I have a system with a static water droplet (diameter = 1 mm) in air. If I reduce the density of the water droplet from 1000 Kg/m${}^3$ to 100 Kg/m${}^3$ for my simulation purposes (because I am facing issues with high density ratios), what would be the effect on surface tension? How does surface tension scale with density (the governing non-dimensional number)? Is there a formula that governs it?
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Surface tension ($\gamma$) can be measured by capillary action of the liquid, as
$$\gamma = h \rho g \frac{r}{2 cos \theta} $$
where
$h$ is the height the liquid rises to
$\rho$ is the density of the liquid
$r$ is the radius of the capillary
$g$ is acceleration due to gravity
$\theta$ is the contact angle between the liquid surface and the capillary walls, $0^\circ$ being vertically upwards, $90^\circ$ being horizontal, $180^\circ$ being vertically downwards)
Thus in answer to your question $$\gamma \propto \rho $$
Noting that water is nearly incompressible, although density can be varied by adding impurities.
