In fluid mechanics, 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 ε2 and ε2Re are negligible, where ε ≪ 1 is the aspect ratio and Re is the Reynolds number.
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.
A generalised thin film equation is discussed in SIAM (Society for Industrial and Applied Mathematics)[5] When
describes the thickness of a thin bridge between two masses of fluid in a Hele-Shaw cell.
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 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.
[5] With the inclusion of phase change at the substrate a form of thin film equation for an arbitrary surface is derived in Physics of Fluids.
[10] A detailed study of the steady-flow of a thin film near a moving contact line is given in another SIAM paper.
[12] For purely surface tension driven flow it is easy to see that one static (time-independent) solution is a paraboloid of revolution 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] where B, h*, m and n are some parameters.
can be approximately related to the equilibrium liquid-solid contact angle
[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
[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.