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
M o = g μ c 4 Δ ρ ρ c 2 σ 3 , {\displaystyle \mathrm {Mo} ={\frac {g\mu _{c}^{4}\,\Delta \rho }{\rho _{c}^{2}\sigma ^{3}}},}
where g is the acceleration of gravity, μ c {\displaystyle \mu _{c}} is the viscosity of the surrounding fluid, ρ c {\displaystyle \rho _{c}} the density of the surrounding fluid, Δ ρ {\displaystyle \Delta \rho } the difference in density of the phases, and σ {\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
M o = g μ c 4 ρ c σ 3 . {\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,
M o = W e 3 F r 2 R e 4 . {\displaystyle \mathrm {Mo} ={\frac {\mathrm {We} ^{3}}{\mathrm {Fr} ^{2}\,\mathrm {Re} ^{4}}}.}
The Froude number in the above expression is defined as
F r 2 = V 2 g d {\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