Wednesday, July 11, 2007

Towards mathematical modeling of a tear drop formation: Part one

I was inspired by an image in this blog ( that roughly shows the shape of a tear. it pushed me to think if I can mathematically modulate the formation of tears, and if one can count how many tear drops can a person make per unit time. Then if I can estimate the volume of the total crying lot!
this is an attempt to use some mathematics as a basic tool to describe any phenomenon around us. And because I couldn't attach mathematical symbols directly to the blog page, I had to write and derive everything in a Word page, then save it in parts as pictures then adding them together here. I'll be glad if anyone announce for errors or wrong formulas. Here we go!

Assuming that tears form on the bottom side of a horizontal surface, the free-falling of a tear drop is similar to that of a water drop. It begins with the first secretion amount that has incomplete oval shape, just like if we turn an image of a rising sun upside down. The mass continues to elongate as it is subject to its weight force acting downward. This force is equal to the mass of the tear multiplied by the gravitational acceleration (W=mg). the figure below shows instant magnitudes of diameter of the oval tear as it decreases from both ends. It also shows that the smaller the diameter of the oval tear the larger the value of elongation it gets. This implies that if we consider the diameter reduction a function of time, then at any given time both the diameter and the length are constrained by the relations: di and li>li+1 respectively.