Asymptotic of the velocity of a dilute suspension of droplets with interfacial tension
Authors:
Eric Bonnetier, David Manceau and Faouzi Triki
Journal:
Quart. Appl. Math. 71 (2013), 89-117
MSC (2010):
Primary 35R30
DOI:
https://doi.org/10.1090/S0033-569X-2012-01275-7
Published electronically:
October 12, 2012
MathSciNet review:
3075537
Full-text PDF Free Access
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Abstract: In this paper we derive the asymptotic expansion of the velocity field of a small deformable droplet immersed in an incompressible Newtonian fluid. Using an appropriate physical scaling of the surface tension with respect to the droplet volume, we show that the first order of the asymptotic can be expressed in terms of the velocity field in the absence of the droplet and a new kind of moment tensor, called the curvature moment tensor. Our asymptotic formula extends those already derived for rigid droplets and aimed to obtain simplified macroscale properties of a dilute suspension composed of identical droplets dispersed in an incompressible Newtonian fluid from knowledge of its microscopic properties. We finally determine explicitly the curvature moment tensor for ellipses and ellipsoids.
References
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References
- H. Ammari, An introduction to mathematics of emerging biomedical imaging, Math. & Applications, Vol. 62, Springer-Verlag, Berlin, 2008. MR 2440857 (2010j:44002)
- H. Ammari, Y. Capdeboscq, H. Kang, H. Lee, G. Milton, and H. Zribi, Progress on the strong Eshelby’s Conjecture and extremal structures for the elastic moment tensor, Jour. Math. Pures Appl., 94 (2010), 93-106. MR 2653981
- H. Ammari, P. Garapon, H. Kang and H. Lee, A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements, Quart. Appl. Math., 66 (2008) 139-175. MR 2396655 (2009c:35468)
- H. Ammari, P. Garapon, H. Kang and H. Lee, Effective viscosity properties of dilute suspensions of arbitrarily shaped particles, to appear in Asymptotic Analysis.
- H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Volume 1846, Springer-Verlag, Berlin (2004). MR 2168949 (2006k:35295)
- H. Ammari and H. Kang, Polarization and Moment Tensors: with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences Series, Vol. 162, Springer-Verlag, New York, 2007. MR 2327884 (2009f:35339)
- H. Ammari, H. Kang and H. Lee, A boundary integral method for computing elastic moment tensors for ellipses and ellipsoids, J. Comp. Math. 25 (2007) 2-12. MR 2292423 (2007k:74011)
- H. Ammari, H. Kang, and M. Lim, Effective parameters of elastic composites, Indiana University Mathematics Journal, 55 (2006), 903-922. MR 2244590 (2008g:74086)
- H. Ammari, H. Kang, and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptotic Analysis, 41 (2005), 119-140. MR 2129229 (2006b:35333)
- E. Bonnetier and F. Triki, Asymptotics in the presence of inclusions of small volume for a conduction equation: A case with a non-smooth reference potential, AMS, Contemporary Math. Series, volume 494, pages 95-112, 2009. MR 2581768
- G.K. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech. 41 (1970) 545–570.
- Y. Capdeboscq and M. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Math. Modelling Num. Anal., 37 (2003) 159–173. MR 1972656 (2004b:35334)
- L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Padova, 31 (1961) 308-340. MR 0138894 (25:2334)
- D.J. Cedio-Fengya, S. Moskow and M. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998) 553-595. MR 1629995 (99d:78011)
- M. Do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976. MR 0394451 (52:15253)
- A. Einstein, Eine neue Bestimmung der Molekuldimensionen, Annalen der Physik, 19 (1906) 289–306.
- N.E. Jackson, and C. L. Tucker, A model for large deformation of an ellipsoid droplet with interfacial tension, J. Rheol. 47 (2003) 659-682.
- H. Kang and G. Milton, Solutions to the Pólya-Szegö conjecture and the weak Eshelby conjecture, Arch. Rational Mech. Anal. 188 (2008), 93-116. MR 2379654 (2009j:74046)
- O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English Edition, Gordon and Breach, New York, 1969. MR 0254401 (40:7610)
- R. Lipton, On the effective elasticity of a two-dimensional homogenized incompressible elastic composite, Proc. Roy. Soc. Edinburgh Sect. A, 110 (1988) 45-61. MR 963839 (91g:73002)
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- P.L. Maffettone and M. Minale, Equation of change for ellipsoidal drops in viscous flow, J. Non-Newtonian Fluid Mech., 78 (1998) 227-241.
- G. Milton, The theory of composites, Cambridge Monographs on Applied and Computational Mathematics, Vol. 6, Cambridge University Press, Cambridge (2002). MR 1899805 (2003d:74077)
- E. Sánchez-Palencia, Einstein-like approximation for homogenization with small concentration. I. Elliptic problems, Nonlinear Anal., 9 (1985) 1243-1254. MR 813656 (87a:35153)
- G.I. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. R. Soc. Lond. Ser. A 138 (1932) 41–48.
- R. Temam, Navier-Stokes equations, theory and numerical analysis, North-Holland, Amsterdam, 1979. MR 603444 (82b:35133)
- R. C. Tolman, The effect of droplet size on surface tension. Journal of Chemical Physics, 17 (3) (1949) 333-337.
- W. Yu and M. Bousmina, Ellipsoidal model for droplet deformation in emulsions, Journal of Rheology, 47 (2003) 1011-1039.
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Additional Information
Eric Bonnetier
Affiliation:
Laboratoire Jean Kuntzmann, Université de Joseph Fourier & CNRS, 38041 Grenoble Cedex 9, France
Email:
eric.bonnetier@imag.fr
David Manceau
Affiliation:
Laboratoire de Mathématiques appliquées du Havre, Université du Havre, 25 rue Philippe Lebon, BP 540, 76058 Le Havre cedex, France
MR Author ID:
829352
Email:
david.manceau@univ-lehavre.fr
Faouzi Triki
Affiliation:
Laboratoire Jean Kuntzmann, Université de Joseph Fourier & CNRS, 38041 Grenoble Cedex 9, France
MR Author ID:
733264
Email:
faouzi.triki@imag.fr
Keywords:
Stokes system,
asymptotic expansion,
surface tension,
viscous moment tensor,
droplets
Received by editor(s):
March 30, 2011
Published electronically:
October 12, 2012
Article copyright:
© Copyright 2012
Brown University