Summary
The influence of surface viscosity on the rate of thinning and on the critical thickness of rupture of foam films has been studied. It is shown that this effect depends strongly on the length-scale of the flow. Therefore it is possible to neglect it when calculating the rate of thinning, but to assume for the wave motion, caused by the thermic fluctuations, that the surface viscosity is infinitely great. The expression for the critical thickness, obtained in this way, agrees relatively well with the experimental results. A possible explanation of the dependence of critical thickness on surfactant concentration is suggested.
Zusammenfassung
Der Einfluß der Oberflächenviskosität auf die Verdünnungsgeschwindigkeit und die kritische Zerreißdicke von Schaumfilmen wird untersucht. Es zeigt sich, daß dieser Effekt stark von der charakteristischen Flußgröße abhängt. Folglich kann man bei der Berechnung der Verdünnungsgeschwindigkeit die Oberflächenviskosität unberücksichtigt lassen und sie bei der von den thermischen Fluktuationen verursachten wellenartigen Bewegung als unendlich groß annehmen. Die auf diese Art erhaltenen Gleichungen für die kritische Dicke stimmen verhältnismäßig gut mit den experimentellen Daten überein. Eine mögliche Erklärung der Abhängigkeit der kritischen Dicke von der Konzentration des Tensids wird vorgeschlagen.
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Abbreviations
- A n :
-
maximum amplitude of thenth wave
- a :
-
Szyskowiski's constant
- a n :
-
Fourrier-Bessel's coefficients
- c :
-
local concentration
- c 0 :
-
equilibrium concentration
- D :
-
bulk diffusion coefficient
- D s :
-
surface diffusion coefficient
- h :
-
film thickness
- h cr :
-
critical thickness of rupture
- h 0 :
-
transition thickness
- J 0,J 1,J 2 :
-
Bessel's functions of first kind; zero, first and second order
- k m :
-
wave number of the critical wave
- k n =λ n /R :
-
wave number
- l m :
-
critical wave length
- l n :
-
wave length
- P :
-
pressure
- P σ :
-
capillary pressure of the meniscus
- ΔP :
-
P σ-Π
- P :
-
equilibrium film pressure
- q n :
-
σk 2 n - 2dΠ/dh
- R :
-
film radius
- r :
-
coordinate
- T :
-
temperature
- t :
-
time
- V :
-
-dh/dt-velocity of film thinning
- V 0 :
-
Reynolds velocity of thinning
- υr, υz :
-
velocity components
- υs :
-
tangential surface velocity
- υζ :
-
∂ζ/∂t - velocity of local thinning
- z :
-
coordinate
- α:
-
\( = - \frac{{3D\mu }}{{\Gamma _0 (\partial \sigma _0 /\partial c_0 )}}\left( {1 + \frac{{2D_s (\partial \Gamma _0 /\partial c_0 )}}{{Dh}}} \right)\)
- Γ :
-
local surface concentration
- Γ 0 :
-
equilibrium surface concentration
- ξ n :
-
vertical displacement of the surface
- γ n :
-
nth root of the equationJ 0 (γ n ) = 0
- μ :
-
bulk viscosity
- μ s :
-
surface shear viscosity
- h :
-
disjoining pressure
- σ :
-
local surface tension
- σ0 :
-
equilibrium surface tension
- ω :
-
angular velocity
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Ivanov, I.B., Dimitrov, D.S. Hydrodynamics of thin liquid films. Colloid & Polymer Sci 252, 982–990 (1974). https://doi.org/10.1007/BF01566619
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DOI: https://doi.org/10.1007/BF01566619