rate.) Furthermore, the magnitude of the shear rate is given by :, \dot, =\sqrt. The magnitude of the shear rate is an
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
approximation, and it is coupled with the second
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
of the rate-of-strain tensor :II_E = tr(E_ E^) = E_E^.


Channel flow

A frequently-encountered situation in experiments is
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
-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 Rheology (; ) is the study of the flow of matter, primarily in a fluid (liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an appli ...
model, reduce to a single equation: :\frac=\frac\left(\mu\frac\right)\,\,\, =\begin\mu_0\frac,&\left, \frac\<\gamma_0\\ \\\frac\left \frac\^+\tau_0\left, \frac\^\right)\frac\right&\left, \frac\\geq\gamma_0\end 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\frac, which is negative for flow from left to right, and the Bingham number: :Bn=\frac\left(\frac\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 \frac\frac\left left(\pi_0\left(z-z_1\right)+\gamma_0^n\right)^-\left(-\pi_0z_1+\gamma_0^n\right)^\right&z\in\left ,z_1\right\ \frac\left(z^2-z\right)+k,&z\in\left _1,z_2\right\\ \frac\frac\left left(-\pi_0\left(z-z_2\right)+\gamma_0^n\right)^-\left(-\pi_0\left(1-z_2\right)+\gamma_0^n\right)^\right&z\in\left _2,1\right\ \end 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_=\tfrac\pm\delta, where \delta=\frac\leq \tfrac. Since \mu_0=\gamma_0^+Bn/\gamma_0, for a given \left(\gamma_0,Bn\right) pair, there is a critical pressure gradient , \pi_, =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 Laminar means "flat". Laminar may refer to: Terms in science and engineering: * Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules * Laminar armour or "band ...
flow Chilton and Stainsby 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 = \frac \left( \frac \right) ^ n \left( \frac \right) ^ n \frac \left( \frac \right) ^ n : X = \frac : a = \frac : b = \frac : c = \frac :For
turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
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 Hathoot, HM, 2004, "Minimum-cost design of pipelines transporting non-Newtonian fluids", ''Alexandrian Engineering Journal'', 43(3) 375 - 382 : R = \frac : : \mu_ = \tau_^ \left( \frac \right) ^ : : \tau_ = \frac :All units are SI :\Delta P Pressure drop, Pa. :L Pipe length, m :D Pipe diameter, m :V Mean fluid velocity, m/s :Chilton and Stainsby state that defining the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
as :Re = \frac allows standard Newtonian friction factor correlations to be used. The pressure drop can then be calculated, given a suitable friction factor correlation. An iterative procedure is required, as the pressure drop is required to initiate the calculations as well as be the outcome of them.


See also

*
Viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...


References


External links


Description of Herschel–Bulkley fluid; graphical comparison between rheological models
{{DEFAULTSORT:Herschel-Bulkley fluid Rheology Non-Newtonian fluids Equations of fluid dynamics