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December  2018, 11(6): 1443-1474. doi: 10.3934/krm.2018057

Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles

1. 

Lehrstuhl Angewandte Mathematik Ⅰ, FAU Erlangen-Nürnberg, D-91058 Erlangen, Germany

2. 

FB IV - Mathematik, University of Trier, D-54286 Trier, Germany

* Corresponding author: A. Vibe

Received  July 2015 Revised  December 2017 Published  June 2018

In this paper we derive asymptotically the macroscopic bulk stress of a suspension of small inertial particles in an incompressible Newtonian fluid. We apply the general asymptotic framework to the special case of ellipsoidal particles and show the resulting modification due to inertia on the well-known particle-stresses based on the theory by Batchelor and Jeffery.

Citation: Alexander Vibe, Nicole Marheineke. Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles. Kinetic and Related Models, 2018, 11 (6) : 1443-1474. doi: 10.3934/krm.2018057
References:
[1]

G. K. Batchelor, The stress system in a suspension of force-free particles, Journal of Fluid Mechanics, 41 (1970), 545-570. 

[2]

G. K. Batchelor, The stress generated in a non-dilute suspension of elongated particles by pure straining motion, Journal of Fluid Mechanics, 46 (1971), 813-829.  doi: 10.1017/S0022112071000879.

[3]

M. Berezhnyi and E. Khruslov, Asymptotic behavior of a suspension of oriented particles in a viscous incompressible fluid, Asymptotic Analysis, 83 (2013), 331-353.  doi: 10.3233/ASY-131162.

[4]

S. M. Dinh, On the Rheology of Concentrated Fiber Suspensions, PhD thesis, Massachusetts Institute of Technology, Cambridge, USA, 1981.

[5]

J. DupireM. Socol and A. Viallat, Full dynamics of a red blood cell in shear flow, Proceedings of the National Academy of Sciences of the United States of America, 109 (2012), 20808-20813.  doi: 10.1073/pnas.1210236109.

[6]

D. Edwardes, Steady motion of a viscous liquid in which an ellipsoid is constrained to rotate about a principal axis, The Quarterly Journal of Pure and Applied Mathematics, 26 (1893), 70-78. 

[7]

A. Einstein, Eine neue Bestimmung der Moleküldimensionen, Annalen der Physik, 324 (1906), 289-306.  doi: 10.1002/andp.19063240204.

[8]

F. Folgar and C. L. Tucker, Orientation behavior of fibers in concentrated suspensions, Journal of Reinforced Plastics and Composites, 3 (1984), 98-119.  doi: 10.1177/073168448400300201.

[9]

H. Giesekus, Elasto-viskose Flüssigkeiten, für die in stationären Schichtströmungen sämtliche Normalspannungskomponenten verschieden großsind, Rheologica Acta, 2 (1962), 50-62.  doi: 10.1007/BF01972555.

[10]

M. Hillairet, On the homogenization of the stokes problem in a perforated domain, preprint, arXiv: 1604.04379.

[11]

G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 102 (1922), 161-179.  doi: 10.1098/rspa.1922.0078.

[12]

M. Junk and R. Illner, A new derivation of Jeffery's equation, Journal of Mathematical Fluid Mechanics, 9 (2007), 455-488.  doi: 10.1007/s00021-005-0208-0.

[13]

E. Khruslov and L. Berlyand, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid, SIAM Journal on Applied Mathematics, 64 (2004), 1002-1034.  doi: 10.1137/S0036139902403913.

[14]

S. Kim and S. J. Karilla, Microhydrodynamics. Principles and Selected Applications, Dover Publications, Inc., Mineola, New York, 2005.

[15]

L. G. Leal and E. J. Hinch, The effect of weak Brownian rotations on particles in shear flow, Journal of Fluid Mechanics, 46 (1971), 685-703.  doi: 10.1017/S0022112071000788.

[16]

L. G. Leal and E. J. Hinch, Theoretical studies of a suspension of rigid particles affected by Brownian couples, Rheologica Acta, 12 (1973), 127-132. 

[17]

S. B. Lindström and T. Uesaka, Simulation of semidilute suspensions of non-Brownian fibers in shear flow, The Journal of Chemical Physics, 128 (2008), 024901. doi: 10.1063/1.2815766.

[18]

A. Oberbeck, Ueber stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung, Journal für die reine und angewandte Mathematik, 81 (1876), 62-80.  doi: 10.1515/crll.1876.81.62.

[19]

N. PatankarP. SinghD. JosephR. Glowinski and T.-W. Pan, A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows, International Journal of Multiphase Flow, 26 (2000), 1509-1524. 

[20]

N. Phan-Thien and A. L. Graham, A new constitutive model for fibre suspensions: Flow past a sphere, Rheologica Acta, 30 (1991), 44-57.  doi: 10.1007/BF00366793.

[21]

A. Prosperetti, The average stress in incompressible disperse flow, International Journal of Multiphase Flow, 30 (2004), 1011-1036.  doi: 10.1016/j.ijmultiphaseflow.2004.05.003.

[22]

A. ProsperettiQ. Zhang and K. Ichiki, The stress system in a suspension of heavy particles: Antisymmetric contribution, Journal of Fluid Mechanics, 554 (2006), 125-146.  doi: 10.1017/S0022112006009402.

[23]

W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1017/CBO9780511608810.

[24]

A. Vibe, Kinetische Modellierung ausgedehnter Partikel in Strömungen, Master's thesis, Friedrich-Alexander University Erlangen-Nürnberg, Germany, 2014.

[25]

D. C. Wilcox, Turbulence Modeling for CFD, DCW Industries, La Ca˜nada, Calif., 1993.

[26]

Q. Zhang and A. Prosperetti, Physics-based analysis of the hydrodynamic stress in a fluid-particle system, Physics of Fluids, 22 (2010), 033306. doi: 10.1063/1.3365950.

show all references

References:
[1]

G. K. Batchelor, The stress system in a suspension of force-free particles, Journal of Fluid Mechanics, 41 (1970), 545-570. 

[2]

G. K. Batchelor, The stress generated in a non-dilute suspension of elongated particles by pure straining motion, Journal of Fluid Mechanics, 46 (1971), 813-829.  doi: 10.1017/S0022112071000879.

[3]

M. Berezhnyi and E. Khruslov, Asymptotic behavior of a suspension of oriented particles in a viscous incompressible fluid, Asymptotic Analysis, 83 (2013), 331-353.  doi: 10.3233/ASY-131162.

[4]

S. M. Dinh, On the Rheology of Concentrated Fiber Suspensions, PhD thesis, Massachusetts Institute of Technology, Cambridge, USA, 1981.

[5]

J. DupireM. Socol and A. Viallat, Full dynamics of a red blood cell in shear flow, Proceedings of the National Academy of Sciences of the United States of America, 109 (2012), 20808-20813.  doi: 10.1073/pnas.1210236109.

[6]

D. Edwardes, Steady motion of a viscous liquid in which an ellipsoid is constrained to rotate about a principal axis, The Quarterly Journal of Pure and Applied Mathematics, 26 (1893), 70-78. 

[7]

A. Einstein, Eine neue Bestimmung der Moleküldimensionen, Annalen der Physik, 324 (1906), 289-306.  doi: 10.1002/andp.19063240204.

[8]

F. Folgar and C. L. Tucker, Orientation behavior of fibers in concentrated suspensions, Journal of Reinforced Plastics and Composites, 3 (1984), 98-119.  doi: 10.1177/073168448400300201.

[9]

H. Giesekus, Elasto-viskose Flüssigkeiten, für die in stationären Schichtströmungen sämtliche Normalspannungskomponenten verschieden großsind, Rheologica Acta, 2 (1962), 50-62.  doi: 10.1007/BF01972555.

[10]

M. Hillairet, On the homogenization of the stokes problem in a perforated domain, preprint, arXiv: 1604.04379.

[11]

G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 102 (1922), 161-179.  doi: 10.1098/rspa.1922.0078.

[12]

M. Junk and R. Illner, A new derivation of Jeffery's equation, Journal of Mathematical Fluid Mechanics, 9 (2007), 455-488.  doi: 10.1007/s00021-005-0208-0.

[13]

E. Khruslov and L. Berlyand, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid, SIAM Journal on Applied Mathematics, 64 (2004), 1002-1034.  doi: 10.1137/S0036139902403913.

[14]

S. Kim and S. J. Karilla, Microhydrodynamics. Principles and Selected Applications, Dover Publications, Inc., Mineola, New York, 2005.

[15]

L. G. Leal and E. J. Hinch, The effect of weak Brownian rotations on particles in shear flow, Journal of Fluid Mechanics, 46 (1971), 685-703.  doi: 10.1017/S0022112071000788.

[16]

L. G. Leal and E. J. Hinch, Theoretical studies of a suspension of rigid particles affected by Brownian couples, Rheologica Acta, 12 (1973), 127-132. 

[17]

S. B. Lindström and T. Uesaka, Simulation of semidilute suspensions of non-Brownian fibers in shear flow, The Journal of Chemical Physics, 128 (2008), 024901. doi: 10.1063/1.2815766.

[18]

A. Oberbeck, Ueber stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung, Journal für die reine und angewandte Mathematik, 81 (1876), 62-80.  doi: 10.1515/crll.1876.81.62.

[19]

N. PatankarP. SinghD. JosephR. Glowinski and T.-W. Pan, A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows, International Journal of Multiphase Flow, 26 (2000), 1509-1524. 

[20]

N. Phan-Thien and A. L. Graham, A new constitutive model for fibre suspensions: Flow past a sphere, Rheologica Acta, 30 (1991), 44-57.  doi: 10.1007/BF00366793.

[21]

A. Prosperetti, The average stress in incompressible disperse flow, International Journal of Multiphase Flow, 30 (2004), 1011-1036.  doi: 10.1016/j.ijmultiphaseflow.2004.05.003.

[22]

A. ProsperettiQ. Zhang and K. Ichiki, The stress system in a suspension of heavy particles: Antisymmetric contribution, Journal of Fluid Mechanics, 554 (2006), 125-146.  doi: 10.1017/S0022112006009402.

[23]

W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1017/CBO9780511608810.

[24]

A. Vibe, Kinetische Modellierung ausgedehnter Partikel in Strömungen, Master's thesis, Friedrich-Alexander University Erlangen-Nürnberg, Germany, 2014.

[25]

D. C. Wilcox, Turbulence Modeling for CFD, DCW Industries, La Ca˜nada, Calif., 1993.

[26]

Q. Zhang and A. Prosperetti, Physics-based analysis of the hydrodynamic stress in a fluid-particle system, Physics of Fluids, 22 (2010), 033306. doi: 10.1063/1.3365950.

Figure 1.  Lagrangian description, bijective mapping between reference state and the actual time-dependent state.
Table 1.  Classification of inertial types by means of the density scaling function, cf. (4).
$\alpha_{\rho}(\epsilon)=\epsilon^r$ Name of type Behavior of density ratio
$r\geq1$ light weighted tracer particles $\rho_\epsilon\to0$
$r=0$ normal tracer particles $\rho_\epsilon\equiv const$
$r=-1$ heavy tracer particles $\rho_\epsilon\to\infty$
$\alpha_{\rho}(\epsilon)=\epsilon^r$ Name of type Behavior of density ratio
$r\geq1$ light weighted tracer particles $\rho_\epsilon\to0$
$r=0$ normal tracer particles $\rho_\epsilon\equiv const$
$r=-1$ heavy tracer particles $\rho_\epsilon\to\infty$
Table 2.  Balances of the accelerations with the integral terms (given in Remark 2) in (15) for the different inertial regimes. Each row shows the $\mathcal{O}(\epsilon^\ell)$-correction of the momentum equations and each column corresponds to the choice of the density scaling function $\alpha_\rho(\epsilon) = \epsilon^r$.
$\ell\backslash r$ $-3$ $-2$ $-1$ $0$ 1
$-2$ ${\bf{k}}_0={\bf{0}}$
$-1$ ${\bf{k}}_1={\bf{0}}$ ${\bf{k}}_0={\bf{0}}$
$0$ ${\bf{k}}_2={\bf{m}}_1^v$ ${\bf{k}}_1={\bf{m}}_1^v$ ${\bf{k}}_0={\bf{m}}_1^v$ ${\bf{0}}={\bf{m}}_1^v$ ${\bf{0}}={\bf{m}}_1^v$
$1$ ${\bf{k}}_3={\bf{m}}_2^v$ ${\bf{k}}_2={\bf{m}}_2^v$ ${\bf{k}}_1={\bf{m}}_2^v$ ${\bf{k}}_0={\bf{m}}_2^v$ ${\bf{0}}={\bf{m}}_2^v$
$\ell\backslash r$ $-3$ $-2$ $-1$ $0$ $1$
$-1$ $\mathit{\boldsymbol{\ell}}_0={\bf{0}}$
$0$ $\mathit{\boldsymbol{\ell}}_1={\bf{m}}_1^{\omega}$ $\mathit{\boldsymbol{\ell}}_0={\bf{m}}_1^{\omega}$ ${\bf{0}}={\bf{m}}_1^{\omega}$ ${\bf{0}}={\bf{m}}_1^{\omega}$ ${\bf{0}}={\bf{m}}_1^{\omega}$
$1$ $\mathit{\boldsymbol{\ell}}_2={\bf{m}}_2^{\omega}$ $\mathit{\boldsymbol{\ell}}_1={\bf{m}}_2^{\omega}$ $\mathit{\boldsymbol{\ell}}_0={\bf{m}}_2^{\omega}$ ${\bf{0}}={\bf{m}}_2^{\omega}$ ${\bf{0}}={\bf{m}}_2^{\omega}$
$\ell\backslash r$ $-3$ $-2$ $-1$ $0$ 1
$-2$ ${\bf{k}}_0={\bf{0}}$
$-1$ ${\bf{k}}_1={\bf{0}}$ ${\bf{k}}_0={\bf{0}}$
$0$ ${\bf{k}}_2={\bf{m}}_1^v$ ${\bf{k}}_1={\bf{m}}_1^v$ ${\bf{k}}_0={\bf{m}}_1^v$ ${\bf{0}}={\bf{m}}_1^v$ ${\bf{0}}={\bf{m}}_1^v$
$1$ ${\bf{k}}_3={\bf{m}}_2^v$ ${\bf{k}}_2={\bf{m}}_2^v$ ${\bf{k}}_1={\bf{m}}_2^v$ ${\bf{k}}_0={\bf{m}}_2^v$ ${\bf{0}}={\bf{m}}_2^v$
$\ell\backslash r$ $-3$ $-2$ $-1$ $0$ $1$
$-1$ $\mathit{\boldsymbol{\ell}}_0={\bf{0}}$
$0$ $\mathit{\boldsymbol{\ell}}_1={\bf{m}}_1^{\omega}$ $\mathit{\boldsymbol{\ell}}_0={\bf{m}}_1^{\omega}$ ${\bf{0}}={\bf{m}}_1^{\omega}$ ${\bf{0}}={\bf{m}}_1^{\omega}$ ${\bf{0}}={\bf{m}}_1^{\omega}$
$1$ $\mathit{\boldsymbol{\ell}}_2={\bf{m}}_2^{\omega}$ $\mathit{\boldsymbol{\ell}}_1={\bf{m}}_2^{\omega}$ $\mathit{\boldsymbol{\ell}}_0={\bf{m}}_2^{\omega}$ ${\bf{0}}={\bf{m}}_2^{\omega}$ ${\bf{0}}={\bf{m}}_2^{\omega}$
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