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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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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.  Google Scholar

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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