d-square law

The d-square law or ${\displaystyle d^{2}}$-law is a relationship between diameter and time for an isolated, spherical droplet when it evaporates quasi-steadily, which was first observed by Boris Sreznevsky in 1882,^[1] and was explained by Irving Langmuir in 1918.^[2] If ${\displaystyle d(t)}$ and ${\displaystyle t}$ are the droplet diameter and time, then ${\displaystyle d^{2}}$-law pertains to the relation^[3][4]

${\displaystyle d_{0}^{2}-d^{2}=K(t-t_{0}),}$

where ${\displaystyle t_{0}}$ is the initial time, ${\displaystyle d_{0}=d(t_{0})}$ is the initial droplet diameter and ${\displaystyle K}$ is called the evaporation constant.

Crespo–Liñán correction refers to a small correction of the order ${\displaystyle 1/{\sqrt {\varepsilon }}}$ to the d-square law in terms of the small parameter ${\displaystyle \varepsilon =\rho _{g}/\rho _{l}\ll 1}$, the ratio of gas to liquid density. Antonio Crespo and Amable Liñán^[5] showed that the quasi-steady approximation is correct in the inner zone having the size on the order of droplet diameter, but becomes invalid in an outer zone with a size larger than the droplet diameter by the factor of ${\displaystyle 1/{\sqrt {\varepsilon }}}$, where the unsteady terms cannot be neglected.^[6]