Morton number

From HandWiki

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

[math]\displaystyle{ \mathrm{Mo} = \frac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3}, }[/math]

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

[math]\displaystyle{ \mathrm{Mo} = \frac{g\mu_c^4}{\rho_c \sigma^3}. }[/math]

Relation to other parameters

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

[math]\displaystyle{ \mathrm{Mo} = \frac{\mathrm{We}^3}{\mathrm{Fr}^2\, \mathrm{Re}^4}. }[/math]

The Froude number in the above expression is defined as

[math]\displaystyle{ \mathrm{Fr^2} = \frac{V^2}{g d} }[/math]

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, https://archive.org/details/experimentalinve00habe 
  3. Pfister, Michael; Hager, Willi H. (May 2014). "History and significance of the Morton number in hydraulic engineering". Journal of Hydraulic Engineering 140 (5): 02514001. doi:10.1061/(asce)hy.1943-7900.0000870. http://infoscience.epfl.ch/record/198760/files/2014_971_Pfister_Hager_history_and_significance_Morton_number_in_hydraulic_engineering.pdf.