Modelling of Miscible Liquids with the Korteweg Stress
Ilya Kostin; Martine Marion; Rozenn Texier-Picard; Vitaly A. Volpert
ESAIM: Mathematical Modelling and Numerical Analysis (2010)
- Volume: 37, Issue: 5, page 741-753
- ISSN: 0764-583X
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topKostin, Ilya, et al. "Modelling of Miscible Liquids with the Korteweg Stress." ESAIM: Mathematical Modelling and Numerical Analysis 37.5 (2010): 741-753. <http://eudml.org/doc/194189>.
@article{Kostin2010,
abstract = {
When two miscible fluids, such as glycerol (glycerin) and water,
are brought in contact, they immediately diffuse in each other.
However if the diffusion is sufficiently slow, large concentration gradients exist
during some time. They can lead to the appearance of an
“effective interfacial tension”. To study these phenomena we
use the mathematical model
consisting of the diffusion equation with convective terms and of
the Navier-Stokes equations with the Korteweg stress.
We prove the global existence and uniqueness of the solution for the
associated initial-boundary value problem in a two-dimensional bounded domain.
We study the longtime behavior of the solution and show that it converges
to the uniform composition distribution with zero velocity field.
We also present numerical simulations of miscible drops and show how
transient interfacial phenomena can change their shape.
},
author = {Kostin, Ilya, Marion, Martine, Texier-Picard, Rozenn, Volpert, Vitaly A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Miscible liquids; Korteweg stress; drops.},
language = {eng},
month = {3},
number = {5},
pages = {741-753},
publisher = {EDP Sciences},
title = {Modelling of Miscible Liquids with the Korteweg Stress},
url = {http://eudml.org/doc/194189},
volume = {37},
year = {2010},
}
TY - JOUR
AU - Kostin, Ilya
AU - Marion, Martine
AU - Texier-Picard, Rozenn
AU - Volpert, Vitaly A.
TI - Modelling of Miscible Liquids with the Korteweg Stress
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 5
SP - 741
EP - 753
AB -
When two miscible fluids, such as glycerol (glycerin) and water,
are brought in contact, they immediately diffuse in each other.
However if the diffusion is sufficiently slow, large concentration gradients exist
during some time. They can lead to the appearance of an
“effective interfacial tension”. To study these phenomena we
use the mathematical model
consisting of the diffusion equation with convective terms and of
the Navier-Stokes equations with the Korteweg stress.
We prove the global existence and uniqueness of the solution for the
associated initial-boundary value problem in a two-dimensional bounded domain.
We study the longtime behavior of the solution and show that it converges
to the uniform composition distribution with zero velocity field.
We also present numerical simulations of miscible drops and show how
transient interfacial phenomena can change their shape.
LA - eng
KW - Miscible liquids; Korteweg stress; drops.
UR - http://eudml.org/doc/194189
ER -
References
top- D.M. Anderson, G.B. McFadden and A.A. Wheeler, Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech.30 (1998) 139-165.
- L.K. Antanovskii, Microscale theory of surface tension. Phys. Rev. E54 (1996) 6285-6290.
- J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial Free Energy. J. Chem. Phys.28 (1958) 258-267.
- D. Joseph and M. Renardy, Fundamentals of two-fluid dynamics, Vol. II. Springer, New York (1992).
- D.J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais connues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité. Arch. Néerl. Sci. Exactes Nat. Ser. II6 (1901) 1-24.
- O.A. Ladyzhenskaya, Mathematical theory of viscous incompressible flow. Gordon and Breach (1963).
- J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars, Paris (1969).
- J. Pojman, N. Bessonov, R. Texier, V. Volpert and H. Wilke, Numerical simulations of transient interfacial phenomena in miscible fluids, in Proceedings AIAA, Reno, USA (January 2002).
- J. Pojman, Y. Chekanov, J. Masere, V. Volpert, T. Dumont and H. Wilke, Effective interfacial tension induced convection in miscible fluids, in Proceedings of the 39th AIAA Aerospace Science Meeting, Reno, USA (January 2001).
- P. Petitjeans, Une tension de surface pour les fluides miscibles. C. R. Acad. Sci. Paris Sér. I Math.322 (1996) 673-679.
- R. Temam, Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam-New York, Stud. Math. Appl. 2 (1979).
- R. Temam, Navier-Stokes equations and nonlinear functional analysis. SIAM (1983).
- J.S. Rowlinson, Translation of J.D. van der Waals' ``The thermodynamic theory of capillarity under hypothesis of a continuous variation of density''. J. Statist. Phys.20 (1979) 197.
- V. Volpert, J. Pojman and R. Texier-Picard, Convection induced by composition gradients in miscible liquids. C. R. Acad. Sci. Paris Sér. I Math.330 (2002) 353-358.
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