III)Deformation

To see how much drops and bubbles deform from its spherical form while rising in a fluid, the ratio of dynamic pressure $P_{dyn}$ to that of the surface tension stresses \frac{\sigma}{2d} is used. The surface tension forces will drive a drop or bubble to a spherical shape while the dynamic forces, due to the non-symmetri of them will drive it to a non spherical shpae. This dimesnionless ratio is called the weber number $W_e$

$W_e=\frac{\rho_fw^2d}{\sigma}$

where $\sigma$ represents the surface tension and is expressed in $\frac{N}{m}$. for values of $W_e<<1$ the particles are approximately of a spherical form, for values of $W_e=1$ there is a moderate deviation, while for $W_e>>1$ there are large deviations. However sicne the velocity is to be known a priori to know how much a drop or bubble will deform, the dimensionless BONd number is used. It represents the ratio of the effective gravitational forces to tension forces & is expressed as follows

$B_O=\frac{gd^2(\rho_p\rho_f)}{\rho_{f}^{2}\sigma^3}$

To relate the bond, Weber and reynolds number in a satisfying way, the Mortion number is introduced. This number has no phsical interpretation but is convenient and can be calculated easily. It is defined as 

$M_O=\frac{g\mu_{f}^{4}(\rho_p-\rho_f))}{\rho_{f}^{2}\sigma^3}$ 

Which yields

​$\frac{W_{e_{terminal}}^{2}}{R_{e_{terminal}}^{4}}=\frac{M_O}{B_O}$

And make it possible to study the deformation by keeping $M_O$ constant and varying $B_O$ or viceversa. Also solving it further :

$W_{e_{terminal}}=\frac{4B_O}{3C_{D_{terminal}}}$

Though the dimesnionless parameters $B_O$ and $M_O$ seem convenient, they are only to be used for the case of terminal velocity $u_{max}$