# Thin-film equation

In physics and engineering, the thin-film equation is a partial differential equation that approximately predicts the time evolution of the x- and y-dependent local thickness h(x,y,t) of a liquid film (puddle, flat drop of liquid) that lies on a flat surface which coincides with the xy-plane. In lubrication theory in fluid mechanics, a body of liquid is assumed to be a thin film, i.e. considerably thinner in one Cartesian direction than in the two directions orthogonal to that. That is one of many problems that can be simplified if one can make some assumptions about the length scales that are relevant for the physical system.

## Definition

The simplest form of a 2-dimensional thin film equation reads[1][2]

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

where μ is the viscosity of the liquid, h(x,y,t) is the x-, y- and t-dependent film thickness and γ is the interfacial tension between the liquid and the gas phase above it. The parameter γ can also be a function of x and y, which is why it has been left inside the differentiation and not included in the numerator of the factor −1/(3μ).

The equation can be modified in many ways to fit different situations, and the most important of these is the addition of a disjoining pressure Π(h) in the equation,[3] 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

From the simpler form of the equation, which doesn't contain the term Π(h), 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[3]

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

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[3][4]

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

The thin film equation can be used to simulate several behaviors of liquids, such as the fingering instability in gravity driven flow.[5]

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 ${\displaystyle \rho }$.[5] This is somewhat similar to the situation with Washburn's equation, which describes the capillarity-driven flow of a liquid in a thin tube.