= Solution
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 becomes
$$
\boxed{C_t+\partial_x(Cu_s)=0,\qquad u_s=u(x,h,t).}
$$
The local <hydrostatic pressure> and capillary normal <stress> give
$$
p_x=\rho g h_x-\partial_x(\gamma h_{xx}).
$$
Solve $\mu u_{zz}=p_x$ with <no-slip boundary condition> at $z=0$ and <Marangoni stress> $\mu u_z(h)=\gamma_x=-AC_x$. The <velocity>, surface <velocity> and <volume flux per unit width> are
$$
\begin{aligned}
u(z)&=\frac{p_x}{2\mu}(z^2-2hz)+\frac{\gamma_x}{\mu}z,\\
u_s&=-\frac{h^2p_x}{2\mu}-\frac{AhC_x}{\mu},\\
q&=-\frac{h^3p_x}{3\mu}-\frac{Ah^2C_x}{2\mu}.
\end{aligned}
$$
<Mass conservation>, $h_t+q_x=0$, yields the <thin-film equations with insoluble surfactant>:
$$
\boxed{h_t=\frac{A}{2\mu}\partial_x(h^2C_x)+\frac{\rho g}{3\mu}\partial_x(h^3h_x)-\frac1{3\mu}\partial_x\left[h^3\partial_x(\gamma h_{xx})\right],}
$$
and the corresponding <surfactant> equation is
$$
\boxed{C_t=\frac{A}{\mu}\partial_x(hCC_x)+\frac{\rho g}{2\mu}\partial_x(Ch^2h_x)-\frac1{2\mu}\partial_x\left[Ch^2\partial_x(\gamma h_{xx})\right].}
$$
The 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 $u_s\sim AMh_0/(\mu\ell^2)$. Equating $\ell/t$ to this speed gives
$$
\boxed{\ell(t)\sim\left(\frac{AMh_0t}{\mu}\right)^{1/3}.}
$$
Set $\ell=(AMh_0t/\mu)^{1/3}$ exactly as a similarity scale, and write
$$
\eta=x/\ell,\qquad h=h_0H(\eta),\qquad C=(M/\ell)\Gamma(\eta).
$$
On the positive half of the pool, substitution produces two <ordinary differential equations>:
$$
\boxed{-\frac23\eta H'=(H^2\Gamma')',\qquad-\frac13(\eta\Gamma)'=(H\Gamma\Gamma')'.}
$$
Symmetry fixes the flux integration constant in the second equation to zero, giving $H\Gamma'=-\eta/3$ wherever $\Gamma>0$. Substitution into the first equation gives $\eta H'=H$. Thus the <linear similarity profiles for surfactant spreading> have the form
$$
H=k\eta,\qquad\Gamma=\frac{\eta_N-\eta}{3k},\qquad0\leq\eta\leq\eta_N.
$$
The 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:
$$
\boxed{2\int_0^{\eta_N}\Gamma\,d\eta=1,\qquad\int_0^{\eta_N}(H-1)\,d\eta=0.}
$$
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 is
$$
\boxed{x_N=\left(\frac{6AMh_0t}{\mu}\right)^{1/3},\quad h=\frac{2h_0|x|}{x_N},\quad C=\frac{M}{x_N}\left(1-\frac{|x|}{x_N}\right)\quad(|x|<x_N).}
$$
Outside the pool the leading outer solution has $h=h_0$, $C=0$. In particular, \b[$h(x_{N-},t)=2h_0$]: 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 $AMh_0^2/(\mu x_N^2)$. At $g=0$, its balance with the capillary flux $\gamma_0h_0^4/(\mu\Delta^3)$ gives <capillary smoothing of a Marangoni front>:
$$
\boxed{\Delta\sim\left(\frac{\gamma_0h_0^2x_N^2}{AM}\right)^{1/3}\propto t^{2/9}.}
$$
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 $\rho gh_0^4/(\mu\Delta)$. The same balance gives <gravity smoothing of a Marangoni front>:
$$
\boxed{\Delta\sim\frac{\rho gh_0^2x_N^2}{AM}\propto t^{2/3}.}
$$
The gravity-dominated condition is $\Delta^2\gg\gamma_0/(\rho g)$, the square of the <capillary length>. \b[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 when
$$
\boxed{x_N^*\sim\frac{AM}{\rho gh_0^2},\qquad t^*\sim\frac{\mu(AM)^2}{(\rho g)^3h_0^7}.}
$$
The time is a scaling estimate, so numerical factors cannot be fixed by the width balance. For $t\gg t^*$, 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>, $q\simeq0$ gives $h_x\simeq-3AC_x/(2\rho gh_0)$, so the fractional height variation is of order $AM/(\rho gh_0^2x_N)\ll1$.
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