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The material derivative follows surfactant carried by the moving interface. Tangential surface divergence changes the local surface area and hence dilutes or concentrates insoluble material. Normal motion changes area through the curvature: an expanding sphere has positive $u_n{\nabla }_s\cdot \mathbf{n}$ and its concentration decreases. The last term is surface diffusion, which smooths concentration gradients. There is no exchange with the bulk; this is conservation of insoluble surfactant on a moving interface.

The interfacial speed is of order $a|\mathbf{F}|$. Diffusion dominates redistribution when the surface Péclet number satisfiesThen $C^{\prime }/C_0=O({\mathrm{Pe}}_s)$, and advection of $C^{\prime }$ is a second-order effect. Write $S=\mathbf{n}\cdot \mathbf{Fn}$. Since ${\mathbf{u}}_s=\mathbf{Fx}-S\mathbf{x}$, the projected divergence of the first term is $\mathrm{tr}\mathbf{F}-S=-S$, whereas the second has divergence $2S$; its gradient contracted with $\mathbf{x}$ is zero. Hence ${\nabla }_s\cdot {\mathbf{u}}_s=-3S$.

The leading steady transport equation is $D_s{\Delta }_s{C}^{\prime }=C_0{\nabla }_s\cdot {\mathbf{u}}_s=-3C_{0}S$. Trace-freeness and the supplied surface Laplacian identity give ${\Delta }_{s}S=-6S/a^2$. Fixing the conserved total surfactant removes the uniform null mode and yieldsThis is quadrupolar surfactant distribution on a spherical interface; its spherical mean is zero because $⟨n_i{n}_j⟩={\delta }_{ij}/3$.

With $+$ outside, $-$ inside, and the normal directed out of the bubble, the interfacial stress balance with variable surface tension isNow ${\nabla }_{s}S=2I_s\mathbf{Fn}/a$, so $-{\nabla }_s\gamma =2A{\gamma }_1{I}_s\mathbf{Fn}/a$. Multiplying the curvature by the tension and retaining linear terms givesThe undeformed isotropic pressure jump is kept with its normal; the perturbation terms are evaluated on the reference sphere. Products of deformation and concentration perturbations are beyond the approximation.

The imposed tensor is symmetric and traceless, so $\mathbf{x}\cdot \mathbf{Ex}$ is a harmonic scalar of degree two. Its decaying partner is $\mathbf{E}:\nabla \nabla (1/r)$; the corresponding harmonic vector uses $\mathbf{E}\nabla (1/r)$. Linearity and rotational symmetry give precisely these tensorial modes, with the growing scalar chosen to reproduce the imposed strain. Thus in the unscaled convention one can takeThe coefficients multiply distinct decaying modes. No other angular harmonic is forced at this order.

No penetration in the steady state gives $P=3Q-1$. The given tangential surface velocity implies $\alpha =1+2Q$. The tangential stress balance, with $M=A{\gamma }_1/(\mu a)$, gives $1+P-8Q=M\alpha$, or $-5Q=M(1+2Q)$. ThereforeThe nonuniform normal stress is unaffected by the uniform inviscid interior pressure. Its balance isSubstitution and rearrangement giveAs $M\to \infty$, $\alpha \to 0$: the concentration-induced Marangoni stress suppresses tangential surface motion, making the exterior boundary behave like an immobile surface despite the inviscid interior. The deformation coefficient tends to $5\mu a/(2{\gamma }_0)$; the clean limit is $2\mu a/{\gamma }_0$. Both small surface Péclet number and small deformation, of order $\mu a|\mathbf{E}|/{\gamma }_0$, are required. This is Marangoni immobilization of a bubble in straining flow.

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.

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 material derivative follows the interfacial surfactant. The term $-C{\nabla }_s\cdot {\mathbf{u}}_s$ describes dilution by tangential expansion and concentration by compression. The term $-C(\mathbf{u}\cdot \mathbf{n}){\nabla }_s\cdot \mathbf{n}$ accounts for changing surface area due to normal motion of a curved interface. Finally, $D_s{\nabla }_s^{2}C$ describes surface diffusion. Together these give conservation of insoluble surfactant on a moving interface.

The relevant surface Péclet number isWrite $S=\mathbf{n}\cdot \mathbf{A}\cdot \mathbf{n}$ and ${\mathbf{u}}_s=\mathbf{Ax}-S\mathbf{x}$. Since $\mathbf{A}$ has zero trace and ${\nabla }_s\mathbf{n}={\mathbf{I}}_s/a$, the surface divergence isThere is no normal motion. To first order in ${\mathrm{Pe}}_s$, the steady surfactant balance becomes $D_s{\nabla }_s^2{C}^{\prime }=C_0{\nabla }_s\cdot {\mathbf{u}}_s=-3C_{0}S$. The traceless quadratic $S$ is a degree-two spherical harmonic, so ${\nabla }_s^{2}S=-6S/a^2$. The mean of $C^{\prime }$ is zero by total surfactant conservation; consequentlyThe discarded advective term ${\mathbf{u}}_s\cdot {\nabla }_s{C}^{\prime }$ is smaller by ${\mathrm{Pe}}_s$. This is the quadrupolar surfactant distribution on a spherical interface.

With the unit normal directed from the bubble into the exterior, the interfacial stress balance with variable surface tension isUsing the specified first-order curvature and $\gamma ={\gamma }_0-K{\gamma }_{1}S$, together with ${\nabla }_{s}S=2{\mathbf{I}}_s\mathbf{An}/a$, givesThe first term is the spherical capillary pressure; the last contains the normal tension correction and the tangential Marangoni stress.

The ambient Stokes flow has the symmetry of the symmetric traceless tensor $\mathbf{E}$. In the Unscaled Papkovich–Neuber representation, ${\chi }_{\infty }=\mathbf{x}\cdot \mathbf{E}\cdot \mathbf{x}/2$ is harmonic and yields $\mathbf{Ex}$. The decaying vector potential must have the form $\mathbf{E}\nabla (1/r)$, and the scalar disturbance must have the degree-two form $\mathbf{E}:\nabla \nabla (1/r)$. They are harmonic outside the bubble and have precisely the required rotational covariance. Dimensionless amplitudes may therefore be writtenFor a steady bubble, the no-penetration boundary condition gives $1+P-3Q=0$. Its tangential velocity gives $\alpha =1+2Q$. The tangential stress boundary condition then yieldsHence $\alpha =5/(5+2M)$, $Q=-M/(5+2M)$ and $P=-(5+5M)/(5+2M)$. The inviscid interior supplies only the constant pressure balancing $2{\gamma }_0/a$. Matching the remaining normal stress givesEliminating $P,Q,\alpha$ givesAs $M\to \infty$, $\alpha \to 0$: the Marangoni stress suppresses tangential motion and effectively immobilizes the interface. This limit must retain the small surface Péclet number and small-deformation assumptions; $M$ can grow through increasing ${\gamma }_1$ without invalidating the linear concentration approximation.

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.