# Modelling of Miscible Liquids with the Korteweg Stress

• Volume: 37, Issue: 5, page 741-753
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Kostin, 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

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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).
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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.

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