Exact solutions for viscous Marangoni spreading

When surface-active molecules are released at a liquid interface, their spreading dynamics is controlled by Marangoni flows. Though such Marangoni spreading was investigated in different limits, exact solutions remain very few. Here we consider the spreading of an insoluble surfactant along the interface of a deep fluid layer. For two-dimensional Stokes flows, it was recently shown that the nonlinear transport problem can be exactly mapped to a complex Burgers equation [D. Crowdy, SIAM J. Appl. Math. 81, 2526 (2021)]. We first present a very simple derivation of this equation. We then provide fully explicit solutions and find that varying the initial surfactant distribution—pulse, hole, or periodic—results in distinct spreading behaviors. By obtaining the fundamental solution, we also discuss the influence of surface diffusion. We identify situations where spreading can be described as an effective diffusion process but observe that this approximation is not generally valid. Finally, the case of a three-dimensional flow with axial symmetry is briefly considered. Our findings should provide reference solutions for Marangoni spreading that may be tested experimentally with fluorescent or photoswitchable surfactants.

Figure 1

Spreading of an insoluble surfactant above a semi-infinite liquid layer with two-dimensional flow. The inhomogeneous distribution of surfactant $\mathrm{\Gamma }(x,t)$ induces a Marangoni flow with bulk velocity $\mathbf{u}(x,z,t)$ and interfacial velocity $u_{\mathrm{s}}(x,t)$.

Figure 2

Spreading of a surfactant pulse: Concentration and velocity profiles for a Cauchy, circular, and Gaussian pulse from left to right. The typical width is $a=1$ and the total amount of surfactant is the same in all cases. Time is $t=0,0.5,1,2,5,10$ from top to bottom. Note that the vertical scale changes from one graph to the other.

Figure 3

[(a) and (b)] Closing of a surfactant hole: Concentration and velocity profiles for Cauchy and circular holes. Here $\overline{c}=a=\mathcal{A}=1$. Time is $t=0,0.5,1,1.5,2,3,5$ from bottom to top. A singularity occurs at $t^{★}=1$ and $x^{★}=0$. (c) Relaxation of a periodic distribution of surfactant: Concentration and velocity profiles for a sinusoidal initial distribution with $\overline{c}=a=\mathcal{A}=1$. Time is $t=0,0.2,0.5,1,1.5,2,5$. A singularity occurs at $t^{★}=1$ and $x^{★}=\pm \pi$.

Figure 4

Influence of surface diffusion on the transient spreading of a Dirac pulse. Here time is fixed to $t=0.5$ and the concentration profile is shown for $\epsilon =0.1,0.3,1,3,10$ from top to bottom. The dashed lines correspond to Gaussian distributions with variance $2t{D}_{\mathrm{eff}}\left(\epsilon \right)$. Inset: Effective diffusion coefficient $D_{\mathrm{eff}}\left(\epsilon \right)$ as given by Eq. (38). Approximations from Eqs. (39) and (40) are also shown with dashed lines.

Figure 5

Concentration and velocity profiles for the circular solution to transient spreading. Shown are the rescaled concentration $f_{\mathrm{rs}}\equiv \lambda \left(t\right)f(r,t)$ and the rescaled velocity $u_{\mathrm{rs}}\equiv \lambda \left(t\right)u(r,t)$, with a scaling factor $\lambda \left(t\right)={[a\left(t\right)/a]}^d/\mathcal{A}$, $d$ the dimension, $a\left(t\right)$ the radius at time $t$, and $a$ the initial radius. The case $d=2$ (solid lines) and $d=1$ (dashed line) correspond respectively to Eqs. (45) and (46) and Eq. (22). The dotted line is the large distance approximation of velocity from Eqs. (48) and (49) taken at lowest order.

Figure 6

Steady state around a source of surfactant. (a) Without evaporation. The punctual source at the origin is surrounded by two punctual sinks located at $x=l=\pm 1$. The dashed line is the local flux of surfactant $j\left(x\right)=u\left(x\right)f\left(x\right)$. (b) With evaporation. The source at the origin is of Cauchy type with $a=1$ and $\alpha =1$. The dashed lines show the approximations at small and large $x$.