Thin-film equation

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.

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.

• Partial differential equation
• Lubrication theory
• Disjoining pressure

• Viscous Thin Films - Max Planck Institute