Morton number

Physical key figure
Surname	Morton number
Formula symbol	${\displaystyle {\mathit {Mo}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Mo}}={\frac {g\cdot \eta ^{4}\cdot \Delta \rho }{\rho ^{2}\cdot \sigma ^{3}}}}$
${\displaystyle g}$ Gravity acceleration ${\displaystyle \eta }$ dynamic viscosity of the continuous phase ${\displaystyle \Delta \rho }$ Density difference ${\displaystyle \rho }$ Continuous phase density ${\displaystyle \sigma }$ Interfacial tension
Named after	RK Morton
scope of application	dispersive two-phase flows

The Morton number (after Rose Katherine Morton, although it was used three years earlier by B. Rosenberg) is a dimensionless number of fluid mechanics . It is important for the characterization of disperse two-phase flows , since the shape and the rate of rise or fall of gas bubbles and drops in the gravitational field depend on it and on the Eötvös number . ${\displaystyle {\mathit {Mo}}}$

The Morton number measures the ratio of viscous forces to surface tensions and, by definition, depends only on the physical properties of the disperse (inner) and the continuous (outer, surrounding) phase : ${\displaystyle F_{\mathrm {v} }}$ ${\displaystyle F_{\mathrm {O} }}$

${\displaystyle {\mathit {Mo}}={\frac {F_{\mathrm {v} }}{F_{\mathrm {O} }}}={\frac {g\cdot \eta ^{4}\cdot \Delta \rho }{\rho ^{2}\cdot \sigma ^{3}}}}$

With

• ${\displaystyle g}$the acceleration due to gravity
• ${\displaystyle \eta }$the dynamic viscosity of the continuous phase surrounding the bubble
• ${\displaystyle \Delta \rho }$the density difference of the two phases
• ${\displaystyle \rho }$ the density of the continuous phase
• ${\displaystyle \sigma }$the interfacial tension .

In the event that the density of the bubble is negligible, the following applies , so that the equation is simplified accordingly. ${\displaystyle \Delta \rho \to \rho }$

Alternatively, the Morton number can be calculated from the key figures Eötvös number , capillary number and Reynolds number : ${\displaystyle {\mathit {Eo}}}$ ${\displaystyle {\mathit {Ca}}}$ ${\displaystyle {\mathit {Re}}}$

${\displaystyle {\mathit {Mo}}={\frac {{\mathit {Eo}}\cdot {\mathit {Ca}}^{2}}{{\mathit {Re}}^{2}}}}$

See also

• Capillary number

Individual evidence

1. ↑ Haberman, WL; Morton, RK: An experimental investigation of the drag and shape of air bubbles rising in various liquids . David W. Taylor Model Basin, Washington, DC 1953 ( online ). online ( Memento of the original from August 19, 2014 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
2. ↑ ^{a b} Michael Pfister, Willi H. Hager: History and Significance of the Morton Number in Hydraulic Engineering . In: Journal of Hydraulic Engineering . tape 140 , no. 5 , 2014, doi : 10.1061 / (ASCE) HY.1943-7900.0000870 ( online [PDF]).
3. ^ Josef Kunes: Dimensionless Physical Quantities in Science and Engineering . Elsevier, 2012, ISBN 0-12-391458-2 , pp. 254 ( limited preview in Google Book search).