Since a liquid with a high surface tension pulls more strongly on the surrounding liquid than one with a low surface tension, the presence of a gradient in surface tension will naturally cause the liquid to flow away from regions of low surface tension. The surface tension gradient can be caused by concentration gradient or by a temperature gradient (surface tension is an inversely proportional function of temperature). In simple cases, the speed of the flow u \approx \Delta\gamma/\mu, where \Delta\gamma is the difference in surface tension and \mu is the
viscosity of the liquid. Water at room temperature has a surface tension of around 0.07 N/m and a viscosity of approximately 10−3 Pa⋅s. So even variations of a few percent in the surface tension of water can generate Marangoni flows of almost 1 m/s. Thus Marangoni flows are common and easily observed. For the case of a small drop of surfactant dropped onto the surface of water, Roché and coworkers performed quantitative experiments and developed a simple model that was in approximate agreement with the experiments. This described the expansion in the radius r of a patch of the surface covered in surfactant, due to an outward Marangoni flow at a speed u. They found that speed of expansion of the surfactant-covered patch of the water surface occurred at speed of approximately u \approx \frac{(\gamma_\text{w} - \gamma_\text{s})^{2/3}}{(\mu\rho r)^{1/3}} for \gamma_\text{w} the surface tension of water, \gamma_\text{s} the (lower) surface tension of the surfactant-covered water surface, \mu the viscosity of water, and \rho the mass density of water. For (\gamma_\text{w} - \gamma_\text{s}) \approx 10^{-2} N/m, i.e., of order of tens of percent reduction in surface tension of water, and as for water \mu\rho \sim 1 N⋅m−6⋅s3, we obtain u \approx 10^{-2}\,r^3 with
u in m/s and
r in m. This gives speeds that decrease as surfactant-covered region grows, but are of order of cm/s to mm/s. The equation is obtained by making a couple of simple approximations, the first is by equating the stress at the surface due to the concentration gradient of surfactant (which drives the Marangoni flow) with the viscous stresses (that oppose flow). The Marangoni stress \sim (\partial\gamma/\partial r), i.e., gradient in the surface tension due gradient in the surfactant concentration (from high in the centre of the expanding patch, to zero far from the patch). The viscous
shear stress is simply the viscosity times the gradient in shear velocity \sim \mu (u/l), for l the depth into the water of the flow due to the spreading patch. Roché and coworkers assume that the momentum (which is directed radially) diffuses down into the liquid, during spreading, and so when the patch has reached a radius r, l \sim (\nu r/u)^{1/2}, for \nu = \mu/\rho the
kinematic viscosity, which is the diffusion constant for momentum in a fluid. Equating the two stresses, u^{3/2} \approx \frac{(\nu r)^{1/2}}{\mu}\left(\frac{\partial\gamma}{\partial r}\right) \approx \frac{r^{1/2}}{(\mu\rho)^{1/2}}\frac{\gamma_\text{w} - \gamma_\text{s}}{r}, where we approximated the gradient (\partial\gamma/\partial r) \approx (\gamma_\text{w} - \gamma_\text{s})/r. Taking the 2/3 power of both sides gives the expression above. The
Marangoni number, a dimensionless value, can be used to characterize the relative effects of surface tension and viscous forces. == Tears of wine ==