Abstract
Granular matters are collections of a large amount of discrete solid particles with interstices filled with a fluid, and granular flows are granular matters in flowing state. In contrast to simple fluids such as water or air which can be dealt with by classical fluid mechanics, granular flows exhibit significant non-Newtonian features. The evolutions of grain configurations as well as the interstitial fluids influence to a large extent the macroscopic features, which are referred to as the microstructural effects. Granular flows may be macroscopically considered complex rheological fluids, whose features are significantly affected by the microscopic time- and space-dependent internal structures. In other words, granular flows assume multiple time and length scales
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Notes
- 1.
Or Renatus Cartesius in Latin, hence the name Cartesian Coordinates, 1596–1650, a French philosopher, mathematician, and scientist, who is dubbed “Father of Modern Western Philosophy”.
- 2.
Ralph Alger Bagnold, 1896–1990, a British general, who is generally considered to have been a pioneer of desert exploration and laid the foundations for the research on sand transport by wind in his influential book entitled “The Physics of Blown Sand and Desert Dunes”, which is still a main reference in the field of granular matter.
- 3.
Christian Otto Mohr, 1835–1918, a German civil engineer. Charles-Augustin de Coulomb, 1736–1806, a French military engineer and physicist.
- 4.
In 1988, Savage and Lun used the concept of information entropy originating from the statistical thermodynamics to derive a relatively successful theory for particle segregation, known as the random fluctuating sieve mechanism.
- 5.
The term “hard-sphere” does not necessarily imply that the collisions are perfectly elastic. Rather, it means that there is no interpenetration or deformation during impact.
- 6.
The granular temperature is defined as one-third of the mean fluctuation kinetic energy of a granular matter, which was introduced by Blinowski and Ogawa in 1978. See e.g. Ogawa, S., Multitemperature theory of granular materials. In: Cowin, S.C., Satake, M. (eds) Proc. Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials. US-Japan Seminar, Sendai, Japan, 1978.
- 7.
Unlike the first-order phase transition, the chemical composition of a granular matter experiences no change during the solid-fluid transition. Only some parts of the granular body behave like a fluid, while the other parts exhibit a solid-like feature.
- 8.
While the balance of angular momentum can be fulfilled by the prescription of constitutive class, the balance of entropy is an inequality which will be used in the thermodynamic analysis.
- 9.
The exclusion of \(\dot{\theta }\) in Eq. (12.2.17) only leads to \(\lambda ^e=\hat{\lambda }^e(\theta )\). The specific form of \(\lambda ^e=1/\theta \) can only be derived for simple substances. This conjecture is motivated by previous works.
- 10.
From the perspective of practical application, this assumption is justified in most cases, although in most dry granular flows \(\gamma \) can be considered a constant, but \(\nu \) experiences a variation, so that a non-uniform bulk density field presents.
- 11.
In doing so, the internal friction can only enter the Cauchy stress tensor via the prescription of yield stress, hence, the evolution of internal friction is decoupled from other field equations.
- 12.
The DEM simulation results are quoted from Volfson, D., Tsimging, L.D., Aranson, I.S., Partially fluidized shear granular flow: Continuum theory and molecular dynamics simulations, Physics Review E, 68, 021301, 2003.
- 13.
In the Newtonian fluids, it is called the Kolmogorov scale .
- 14.
The simple relation between \(\theta ^M\) and \(\vartheta ^M\) does not hold in general between \(\theta ^T\) and \(\vartheta ^T\) for dry granular systems. It is only understood that \(\vartheta ^T=\hat{\vartheta }^T(\theta ^T,\dot{\theta }^T)\). Thus, either \(\theta ^i\) or \(\vartheta ^i\), \(i=\{M,T\}\), can be introduced as primitive fields.
- 15.
The experimental results are quoted from Perng, A.T.H., Capart, H., Chou, H.T., Granular configurations, motions, and correlations in slow uniform flows driven by an inclined conveyor belt, Granular Matter, 8, 5–17, 2006. The results of zeroth-order model are quoted from Fang, C., Wu, W., On the weak turbulent motions of an isothermal dry granular dense flow with incompressible grains: part II. Complete closure models and numerical simulations, Acta Geotechica, 9(5), 739–752, 2014.
Further Reading
S.J. Antony, W. Hoyle, Y. Ding (eds.), Granular Materials: Fundamentals and Applications (The Royal Society of Chemistry, Cambridge, 2004)
I.S. Aranson, L.S. Tsimring, Granular Pattern (Oxford University Press, Oxford, 2009)
T. Aste, T.D. Matteo, A. Tordesillas (eds.), Granular and Complex Materials (World Scientific, New Jersey, 2007)
D. Bideau, A. Hansen (eds.), Disorder and Granular Media (North-Holland, Amsterdam, 1993)
G. Capriz, P. Giovine, P.M. Mariano (eds.), Mathematical Models of Granular Matter (Springer, Berlin, 2008)
P. Coussot, Mudflows Rheology and Dynamics (A.A. Balkema, Rotterdam, 1997)
D.A. Drew, D.D. Joseph, S.L. Passman (eds.), Particulate Flows: Processing and Rheology (Springer, Berlin, 1998)
J. Duran, Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials (Springer, Berlin, 2000)
C. Fang, Gravity-driven dry granular slow flows down an inclined moving plane: a comparative study between two concepts of the evolution of porosity. Rheological Acta 48, 971–992 (2009)
C. Fang, Rheological characteristics of solid-fluid transition in dry granular dense flows: a thermodynamically consistent constitutive model with a pressure ratio order parameter. Int. J. Numer. Anal. Methods Geomech. 34(9), 881–905 (2010)
C. Fang, A \(k\)-\({\varepsilon }\) turbulent closure model of an isothermal dry granular dense matter. Contin. Mech. Thermodyn. 28(4), 1049–1069 (2016)
K. Hutter, N. Kirchner (eds.), Dynamic Response of Granular and Porous Materials under Large and Catastrophic Deformations (Springer, Berlin, 2003)
K. Hutter, K. Wilmánski (eds.), Kinetic and Continuum Theories of Granular and Porous Media (Springer, Berlin, 1999)
K. Iwashita, M. Oda (eds.), Mechanics of Granular Materials: An Introduction (A.A. Balkema, Rotterdam, 1999)
M. Jakob, O. Hungr (eds.), Debris-Flow Hazards and Related Phenomena (Springer, Berlin, 2005)
D. Kolymbas (ed.), Constitutive Modeling of Granular Materials (Springer, Berlin, 2000)
A. Mehta, Granular Physics (Cambridge University Press, Cambridge, 2007)
T. Pöschel, N. Brilliantov (eds.), Granular Gas Dynamics (Springer, Berlin, 2003)
S. Pudasaini, K. Hutter, Avalanche Dynamics (Springer, Berlin, 2007)
K.K. Rao, P.R. Nott, An Introduction to Granular Flows (Cambridge University Press, Cambridge, 2008)
A.F. Revuzhenko, Mechanics of Granular Media (Springer, Berlin, 2006)
G.H. Ristow, Pattern Formation in Granular Materials (Springer, Berlin, 2000)
L. Schneider, K. Hutter, Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context (Springer, Berlin, 2009)
T. Takahashi, Debris Flows: Mechanics, Prediction and Countermeasures (Taylor & Francis, London, 2007)
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Fang, C. (2019). Granular Flows. In: An Introduction to Fluid Mechanics. Springer Textbooks in Earth Sciences, Geography and Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-91821-1_12
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DOI: https://doi.org/10.1007/978-3-319-91821-1_12
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