View source for Μ(I) rheology

{{Short description|Model of the rheology of a granular flow}}
{{DISPLAYTITLE:<var>μ</var>(<var>I</var>) rheology}}
In [[granular mechanics]], the '''''μ''(''I'') rheology''' is one model of the [[rheology]] of a [[granular flow]]. 

==Details==

The [[inertial number]] of a granular flow is a [[dimensionless quantity]] defined as
<div class="center"><math>I = \frac{||\dot\gamma|| d}{\sqrt{P/\rho}},</math></div>
where <math>\dot\gamma</math> is the shear rate tensor, <math>||\dot\gamma||</math> is its magnitude, ''d'' is the average particle diameter, ''P'' is the isotropic pressure and ''&rho;'' is the density. It is a local quantity and may take different values at different locations in the flow. 

The ''&mu;''(''I'') rheology asserts a [[constitutive relationship]] between the [[Cauchy stress tensor|stress tensor]] of the flow and the rate of strain tensor:
<div class="center"><math> \sigma_{ij} = -P\delta_{ij} + \mu(I)P \frac{\dot\gamma_{ij}}{||\dot\gamma||} </math></div>
where the eponymous ''&mu;''(''I'') is a dimensionless function of ''I''. As with [[Newtonian fluid]]s, the first term -''P&delta;''<sub>''ij''</sub> represents the effect of pressure. The second term represents a [[shear stress]]: it acts in the direction of the shear, and its magnitude is equal to the pressure multiplied by a coefficient of friction ''&mu;''(''I''). This is therefore a generalisation of the standard [[Coulomb friction]] model. The multiplicative term <math>\mu(I)P/||\dot\gamma||</math> can be interpreted as the effective viscosity of the granular material, which tends to infinity in the limit of vanishing shear flow, ensuring the existence of a yield criterion.<ref name = "Jop_Nature_2006">{{cite journal|last1=Jop|first1=Pierre|last2=Forterre|first2=Yoël|last3=Pouliquen|first3=Olivier|title=A constitutive law for dense granular flows|journal=Nature|date=8 June 2006|volume=441|issue=7094|pages=727–730|doi=10.1038/nature04801|pmid=16760972 |arxiv=cond-mat/0612110|bibcode=2006Natur.441..727J}}</ref>

One deficiency of the ''μ''(''I'') rheology is that it does not capture the [[hysteresis|hysteretic]] properties of a granular material.<ref>{{cite journal|last1=Forterre|first1=Yoël|last2=Pouliquen|first2=Olivier|title=Flows of Dense Granular Media|journal=Annual Review of Fluid Mechanics|date=January 2008|volume=40|issue=1|pages=1–24|doi=10.1146/annurev.fluid.40.111406.102142|bibcode=2008AnRFM..40....1F}}</ref> 

==Development==

The ''μ''(''I'') rheology was developed by Pierre Jop ''et al.'' in 2006.<ref name = "Jop_Nature_2006" /><ref>{{cite book|last1=Holyoake|first1=Alex|title=Rapid Granular Flows in an Inclined Chute|date=December 2011|url=http://alexholyoake.com/academic/RapidGranularFlows_Holyoake.pdf|accessdate=21 July 2015}}</ref> Since its initial introduction, many works has been carried out to modify and improve this rheology model.<ref>{{Cite journal |last1=Barker |first1=T. |last2=Gray |first2=J. M. N. T. |date=October 2017 |title=Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow |url=https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/partial-regularisation-of-the-incompressible-irheology-for-granular-flow/EE6FA9B53D176FC0A6647D67E586CE56 |journal=Journal of Fluid Mechanics |language=en |volume=828 |pages=5–32 |doi=10.1017/jfm.2017.428 |issn=0022-1120|hdl=20.500.11820/834cfa72-7cc7-4f75-b0c4-733ad4ed346c |hdl-access=free }}</ref> This model provides an alternative approach to the [[Discrete element method|Discrete Element Method]] (DEM), offering a lower computational cost for simulating granular flows within mixers.<ref>{{Cite journal |last1=Biroun |first1=Mehdi H. |last2=Sorensen |first2=Eva |last3=Hilden |first3=Jon L. |last4=Mazzei |first4=Luca |date=October 2023 |title=CFD modelling of powder flow in a continuous horizontal mixer |url=https://doi.org/10.1016/j.powtec.2023.118843 |journal=Powder Technology |volume=428 |pages=118843 |doi=10.1016/j.powtec.2023.118843 |issn=0032-5910|doi-access=free }}</ref>

==See also==
* [[Dilatancy (granular material)]]

==References==
<references/>

{{DEFAULTSORT:Mu I rheology}}
[[Category:Rheology]]
[[Category:Granularity of materials]]