Marangoni effect

Remarks

Example: let $\sigma(x) = \sigma_0 + \tau x$ with $\tau > 0$ constant, and take a liquid layer $-d < z < 0$ on a solid bottom with the no-slip condition. The steady shear flow $u = \left(\frac{\tau}{\mu}(z + d), 0, 0\right)$ satisfies the Navier–Stokes equations with constant pressure, vanishes at $z = -d$, and has $\mu\,\partial_z u_1 = \tau = \partial_x\sigma$ at $z = 0$. So the surface moves towards larger $\sigma$, with speed $\tau d/\mu$. Non-example: if $\sigma$ is constant, the condition reduces to $\partial_z u_1 = \partial_z u_2 = 0$, the stress-free condition, and there is no Marangoni stress. For thermal Marangoni convection one uses the Marangoni number $\mathrm{Ma} = -\frac{d\sigma}{dT}\frac{\Delta T\,L}{\mu\alpha}$, with $\Delta T$ a temperature difference, $L$ a length and $\alpha$ the thermal diffusivity.