zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Modelling of miscible liquids with the Korteweg stress. (English) Zbl 1201.76045
Summary: 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.

76D05Navier-Stokes equations (fluid dynamics)
Full Text: DOI Numdam EuDML
[1] D.M. Anderson , G.B. McFadden and A.A. Wheeler , Diffuse interface methods in fluid mechanics . Annu. Rev. Fluid Mech. 30 ( 1998 ) 139 - 165 .
[2] L.K. Antanovskii , Microscale theory of surface tension . Phys. Rev. E 54 ( 1996 ) 6285 - 6290 .
[3] J.W. Cahn and J.E. Hilliard , Free energy of a nonuniform system . I. Interfacial Free Energy. J. Chem. Phys. 28 ( 1958 ) 258 - 267 .
[4] D. Joseph and M. Renardy , Fundamentals of two-fluid dynamics , Vol. II. Springer, New York ( 1992 ). Zbl 0784.76003 · Zbl 0784.76003
[5] 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. II 6 ( 1901 ) 1 - 24 . JFM 32.0756.02 · Zbl 32.0756.02
[6] O.A. Ladyzhenskaya , Mathematical theory of viscous incompressible flow . Gordon and Breach ( 1963 ). MR 155093 | Zbl 0121.42701 · Zbl 0121.42701
[7] J.L. Lions , Quelques méthodes de résolution des problèmes aux limites non linéaires . Gauthier-Villars, Paris ( 1969 ). MR 259693 | Zbl 0189.40603 · Zbl 0189.40603
[8] 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).
[9] 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).
[10] P. Petitjeans , Une tension de surface pour les fluides miscibles . C. R. Acad. Sci. Paris Sér. I Math. 322 ( 1996 ) 673 - 679 .
[11] R. Temam , Navier-Stokes equations . Theory and numerical analysis. North-Holland Publishing Co., Amsterdam-New York, Stud. Math. Appl. 2 ( 1979 ). Zbl 0426.35003 · Zbl 0426.35003
[12] R. Temam , Navier-Stokes equations and nonlinear functional analysis . SIAM ( 1983 ). Zbl 0833.35110 · Zbl 0833.35110
[13] 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.
[14] 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 . Zbl 1076.76597 · Zbl 1076.76597 · doi:10.1016/S1631-0721(02)01467-5