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 $W_e$


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


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 


Which yields


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


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



Transient Velocity for non spherical particles

Non-spherical particles can have spherical shapes or completely random and strange shapes that from now on are referred as non-spherical shapes. For axisymmetric shapes Gabitto and Tsouris finf it convient to use the aspect ratio,E , defined as the ratio of length projected on the axis of symmetry to the maximum diameter normal to the axis.

Wadell also proposed the degree of sphericity defined as 


where $A_V$ is the surface of a sphere of the same volume as the non-spherical particle. This sphericity term is considered the best for isometrically shaped particles. Using this parameter, Haider and Levenspiel Presented a $C_D$ vs $R_e$ correlation. Which is as follows


Where the used $C_D$ and $R_e$ are based on the equal volume sphere diameter called volumetric diameter. $K_1$ and $K_2$ are shape factors applicable in sphericity for solids of spherical shape for non-shperical shpaes $K_1$ and $K_2$ are functions of the sphericity and particle orientation.

Chien, re-analysing pre-existing data in the petroleum engineering & processing literature proposed the following expression for the drag coeffiecient 


The later expression can be used to find a transient velocity profile. This together with the transient velocity method yields.



For Drag coefficient by Ganser we have :


For the drag coefficient by chen, which by calculating the constans n_2,n_3 and n_4 would give a perfect fit.