Thin-film equation

A thin-film equation combines a lubrication flux with mass conservation to evolve a free-surface thickness. Gravity, capillary pressure, wall motion, and intermolecular forces may contribute to the flux.

For a liquid film heated below and cooled above, the quasistatic temperature of a conductively cooled film makes a thicker region colder at the surface. If surface tension decreases with temperature, Marangoni stress draws fluid towards thicker regions, competing with smoothing by capillary pressure. Writing ${\gamma }_s={\gamma }_0+bh$ gives $a_c={\gamma }_0/(3\mu )$, $a_m=b/(2\mu )$. A balanced long-wave scaling gives $H_{\tau }+(H^2{H}_X{)}_X+(H^3{H}_{XXX}{)}_X=0$. The dispersion relation $s=k^2-k^4$ has unstable band $0<|k|<1$ and fastest normal mode at $|k|=1/\sqrt{2}$. This thermal mechanism does not require surfactant.

Write the surface tension as ${\gamma }_s={\gamma }_0+Bh$, where $B>0$. The thermocapillary thin-film equation has flux ${\gamma }_0{h}^3{h}_{xxx}/(3\mu )+B{h}^2{h}_x/(2\mu )$. Balancing both terms at thickness scale $h_{\ast }$ gives aspect ratio ${ϵ}^2=3B{h}_{\ast }/(2{\gamma }_0)$ and time scale $4\mu {\gamma }_0/(3B^2{h}_{\ast })$. The fractional surface-tension variation is $B{h}_{\ast }/{\gamma }_0=2{ϵ}^2/3$, so using constant tension in the leading capillary pressure term is consistent with the balanced long-wave limit.

Positive steady solutions of the balanced thermocapillary thin-film equation with zero volume flux satisfy $H_{XX}+\ln H=0$ after choosing the thickness scale. Their first integral has potential $V(H)=H(\ln H-1)$, with minimum $V(1)=-1$ and limiting value $V(0^{+})=0$. At $E=-1$ the film is uniform; for $-1<E<0$ it is positive and periodic. At $E=0$ a limiting drop has peak $e$, half-width $\sqrt{\pi e}$, zero limiting contact angle and divergent curvature at zero thickness. Small periodic profiles have period $2\pi$ and approach the neutral, rather than fastest-growing, normal mode of the thermocapillary film instability.

For a film with fixed lower temperature $T_0$ and upper condition $-\kappa T_z=\alpha (T_s-T_0+\Delta T)$, negligible horizontal thermal diffusion and thermal advection reduce the heat equation to $T_{zz}=0$. Thus $T=T_0-\alpha \Delta Tz/(\kappa +\alpha h)$. The local approximation requires $(h/L{)}^2\ll 1$ and $U{h}^2/(\kappa L)\ll 1$, together with slow time variation relative to $h^2/\kappa$. For $\alpha h/\kappa \ll 1$, $T_s\simeq T_0-\Delta T\alpha h/\kappa$. The coefficient $\kappa$ is thermal diffusivity, and the boundary coefficient $\alpha$ is normalized accordingly.

For an exterior thin film, circumferential and axial curvature together give $h_t+(h^3[h_{zzz}+h_z+G]{)}_z/3=0$ in appropriate dimensionless variables. The $h_z$ term destabilizes long waves through the Rayleigh–Plateau instability, whereas $h_{zzz}$ damps short waves. The linear growth rate is $(k^2-k^4)/3-iGk$, with fastest growth at $|k|=1/\sqrt{2}$. The gravity parameter is $G=\rho g{a}^2/(\gamma ϵ)$ when film thickness is scaled as $ϵah$ and axial distance by cylinder radius $a$.

For a travelling wave of large positive speed in the cylindrical film equation, the leading core is proportional to $1-\cos x$ over an axial interval of length $2\pi$. Edge coordinates $X=(3c{)}^{1/3}(x-x_e)$ reduce the transition equation to $H^3{H}_{XXX}=H-1$. Its real positive exponent gives a monotone trailing approach and its complex stable exponents give damped capillary waves ahead, with physical wavelength $4\pi a/(3^{5/6}{c}^{1/3})$. Matching the two quadratic edge expansions selects the uniform film thickness; this is a matched asymptotic expansion, valid while the pulse itself remains thin relative to the cylinder.

For a no-slip planar liquid film with an insoluble surfactant, lubrication theory gives $q=-h^3{p}_x/(3\mu )+h^2{\gamma }_x/(2\mu )$ and surface velocity $u_s=-h^2{p}_x/(2\mu )+h{\gamma }_x/\mu$. Mass conservation and conservation of insoluble surfactant on a moving interface give $h_t+q_x=0$ and $C_t+(C{u}_s{)}_x=0$. With $\gamma ={\gamma }_0-AC$ and small slopes, $p_x=\rho g{h}_x-(\gamma h_{xx}{)}_x$. The different coefficients in $q$ and $u_s$ are essential: Marangoni stress and hydrostatic return flow transport fluid and surfactant differently.

In the same gravity–Marangoni reduction, zero liquid flux gives $\Gamma =1+(H_0^2-H^2)/3$ and $6JX=(1+H_0^2/3)(H^3-H_0^3)/3-(H^5-H_0^5)/15$. The lower part of the liquid flows backwards while the surface and upper third flow forwards. For zero initial height and concentration drop $\delta$, $J={\delta }^{3/2}(1-3\delta /5)/(2\sqrt{3})$ is largest formally at complete depletion. Since surface velocity is $J/\Gamma$, that endpoint is singular and requires physical regularization by diffusion, capillarity or an edge model. The algebraic maximum is not by itself a regular zero-concentration outflow.

In the steady nondiffusive gravity–Marangoni model with capillarity neglected, zero surfactant flux and positive concentration make the surface velocity zero. With $Q=-H^2{\Gamma }_X/2-H^3{H}_X/3$ and $J=-H\Gamma {\Gamma }_X-H^2\Gamma H_X/2$, integration gives $H^4=H_0^4-48QX$ and $\Gamma =1+(H_0^2-H^2)/4$. A concentration drop $\delta$ over unit length therefore selects $Q=-\delta (H_0^2+2\delta )/6$. The liquid flows opposite to the imposed Marangoni traction, with velocity proportional to $H_{X}Y(Y-H)/2$ and an immobilized surface. This reduction is singular at a dry endpoint; its small-slope assumption does not resolve that edge.

A localized amount $M=\int Cdx$ of insoluble surfactant spreads over a planar film of depth scale $h_0$. Neglect gravity, capillarity and diffusion. Then $C\sim M/\ell$ and Marangoni stress drives $u_s\sim AM{h}_0/(\mu {\ell }^2)$, giving $\ell \sim (AM{h}_{0}t/\mu {)}^{1/3}$. The thin-film equations with insoluble surfactant admit a similarity solution in which the film height and concentration depend on $x/\ell$, conserving total surfactant and fluid volume relative to the undisturbed layer.

When the smoothing width exceeds the capillary length, gravity dominates capillarity. Balancing hydrostatic flux $\rho g{h}_0^4/(\mu \Delta )$ against the Marangoni stress flux gives $\Delta \sim \rho g{h}_0^2{x}_N^2/(AM)\propto t^{2/3}$. The condition is ${\Delta }^2\gg {\gamma }_0/(\rho g)$ in physical units. The smoothing region catches up with the spreading pool at $x_N\sim AM/(\rho g{h}_0^2)$ and $t\sim \mu (AM{)}^2/[(\rho g{)}^3{h}_0^7]$, after which the sharp-front outer similarity solution ceases to apply.

After gravity smoothing of a Marangoni front reaches the whole spreading pool, the film approaches uniform depth while surface transport continues. A small height gradient creates a hydrostatic pressure return flow that almost cancels the net Marangoni stress flux. Setting $q\simeq 0$ in the thin-film equations with insoluble surfactant gives $h_x\simeq -3A{C}_x/(2\rho g{h}_0)$ and fractional depth variation $O(AM/(\rho g{h}_0^2{x}_N))\ll 1$. The depressed region under the surfactant therefore becomes progressively shallower.

The height jump in linear similarity profiles for surfactant spreading is smoothed by surface tension. If $C_x\sim M/x_N^2$ through a region of width $\Delta$, the Marangoni stress flux is $AM{h}_0^2/(\mu x_N^2)$ and the capillary flux is ${\gamma }_0{h}_0^4/(\mu {\Delta }^3)$. Their balance gives $\Delta \sim ({\gamma }_0{h}_0^2{x}_N^2/(AM){)}^{1/3}\propto t^{2/9}$. This estimate assumes gravity is negligible and the smoothing region remains narrow compared with the pool.

For symmetric finite-mass Marangoni spreading on a liquid film, the leading outer profiles are $h=2h_0|x|/x_N$ and $C=(M/x_N)(1-|x|/x_N)$ inside $|x|<x_N$, where $x_N=(6AM{h}_{0}t/\mu {)}^{1/3}$. Outside, $h=h_0$ and $C=0$. The similarity solution follows from $H{\Gamma }^{\prime }=-\eta /3$, $\eta H^{\prime }=H$, $2\int \Gamma d\eta =1$ and $\int (H-1)d\eta =0$. The film reaches height $2h_0$ immediately behind the front, leaving an outer height jump that gravity or surface tension must smooth. These are ideal leading outer profiles, not a description of the microscopic central dry region or the smoothed edge.

Disjoining pressure is the thickness-dependent normal stress caused by intermolecular interactions between nearby interfaces. An attractive law $\Pi (h)=V/h^3$ grows as a film thins and can drive rupture.

For a viscous sheet of half-thickness $h_0$, attractive disjoining pressure $V/h^3$, and surface tension $\gamma$, a symmetric Fourier mode of dimensionless wavenumber $K=k{h}_0$ has a destabilizing molecular contribution and a stabilizing capillary contribution. The ratiosets the cutoff $K={\Gamma }^{-1/2}$.

A sheet with stress-free surfaces supports nearly uniform cross-thickness extensional motion. Its diverging long-wave mobility compensates the weak pressure gradient, leaving the nonzero rate limit in the creeping-flow model. The exactly uniform thickness change is excluded at fixed volume, and unsteady inertia or finite domain size can regularize the limiting prediction. Strong surfactant immobilization instead makes the rate vanish quadratically with wave number.

The thin-film mass flux is the layer thickness times its depth-averaged velocity. In a quasi-steady transition region, mass conservation makes this flux approximately constant along the layer.