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Historical Note on the Stefan–Reynolds Equations
Journal of Colloid and Interface Science 231, 195 (2000) doi:10.1006/jcis.2000.7066, available online at http://www.idealibrary.com on
NOTE Historical Note on the Stefan–Reynolds Equations We comment on the theory on the dynamics of fluid films confined between parallel surfaces established by Stefan and Reynolds over a centry ago. From a historical perspective, the established theory (often referred to as the lubrication approximation) and the derived equations, as used in colloid science, are to be correctly attributed to both Stefan and Reynolds. °C 2000 Academic Press Key Words: Stefan–Reynolds equations; lubrication approximation; dynamics of fluid films.
The rate of mutual approach of surfaces with a thinning liquid film is central to a number of phenomena in colloid science and processes used in many industrial technologies and has attracted much experimental and theoretical research. On a miccroscopic scale, the rate of the mutual approach is strongly influenced by hydrodynamic forces at short separations, leading to an increase of the fluid resistance to the surfaces. In the case of two parallel flat surfaces with a radius R, which approach each other with a relative velocity V at a separation h in a quiesient fluid with a viscosity µ, the hydrodynamic force described by F yields F=
3πµR 4 V . 2h 3
3πµR 4 4F
"µ
1 h2
¶2
µ −
REFERENCES
[1]
This equation has been applied, as the first approximation, to a range of the surface interactions. These include the film thinning between deformable surfaces such as that in foams and emulsions or between a fluid surface and a flat solid surface. If the applied force is independent of the separation, the time t required for the surfaces to reduce the separation distance h 1 to h 2 yields, from the above equation,
t=
in a viscous liquid. Presently, the simplification of the Navier–Stokes equation based on the special geometry of the thin liquid films (often referred to as the lubrication approximation) and the above equations are usually attributed to Reynolds, especially in the colloid literature. From the historical perspective shown above, the lubrication approximation, as used in colloid science, has to be attributed to both Stefan and Reynolds, and the aforementioned equations have to be correctly referred to as the Stefan– Reynolds equations. Even in his publication in 1886, Reynolds also referred to Stefan’s work published in 1874. It should also be noted that the force expression given by Eq. [1] was not implicitly described by Stefan, but it directly follows from his theory. Furthermore, it must be emphasized that the application of Eq. [2] to the approach of interfaces in close proximity is subject to several limitations. The most important limitation is that the electrical double-layer force, the van der Waals forces and other surface forces, as we know today, are not included in Eq. [2]. Such a limitation is however not applied to Eq. [1]. Both equations are also limited to the parallel surfaces that are tangetially immobile. Nevertheless, the framework of the lubrication approximation to the dynamics of fluid films confined between surfaces established by Stefan and Reynolds over a century ago underpins the subject. After all, as August Comte observed, “To understand a science is necessary to know its history.”
1 h1
¶2 # .
[2]
1. Stefan, M. J., Versuch u¨ ber die scheinbere Adh¨asion. “Sitzungsberichte der Mathematisch-naturwissenschaften Klasse der Kaiserlichen Akademie der Wissenschaften, II. Abteilung,” Vol. 69, pp. 713–735. Wienna, 1874. 2. Reynolds, O., Philos. Trans. R. Soc. London 177, 157–234 (1886). A. V. Nguyen1 Department of Chemical Engineering The University of New Castle University Drive Callaghan NSW 2308, Australia Received April 11, 2000; accepted June 19, 2000
This equation was first derived by Stefan (1) in 1874 to explain the force required to separate two circular discs in close proximity and immersed in a viscous fluid. In 1886 in a classical treatment of the theory of hydrodynamic lubrication, Reynolds (2) also deduced the above equation for two parrallel discs immersed
195
1
Fax: +61 4921 6920. E-mail: [email protected].
0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
NOTE Historical Note on the Stefan–Reynolds Equations We comment on the theory on the dynamics of fluid films confined between parallel surfaces established by Stefan and Reynolds over a centry ago. From a historical perspective, the established theory (often referred to as the lubrication approximation) and the derived equations, as used in colloid science, are to be correctly attributed to both Stefan and Reynolds. °C 2000 Academic Press Key Words: Stefan–Reynolds equations; lubrication approximation; dynamics of fluid films.
The rate of mutual approach of surfaces with a thinning liquid film is central to a number of phenomena in colloid science and processes used in many industrial technologies and has attracted much experimental and theoretical research. On a miccroscopic scale, the rate of the mutual approach is strongly influenced by hydrodynamic forces at short separations, leading to an increase of the fluid resistance to the surfaces. In the case of two parallel flat surfaces with a radius R, which approach each other with a relative velocity V at a separation h in a quiesient fluid with a viscosity µ, the hydrodynamic force described by F yields F=
3πµR 4 V . 2h 3
3πµR 4 4F
"µ
1 h2
¶2
µ −
REFERENCES
[1]
This equation has been applied, as the first approximation, to a range of the surface interactions. These include the film thinning between deformable surfaces such as that in foams and emulsions or between a fluid surface and a flat solid surface. If the applied force is independent of the separation, the time t required for the surfaces to reduce the separation distance h 1 to h 2 yields, from the above equation,
t=
in a viscous liquid. Presently, the simplification of the Navier–Stokes equation based on the special geometry of the thin liquid films (often referred to as the lubrication approximation) and the above equations are usually attributed to Reynolds, especially in the colloid literature. From the historical perspective shown above, the lubrication approximation, as used in colloid science, has to be attributed to both Stefan and Reynolds, and the aforementioned equations have to be correctly referred to as the Stefan– Reynolds equations. Even in his publication in 1886, Reynolds also referred to Stefan’s work published in 1874. It should also be noted that the force expression given by Eq. [1] was not implicitly described by Stefan, but it directly follows from his theory. Furthermore, it must be emphasized that the application of Eq. [2] to the approach of interfaces in close proximity is subject to several limitations. The most important limitation is that the electrical double-layer force, the van der Waals forces and other surface forces, as we know today, are not included in Eq. [2]. Such a limitation is however not applied to Eq. [1]. Both equations are also limited to the parallel surfaces that are tangetially immobile. Nevertheless, the framework of the lubrication approximation to the dynamics of fluid films confined between surfaces established by Stefan and Reynolds over a century ago underpins the subject. After all, as August Comte observed, “To understand a science is necessary to know its history.”
1 h1
¶2 # .
[2]
1. Stefan, M. J., Versuch u¨ ber die scheinbere Adh¨asion. “Sitzungsberichte der Mathematisch-naturwissenschaften Klasse der Kaiserlichen Akademie der Wissenschaften, II. Abteilung,” Vol. 69, pp. 713–735. Wienna, 1874. 2. Reynolds, O., Philos. Trans. R. Soc. London 177, 157–234 (1886). A. V. Nguyen1 Department of Chemical Engineering The University of New Castle University Drive Callaghan NSW 2308, Australia Received April 11, 2000; accepted June 19, 2000
This equation was first derived by Stefan (1) in 1874 to explain the force required to separate two circular discs in close proximity and immersed in a viscous fluid. In 1886 in a classical treatment of the theory of hydrodynamic lubrication, Reynolds (2) also deduced the above equation for two parrallel discs immersed
195
1
Fax: +61 4921 6920. E-mail: [email protected].
0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.