References on thin film equation: derivation and properties

Question

Where can I find

• a derivation of the thin film equation $$u_t = - \mathrm{div} (u^m\nabla\Delta u)$$ from a physical model?
• a good introduction to its properties (e.g. conserved quantities and wellposedness of the initial value problem)?

Answer 1

$\bullet$ Physical model: There is no physical model that gives this equation for arbitrary $m$; the values $m=1,2,3$ appear in viscous flow, as summarized in "Viscous Thin Films": For the no-slip boundary condition one has $m=3$, different slip-boundary conditions give $m=2$ (Navier slip) or $m=1$ (Hele-Shaw cell).

Here is a way to understand from dimensional arguments the physics condition $m\leq 3$. Since $u$ has the dimension of length, the gradient $\nabla$ has dimension of 1/length, and the Laplacian $\Delta$ has dimension of 1/length$^2$, the thin-film equation in physical units has the form $$\frac{1}{v_0}\frac{\partial}{\partial t}u=-\lambda^{3-m}\nabla\cdot(u^m\nabla\Delta u).$$ The parameter $v_0$ is a characteristic velocity of the interface, while the parameter $\lambda$ is the slip length at the interface. The no-slip boundary condition has $\lambda=0$, which enforces $m=3$. When there is slip we may have $m<3$, but not $m>3$, because then the right-hand-side would diverge in the limit $\lambda\rightarrow 0$, which is unphysical. (It would imply a divergent interface velocity.)

$\bullet$ Well-posedness: At the end points $m=0$ and $m=3$ the solution is known to be singular (infinite dissipation for $m=3$, infinite interface velocity for $m=0$). Regularity of the solution in the interval $m\in(0,14/5)$ is proven in "Well-Posedness for a Class of Thin-Film Equations with General Mobility in the Regime of Partial Wetting."

$\bullet$ Conserved quantities: Conservation of mass in a one-dimensional geometry dictates that $\int u(x,t)dx$ should be $t$-independent. This implies that the self-similar solution $$u(x,t)=t^{-\alpha}f(xt^{-\beta}),\;\;m\alpha+4\beta=1,$$ must have $\alpha=\beta=1/(m+4)$. The corresponding self-similar solution is constructed in "Some aspects of the thin film equation."

Answer 2

Earlier references on derivation of this equation can be found in https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/on-the-motion-of-a-small-viscous-droplet-that-wets-a-surface/C23CDBDB8AD0C32EEE3BD87FEE771703 (On the motion of a small viscous droplet that wets a surface, by H.P. Greenspan) and https://academic.oup.com/imamat/article-abstract/40/2/73/740053 (High-Order Nonlinear Diffusion , by N.F. Smyth and J.M. Hill). See also a review article http://iopscience.iop.org/article/10.1088/0953-8984/17/9/002 (The thin-film equation: recent advances and some new perspectives, by J. Becker and G. Grün).