Morton number

This article is about fluid mechanics. For the linear ordering of points in the plane, see Morton number (number theory).

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

$\mathrm {Mo} ={\frac {g\mu _{c}^{4}\,\Delta \rho }{\rho _{c}^{2}\sigma ^{3}}},$

where g is the acceleration of gravity, $\mu _{c}$ is the viscosity of the surrounding fluid, $\rho _{c}$ the density of the surrounding fluid, $\Delta \rho$ 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

$\mathrm {Mo} ={\frac {g\mu _{c}^{4}}{\rho _{c}\sigma ^{3}}}.$

Relation to other parameters

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

$\mathrm {Mo} ={\frac {\mathrm {We} ^{3}}{\mathrm {Fr} ^{2}\,\mathrm {Re} ^{4}}}.$

The Froude number in the above expression is defined as

$\mathrm {Fr^{2}} ={\frac {V^{2}}{gd}}$

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