# Soap Bubble gets smaller and smaller !

At $$t=0$$, a spherical soap bubble with surface tension T and radius R is formed at one end of a cylindrical pipe of length L and cross-section radius of $${ r }_{ o }$$. The other end of the pipe is kept open to the atmosphere as shown.
The air that is inside the soap bubble has density $$\rho$$, and coefficient of viscosity $$\eta$$.

Find the time taken for the radius of the sphere to be halved. The time can be expressed as :

$\displaystyle{t\quad =\cfrac { a }{ b } (\cfrac { \eta L{ \rho }^{ c }{ R }^{ d } }{ T{ { (r }_{ o }) }^{ e } } )}.$

Compute the Value of $$a+b+c+d+e$$.

Here $$a,b,c,d,e$$ are non-negative integers.

Assumptions
$$\bullet$$ $$\displaystyle{R\quad >>\quad { r }_{ o }}$$.
$$\bullet$$ Outside The Soap Bubble air is present in atmosphere with constant atmospheric Pressure (1 atm)
$$\bullet$$ Surface Tension is Enough So that It always Maintains the spherical Shape of the Soap bubble!
$$\bullet$$ Length of Pipe is not So long (i.e Initial Volume Capacity of sphere is Sufficiently greater Than that of Pipe )

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