Morton number

In fluid dynamics, the Morton number (Mo) is a dimensionless number used together with the Eötvös number or Bond number to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase, c. It is named after Rose Morton, who described it with W. L. Haberman in 1953.

Definition

The Morton number is defined as

$\mathrm{M}\mathrm{o}=\frac{g{\mu }_c^4\mathrm{\Delta }\rho }{{\rho }_c^2{\sigma }^3},$

where g is the acceleration of gravity, ${\mu }_c$ is the viscosity of the surrounding fluid, ${\rho }_c$ the density of the surrounding fluid, $\mathrm{\Delta }\rho$ the difference in density of the phases, and $\sigma$ is the surface tension coefficient. For the case of a bubble with a negligible inner density the Morton number can be simplified to

$\mathrm{M}\mathrm{o}=\frac{g{\mu }_c^4}{{\rho }_c{\sigma }^3}.$

Relation to other parameters

The Morton number can also be expressed by using a combination of the Weber number, Froude number and Reynolds number,

$\mathrm{M}\mathrm{o}=\frac{{\mathrm{W}\mathrm{e}}^3}{{\mathrm{F}\mathrm{r}}^2{\mathrm{R}\mathrm{e}}^4}.$

The Froude number in the above expression is defined as

$\mathrm{F}{\mathrm{r}}^{\mathrm{2}}=\frac{V^2}{gd}$

where V is a reference velocity and d is the equivalent diameter of the drop or bubble.