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
\( \mathrm{Mo} = \frac{g \mu_c^4 \, \Delta \rho}{\rho_c^2 \sigma^3}, a \)
where g is the acceleration of gravity, μ c {\displaystyle \mu _{c}} \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
\( \mathrm{Mo} = \frac{g\mu_c^4}{\rho_c \sigma^3}.a \)
Relation to other parameters
The Morton number can also be expressed by using a combination of the Weber number, Froude number and Reynolds number,
\( \mathrm{Mo} = \frac{\mathrm{We}^3}{\mathrm{Fr}\, \mathrm{Re}^4}.a \)
The Froude number in the above expression is defined as
\( \mathrm{Fr} = \frac{V^2}{g d}a \)
where V is a reference velocity and d is the equivalent diameter of the drop or bubble.
References
Clift, R.; Grace, J. R.; Weber, M. E. (1978), Bubbles Drops and Particles, New York: Academic Press, ISBN 978-0-12-176950-5
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
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.
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