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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.

Without surfactant, capillary pressure drives a surface flow towards a growing bulge. Advection compresses the insoluble surfactant in converging parts of that motion, increasing concentration and lowering surface tension. The resulting gradient supplies a tangential Marangoni stress that opposes the surface motion. This additional resistance slows capillary-driven redistribution of liquid even after the base-tension decrease has already been accounted for.

In the strong-elasticity limit, concentration gradients suppress tangential interface motion. The surface behaves as effectively immobile at leading order. Setting $u_s\approx 0$ in the velocity formula gives ${\sigma }_x\approx h{p}_x/2$, henceone-quarter of the clean shear-free mobility $h^3/(3\mu )$. This mobility reduction explains the quarter-rate limit; it slows the unstable capillary mode without removing the long-wave unstable band.

Let the positive undisturbed surface tension be $\bar{\sigma }={\sigma }_0-A{\Gamma }_0>0$. An insoluble surfactant perturbation $g$ produces ${\sigma }_1=-Ag$. The tangential Marangoni stress changes the surface condition to $\mu u_z(h)={\sigma }_x$. Solving the lubrication theory momentum equation now givesAt leading order, conservation of insoluble surfactant on a moving interface reduces to ${\Gamma }_t+(\Gamma u_s{)}_x=0$. Corrections from the changing circumference are smaller by $h_0/a$. Likewise the normal-pressure contribution $-{\sigma }_1/a$ gives a coupling smaller by $h_0/a$ than the retained tangential Marangoni stress, so it is omitted at the same order.

For thickness and concentration amplitudes $H,g$ proportional to $e^{\beta t+ikx}$, put $D=a^{-2}-k^2$ and use $p_1=-\bar{\sigma }DH$. The two conservation equations becomeSet $s=k^2{a}^2$, $\lambda =h_0^3\bar{\sigma }/(3\mu a^4)$ and $\alpha =h_{0}A{\Gamma }_0/(\mu a^2)$. The uncoupled capillary growth and elastic relaxation rates are $L=\lambda s(1-s)$ and $S=\alpha s$. The off-diagonal product is $3LS/4$, so the determinant is $(\beta -L)(\beta +S)+3LS/4$. Thus the surfactant stabilization of an annular liquid lining relation isThis proves the dispersion relation before its lettered deductions. The physical assumption $\bar{\sigma }>0$ is needed for the claimed capillary instability band.

An insoluble surfactant is advected with the surface fluid and has no exchange with the bulk. With negligible surface diffusion, its amount on a material surface strip is conserved. Under the small-slope lubrication approximation, surface arclength equals projected horizontal length to leading order, givingThis is the leading planar form of conservation of insoluble surfactant on a moving interface.

Let $y$ measure height above the substrate. The normal-stress and vertical momentum conditions give $p=p_{\mathrm{atm}}-\gamma h_{xx}+\rho g(h-y)$, so $p_x=\rho g{h}_x-(\gamma h_{xx}{)}_x$. Horizontal Stokes flow obeys $\mu u_{yy}=p_x$, with no slip at $y=0$ and Marangoni stress $\mu u_y(h)={\gamma }_x$. ThusUse $h_t+q_x=0$ and ${\gamma }_x=-A{C}_x$ to obtain the thin-film equations with insoluble surfactant:The differing flux coefficients are essential: the surfactant travels at the surface velocity, not at the depth-averaged liquid velocity.

Neglect the two pressure-gradient terms for the spreading outer region. If its half-length is $\ell$, fixed surfactant mass gives $C\sim M/\ell$, $C_x\sim M/{\ell }^2$. The surface velocity is then $u_s\sim AM{h}_0/(\mu {\ell }^2)$. Equating this to $\ell /t$ gives $\ell \propto t^{1/3}$. For the precise finite-mass Marangoni spreading on a liquid film solution choosewith symmetry about zero and support $0\le \eta \le {\eta }_N$ on the positive half-line. The reduced differential equations areTheir boundary conditions are zero flux at the centre and zero concentration at the moving front. The two integral constraints express conserved surfactant mass and zero net change of liquid volume relative to the original uniform layer:The second follows by integrating $h_t+q_x=0$ across the whole layer: the localized release adds surfactant but no liquid, and the flux vanishes at infinity. Outside the spreading region the leading depth remains $h_0$.

Integrating the concentration equation from the centre gives $H\Gamma {\Gamma }^{\prime }=-\eta \Gamma /3$. Where $\Gamma >0$, this is $H{\Gamma }^{\prime }=-\eta /3$. Substituting into the height equation givesTherefore $H=c\eta$, and ${\Gamma }^{\prime }=-1/(3c)$. With $\Gamma ({\eta }_N)=0$, the linear similarity profiles for surfactant spreading are $\Gamma =({\eta }_N-\eta )/(3c)$. The liquid-volume constraint gives $c{\eta }_N=2$, while the surfactant constraint gives ${\eta }_N^2/(3c)=1$. Thus ${\eta }_N^3=6$, andIn physical variables, the leading profiles on the whole line areOutside this interval $h=h_0$ and $C=0$. The formal outer solution reaches zero depth at the central point, and height $2h_0$ immediately inside the front. It is not a microscopic description of the central dry region or of the front transition. As a conservation check, the front velocity equals both the incoming surface velocity and the liquid Rankine–Hugoniot speed $q/(2h_0-h_0)=2AM{h}_0/(\mu x_N^2)={\dot{x}}_N$.

The jump from $2h_0$ to $h_0$ cannot persist as a smooth solution with both hydrostatic and capillary gradients absent. In an edge region of width $\Delta$, a depth change of order $h_0$ produces $h_x\sim h_0/\Delta$ and $h_{xxx}\sim h_0/{\Delta }^3$. Keep the stipulated concentration-gradient scale $M/x_N^2$. The three liquid-flux scales areFor $g=0$, balancing $q_M$ and $q_{\gamma }$ gives the capillary smoothing of a Marangoni front width

Gravity dominates capillarity when $q_g/q_{\gamma }\sim \rho g{\Delta }^2/{\gamma }_0\gg 1$. In physical units the condition is ${\Delta }^2\gg {\gamma }_0/(\rho g)$: the printed ${\gamma }_0/g$ omits density and is dimensionally incomplete unless surface tension has already been divided by density. Balancing the gravity flux with $q_M$ yields the gravity smoothing of a Marangoni front:This layer grows faster than the spreading half-length. It reaches the whole pool whenOnly scaling constants can be fixed by this balance, so an exact numerical coefficient in $t_{\ast }$ is not implied. The gravity-dominant assumption must hold at this crossover as well.

For $t\gg t_{\ast }$, the sharp-front linear-depth outer profile is replaced by a gravity-levelled surfactant film: the height approaches $h_0$, with a small depression under the surfactant and redistributed liquid outside. The rapid hydrostatic return flow almost cancels the net Marangoni liquid flux. Setting $q\simeq 0$ and $h\simeq h_0$ 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 surface still moves outward, with $u_s\simeq -A{h}_0{C}_x/(4\mu )$, so leveling the bulk liquid does not stop surfactant spreading.

Use $y$ for height above the horizontal wall. In the lubrication approximation, vertical momentum is hydrostatic and the interfacial stress balance with variable surface tension givesThe horizontal equation is $\mu u_{yy}=B$, with $u(0)=0$ and $\mu u_y(h)={\gamma }_x=-A{C}_x$. ThusIntegrating across the film and adding surface diffusion to surfactant advection gives the dimensional thin-film mass flux and insoluble surfactant flux:These include Marangoni stress, hydrostatic leveling, capillarity and surface diffusion with their signs fixed by the surface traction. In particular the derivative acts on the product of surface tension and curvature, not just on curvature.

Choose $h_{\ast }=(A{C}_0/(\rho g){)}^{1/2}$ and $u_{\ast }=A{C}_0{h}_{\ast }/(\mu L)$. The dimensionless definitions areTogether with the stated concentration and horizontal scales, these giveIn steady flow $Q,J$ are constants by liquid and surfactant conservation.

With capillarity and diffusion neglected, solve these two linear equations for the gradients, in the region $H>0,\Gamma >0$:For positive $Q,J$, the phase plane nullclines are $H=3Q\Gamma /(2J)$ for ${\Gamma }_X=0$ and $H=2Q\Gamma /J$ for $H_X=0$. Above both lines, trajectories go left and upward; between them they go left and downward; below both they go right and downward. There is no positive-quadrant equilibrium. The axes are singular boundaries of this positive-flux reduction, not regular equilibria.

The phase portrait shows these trajectories for one choice of positive fluxes; the two nullcline slopes rescale with $Q/J$.

At leading order in the lubrication approximation, surface arclength is horizontal distance. An insoluble surfactant is transported by the surface velocity, without bulk exchange or diffusion. Conservation of insoluble surfactant on a moving interface therefore becomesThe local hydrostatic pressure and capillary normal stress giveSolve $\mu u_{zz}=p_x$ with no-slip boundary condition at $z=0$ and Marangoni stress $\mu u_z(h)={\gamma }_x=-A{C}_x$. The velocity, surface velocity and volume flux per unit width areMass conservation, $h_t+q_x=0$, yields the thin-film equations with insoluble surfactant:and the corresponding surfactant equation isThe hydrostatic and capillary terms thus affect both the film flux and the surface transport, with different coefficients.

For finite-mass Marangoni spreading on a liquid film, first neglect both pressure-gradient terms. If the spread has size $\ell$, then $C\sim M/\ell$ and $C_x\sim M/{\ell }^2$, giving surface speed $u_s\sim AM{h}_0/(\mu {\ell }^2)$. Equating $\ell /t$ to this speed givesSet $\ell =(AM{h}_{0}t/\mu {)}^{1/3}$ exactly as a similarity scale, and writeOn the positive half of the pool, substitution produces two ordinary differential equations:Symmetry fixes the flux integration constant in the second equation to zero, giving $H{\Gamma }^{\prime }=-\eta /3$ wherever $\Gamma >0$. Substitution into the first equation gives $\eta H^{\prime }=H$. Thus the linear similarity profiles for surfactant spreading have the formThe concentration vanishes at the moving front. The two integral constraints are total surfactant mass and conservation of the fluid volume relative to the undisturbed layer:The first gives $k={\eta }_N^2/3$; the second gives $k{\eta }_N=2$. Therefore ${\eta }_N^3=6$. In physical variables, the solution isOutside the pool the leading outer solution has $h=h_0$, $C=0$. In particular, $h(x_{N-},t)=2h_0$: the subscript means the one-sided limit just behind the front. As a consistency check, the surface speed there is $x_N/(3t)={\dot{x}}_N$ and the fluid jump satisfies the moving-front mass conservation condition. This ideal outer solution has a height jump from $2h_0$ to $h_0$.

That jump cannot persist in a physical interface with nonzero gravity or surface tension. Across a smoothing region of width $\Delta$, $h$ changes by $O(h_0)$. The Marangoni flux scales as $AM{h}_0^2/(\mu x_N^2)$. At $g=0$, its balance with the capillary flux ${\gamma }_0{h}_0^4/(\mu {\Delta }^3)$ gives capillary smoothing of a Marangoni front:Here the surface tension approaches ${\gamma }_0$ at the surfactant-free edge. The assumed concentration-gradient scale is $M/x_N^2$ even inside the smoothing region.

When gravity dominates capillarity, the hydrostatic flux is $\rho g{h}_0^4/(\mu \Delta )$. The same balance gives gravity smoothing of a Marangoni front:The gravity-dominated condition is ${\Delta }^2\gg {\gamma }_0/(\rho g)$, the square of the capillary length. The printed condition omits $\rho$; the expression above restores dimensional consistency.

This width grows faster than $x_N$. It becomes comparable with the pool size whenThe time is a scaling estimate, so numerical factors cannot be fixed by the width balance. For $t\gg t^{\ast }$, the sharp-front similarity profile no longer describes the film: hydrostatic pressure levels the layer over the whole pool, and its thickness tends towards $h_0$ with an increasingly small depression beneath the surfactant. The surface can still spread by Marangoni stress while a hydrostatic-pressure-driven return flow nearly cancels the net film flux. In this gravity-levelled surfactant film, $q\simeq 0$ gives $h_x\simeq -3A{C}_x/(2\rho g{h}_0)$, so the fractional height variation is of order $AM/(\rho g{h}_0^2{x}_N)\ll 1$.

The expression in braces in the first equation is the net axial force. The term $-\pi a^2\gamma /a$ is the compressive force from the capillary pressure $\gamma /a$, $3\pi \mu a^2{w}_z$ is the Newtonian extensional tension with Trouton ratio three, and $2\pi a\gamma$ is the axial pull of surface tension around the circumference. Its $z$-derivative vanishes because axial force is conserved.

The final equation is conservation of an insoluble surfactant. The surface divergence $w_z+u/a$ is the sum of axial and circumferential extension rates. Positive surface divergence increases interfacial area and dilutes $C$; negative divergence concentrates it. The absence of a diffusion term expresses the assumption of negligible surface diffusion.

Linearizing the area and surfactant equations givesThusand hencewhere $\psi$ is fixed by the initial data.

Linearizing the displayed net axial force and using ${\gamma }^{\prime }=-A{C}^{\prime }$ givesEliminating $w_z$ and $C^{\prime }$ therefore gives directlyThus the three displayed evolution equations implyThe target formula printed later in the paper contains an additional factor of $\pi$ in both denominators. That factor does not follow from the displayed equations because every term in the axial-force balance contains the same factor $\pi$. If the target formula is adopted as the intended normalization, its corresponding value is $s=({\gamma }_0-A{C}_0)/(6\pi \mu a_0)$.

Initially $\eta =0$ and $C^{\prime }>0$, so $\psi >0$ and ${\eta }_t<0$: a surfactant-rich, low-tension region begins to neck as neighboring higher tension pulls fluid away. The accompanying axial extension dilutes the surfactant.

If $A<{\gamma }_0/C_0$, then $s>0$. The Rayleigh–Plateau instability overwhelms the weak surface-elastic response: the necking perturbation grows in the linear model while the original concentration excess is diluted and eventually changes sign.

If $A>{\gamma }_0/C_0$, then $s<0$. Strong surface elasticity arrests the disturbance atThe concentration perturbation becomes negative, raising the local surface tension until its axial force balances that of the wider regions. This is stabilization by a surfactant-induced Marangoni stress.

For insoluble surfactant with $\sigma ={\sigma }_0-A\Gamma$ and positive base tension, let $q=k^2{a}^2$, $\lambda =h_0^3({\sigma }_0-A{\Gamma }_0)/(3\mu a^4)$ and $\alpha =h_{0}A{\Gamma }_0/(\mu a^2)$. The coupled film-height and advected-surfactant amplitudes give the displayed growth equation. The unstable band remains $0<q<1$. Weak surfactant elasticity gives ${\beta }_{\max }=\lambda /4-3\alpha /8+O({\alpha }^2/\lambda )$; strong elasticity gives ${\beta }_{\max }\sim \lambda /16$. The Marangoni stress reduces tangential surface motion; strong elasticity reduces lubrication mobility to the value for an effectively immobile interface.

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.