// Workers AI · dad joke modeWhy did D-square law go to therapy? It had squared away issues.

The d-square law or $d^{2}$-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, and was explained by Irving Langmuir in 1918. If $d(t)$ and $t$ are the droplet diameter and time, then $d^{2}$-law pertains to the relation

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

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

Crespo–Liñán correction

Crespo–Liñán correction refers to a small correction of the order $1/{\sqrt {\varepsilon }}$ to the d-square law in terms of the small parameter $\varepsilon =\rho _{g}/\rho _{l}\ll 1$, the ratio of gas to liquid density. Antonio Crespo and Amable Liñán 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 $1/{\sqrt {\varepsilon }}$, where the unsteady terms cannot be neglected.