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. 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.
Definition The basic form of a 2-dimensional thin film equation is
∂ h ∂ t = − ∇ ⋅ Q {\displaystyle {\frac {\partial h}{\partial t}}=-\nabla \cdot \mathbf {Q} }
where the fluid flux Q {\displaystyle \mathbf {Q} } is
Q = h 3 3 μ [ ∇ ( γ ∇ 2 h + ρ g ⋅ e ^ n ) + ρ g ⋅ e ^ i ] + h 2 2 μ A {\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, ρ {\displaystyle \rho } is the liquid density and A {\displaystyle \mathbf {A} } the surface shear. The surface shear could be caused by flow of the overlying gas or surface tension gradients. The vectors e ^ i {\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 e ^ n {\displaystyle \mathbf {{\hat {e}}_{n}} } is the unit vector perpendicular to the surface. A generalised thin film equation is discussed in SIAM (Society for Industrial and Applied Mathematics)
∂ h ∂ t = − 1 3 μ ∇ ⋅ ( h n ∇ ( γ ∇ 2 h ) ) {\displaystyle {\frac {\partial h}{\partial t}}=-{\frac {1}{3\mu }}\nabla \cdot \left(h^{n}\,\nabla \left(\gamma \,\nabla ^{2}h\right)\right)} . When n < 3 {\displaystyle n<3} this may represent flow with slip at the solid surface while n = 1 {\displaystyle n=1} describes the thickness of a thin bridge between two masses of fluid in a Hele-Shaw cell. The value n = 3 {\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, as in
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