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

Alexander Vibe; Nicole Marheineke

doi:10.3934/krm.2018057

Article Contents

2018, Volume 11, Issue 6: 1443-1474. Doi: 10.3934/krm.2018057

This issue Previous Article Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus Next Article Linear Boltzmann dynamics in a strip with large reflective obstacles: Stationary state and residence time

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

Alexander Vibe^1, , and
Nicole Marheineke^2,

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
* Corresponding author: A. Vibe

Received: June 30, 2015

Revised: November 30, 2017

Published: June 2018

Abstract Full Text(HTML) Figure(1) / Table(2) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 76Axx, 76Dxx, 76Mxx, 76Txx.

Citation:

Full Text(HTML)

Figure 1. Lagrangian description, bijective mapping between reference state and the actual time-dependent state.

Download: Full-size image PowerPoint slide

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	$\rho_\epsilon\to\infty$

| Show Table

DownLoad: CSV

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
$-2$	${\bf{k}}_0={\bf{0}}$
$-1$	${\bf{k}}_1={\bf{0}}$	${\bf{k}}_0={\bf{0}}$
$0$	${\bf{k}}_2={\bf{m}}_1^v$	${\bf{k}}_1={\bf{m}}_1^v$	${\bf{k}}_0={\bf{m}}_1^v$	${\bf{0}}={\bf{m}}_1^v$	${\bf{0}}={\bf{m}}_1^v$
$1$	${\bf{k}}_3={\bf{m}}_2^v$	${\bf{k}}_2={\bf{m}}_2^v$	${\bf{k}}_1={\bf{m}}_2^v$	${\bf{k}}_0={\bf{m}}_2^v$	${\bf{0}}={\bf{m}}_2^v$
$\ell\backslash r$	$-3$	$-2$	$-1$	$0$	$1$
$-1$	$\mathit{\boldsymbol{\ell}}_0={\bf{0}}$
$0$	$\mathit{\boldsymbol{\ell}}_1={\bf{m}}_1^{\omega}$	$\mathit{\boldsymbol{\ell}}_0={\bf{m}}_1^{\omega}$	${\bf{0}}={\bf{m}}_1^{\omega}$	${\bf{0}}={\bf{m}}_1^{\omega}$	${\bf{0}}={\bf{m}}_1^{\omega}$
$1$	$\mathit{\boldsymbol{\ell}}_2={\bf{m}}_2^{\omega}$	$\mathit{\boldsymbol{\ell}}_1={\bf{m}}_2^{\omega}$	$\mathit{\boldsymbol{\ell}}_0={\bf{m}}_2^{\omega}$	${\bf{0}}={\bf{m}}_2^{\omega}$	${\bf{0}}={\bf{m}}_2^{\omega}$

| Show Table

DownLoad: CSV

Related Papers

Cited by

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. Dupire, M. 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. Patankar, P. Singh, D. Joseph, R. 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. Prosperetti, Q. 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.