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.^[1] It is named after Rose Morton, who described it with W. L. Haberman in 1953.^[2][3]

Definition

The Morton number is defined as

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

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

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

Relation to other parameters

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

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

The Froude number in the above expression is defined as

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

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

References

1. Clift, R.; Grace, J. R.; Weber, M. E. (1978), Bubbles Drops and Particles, New York: Academic Press, ISBN 978-0-12-176950-5
2. Haberman, W. L.; Morton, R. K. (1953), An experimental investigation of the drag and shape of air bubbles rising in various liquids, Report 802, Navy Department: The David W. Taylor Model Basin
3. Pfister, Michael; Hager, Willi H. (May 2014). "History and significance of the Morton number in hydraulic engineering" (PDF). Journal of Hydraulic Engineering. 140 (5): 02514001. doi:10.1061/(asce)hy.1943-7900.0000870.