Thin-film equation

In physics and engineering, the thin-film equation is a partial differential equation that approximately predicts the time evolution of the thickness h of a liquid film that lies on a surface. The equation is derived via lubrication theory which is based on the assumption that the length-scales in the surface directions are significantly larger than in the direction normal to the surface. In the non-dimensional form of the Navier-Stokes equation the requirement is that terms of order ${\displaystyle \epsilon ^{2}}$ and ${\displaystyle \epsilon ^{2}Re}$ are negligible, where ${\displaystyle \epsilon \ll 1}$ is the aspect ratio and ${\displaystyle Re}$ is the Reynold's number. This significantly simplifies the governing equations. However lubrication theory, as the name suggests, is typically derived for flow between two solid surfaces hence the liquid forms a lubricating layer. The thin-film equation holds when there is a single free surface. With two free surfaces the flow must be treated as a viscous sheet.[1][2]

Definition

The basic form of a 2-dimensional thin film equation is[3][4][5]

${\displaystyle {\frac {\partial h}{\partial t}}=-\nabla \cdot \mathbf {Q} }$

where the fluid flux ${\displaystyle \mathbf {Q} }$ is

${\displaystyle \mathbf {Q} ={\frac {h^{3}}{3\mu }}\left[\nabla \right(\gamma \nabla ^{2}h+\rho \mathbf {g} \cdot \mathbf {{\hat {e}}_{n}} )+\rho \mathbf {g} \cdot \mathbf {{\hat {e}}_{i}} ]+{\frac {h^{2}}{2\mu }}\mathbf {A} }$ ,

and μ is the viscosity (or dynamic viscosity) of the liquid, h(x,y,t) is film thickness, γ is the interfacial tension between the liquid and the gas phase above it, ${\displaystyle \rho }$ is the liquid density and ${\displaystyle \mathbf {A} }$ the surface shear. The surface shear could be caused by flow of the overlying gas or surface tension gradients.[6][7] The vectors ${\displaystyle \mathbf {{\hat {e}}_{i}} }$ represent the unit vector in the surface co-ordinate directions, the dot product serving to identify the gravity component in each direction. The vector ${\displaystyle \mathbf {{\hat {e}}_{n}} }$ is the unit vector perpendicular to the surface.

A generalised thin film equation is discussed in [5]

${\displaystyle {\frac {\partial h}{\partial t}}=-{\frac {1}{3\mu }}\nabla \cdot \left(h^{n}\,\nabla \left(\gamma \,\nabla ^{2}h\right)\right)}$ .

When ${\displaystyle n<3}$ this may represent flow with slip at the solid surface whole ${\displaystyle n=1}$ describes the thickness of a thin bridge between two masses of fluid in a Hele-Shaw cell.[8] The value ${\displaystyle n=3}$ represents surface tension driven flow.

A form frequently investigated with regard to the rupture of thin liquid films involves the addition of a disjoining pressure Π(h) in the equation,[9] as in

${\displaystyle {\frac {\partial h}{\partial t}}=-{\frac {1}{3\mu }}\nabla \cdot \left(h^{3}\nabla \left(\gamma \,\nabla ^{2}h-\Pi (h)\right)\right)}$

where the function Π(h) is usually very small in value for moderate-large film thicknesses h and grows very rapidly when h goes very close to zero.

Properties

Physical applications, properties and solution behaviour of the thin-film equation are reviewed in.[3][5] With the inclusion of phase change at the substrate a form of thin film equation for an arbitrary surface is derived in.[10] A detailed study of the steady-flow of a thin film near a moving contact line is given in.[11] For a yield-stress fluid flow driven by gravity and surface tension is investigated in.[12]

For purely surface tension driven flow it is easy to see that one static (time-independent) solution is a paraboloid of revolution

${\displaystyle h(x,y)=A-B(x^{2}+y^{2})\,}$

and this is consistent with the experimentally observed spherical cap shape of a static sessile drop, as a "flat" spherical cap that has small height can be accurately approximated in second order with a paraboloid. This, however, does not handle correctly the circumference of the droplet where the value of the function h(x,y) drops to zero and below, as a real physical liquid film can't have a negative thickness. This is one reason why the disjoining pressure term Π(h) is important in the theory.

One possible realistic form of the disjoining pressure term is[9]

${\displaystyle \Pi (h)=B\left[\left({\frac {h_{*}}{h}}\right)^{n}-\left({\frac {h_{*}}{h}}\right)^{m}\right]}$

where B, h_*, m and n are some parameters. These constants and the surface tension ${\displaystyle \gamma }$ can be approximately related to the equilibrium liquid-solid contact angle ${\displaystyle \theta _{e}}$ through the equation[9][13]

${\displaystyle B\approx {\frac {(m-1)(n-1)}{h_{*}(n-m)}}\gamma (1-\cos \theta _{e})}$.

The thin film equation can be used to simulate several behaviors of liquids, such as the fingering instability in gravity driven flow.[14]

The lack of a second-order time derivative in the thin-film equation is a result of the assumption of small Reynold's number in its derivation, which allows the ignoring of inertial terms dependent on fluid density ${\displaystyle \rho }$.[14] This is somewhat similar to the situation with Washburn's equation, which describes the capillarity-driven flow of a liquid in a thin tube.

See also

• Partial differential equation
• Lubrication theory
• Disjoining pressure

References

1. Fliert, B. W. Van De; Howell, P. D.; Ockenden, J. R. (June 1995). "Pressure-driven flow of a thin viscous sheet". Journal of Fluid Mechanics. 292: 359–376. doi:10.1017/S002211209500156X. ISSN 1469-7645.
2. Buckmaster, J. D.; Nachman, A.; Ting, L. (May 1975). "The buckling and stretching of a viscida". Journal of Fluid Mechanics. 69 (1): 1–20. doi:10.1017/S0022112075001279. ISSN 1469-7645.
3. A. Oron, S. H. Davis, S. G. Bankoff, "Long-scale evolution of thin liquid films", Rev. Mod. Phys., 69, 931–980 (1997)
4. H. Knüpfer, "Classical solutions for a thin-film equation", PhD thesis, University of Bonn.
5. Myers, T. G. (January 1998). "Thin Films with High Surface Tension". SIAM Review. 40 (3): 441–462. doi:10.1137/S003614459529284X. ISSN 0036-1445.
6. O'Brien, S. B. G. M. (September 1993). "On Marangoni drying: nonlinear kinematic waves in a thin film". Journal of Fluid Mechanics. 254: 649–670. doi:10.1017/S0022112093002290. ISSN 0022-1120.
7. Myers, T. G.; Charpin, J. P. F.; Thompson, C. P. (January 2002). "Slowly accreting ice due to supercooled water impacting on a cold surface". Physics of Fluids. 14 (1): 240–256. doi:10.1063/1.1416186. ISSN 1070-6631.
8. Constantin, Peter; Dupont, Todd F.; Goldstein, Raymond E.; Kadanoff, Leo P.; Shelley, Michael J.; Zhou, Su-Min (1993-06-01). "Droplet breakup in a model of the Hele-Shaw cell". Physical Review E. 47 (6): 4169–4181. doi:10.1103/PhysRevE.47.4169. ISSN 1063-651X.
9. L. W. Schwartz, R. V. Roy, R. R. Eley, S. Petrash, "Dewetting patterns in a drying liquid film", Journal of Colloid and Interface Science, 243, 363374 (2001).
10. Myers, T. G.; Charpin, J. P. F.; Chapman, S. J. (August 2002). "The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface". Physics of Fluids. 14 (8): 2788–2803. doi:10.1063/1.1488599. hdl:2117/102903. ISSN 1070-6631.
11. Tuck, E. O.; Schwartz, L. W. (September 1990). "A Numerical and Asymptotic Study of Some Third-Order Ordinary Differential Equations Relevant to Draining and Coating Flows". SIAM Review. 32 (3): 453–469. doi:10.1137/1032079. ISSN 0036-1445.
12. Balmforth, Neil; Ghadge, Shilpa; Myers, Tim (March 2007). "Surface tension driven fingering of a viscoplastic film". Journal of Non-Newtonian Fluid Mechanics. 142 (1–3): 143–149. doi:10.1016/j.jnnfm.2006.07.011.
13. N.V. Churaev, V.D. Sobolev, Adv. Colloid Interface Sci. 61 (1995) 1-16
14. L. Kondic, "Instabilities in gravity driven flow of thin liquid films", SIAM Review, 45, 95–115 (2003)

External links

• Viscous Thin Films - Max Planck Institute