modeling the droplet break-up based on drawing the shape using velocity gradient and normal vector

The present study developed a model to deform a viscous droplet in a viscous matrix by a shear flow based on changing the normal vector. the initial cross-section was assumed to be a regular polygon with 1000 sides instead of a circle or ellipsoid, and also this model was independent of the initial polygon shape. changing the normal vector and the length of each side of the droplet was a function of the velocity gradient. to calculate the velocity gradient over each side of the shape, the equations of tangential and normal stress, the conservation of mass equation, and the absence of mass transfer equation between two phases were solved simultaneously. by knowing the velocity gradient, normal vectors and the length of each side are calculated; therefore, the new shape can be plotted by drawing sides one after another. the results displayed that the time of the break-up, which this model predicts, coincides with the experimental results. on the other hand, the predicted shape of the droplet at the break-up has logically coincided with the experimental results in the middle range of the capillary number ratio (1.4 -2.6 critical capillary number). the drop’s dimensions show less than 30% deviation and its rotation less than 20%. additionally, the dimension of the end bubble also shows a deviation of less than 40%.