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86. Problem 18.43 (RHK) A soap bubble of radius 38.2 mm is blown on the end of
a narrow tube of length 11.2 cm and internal diameter 1.08 mm. The
other end of the tube is exposed to atmosphere. We have to find the time taken
for the bubble radius to fall to 21.6 mm. We can assume Poiseuille flow
in the tube. Surface tension of the soap bubble solution is 2.50 N m-1;
and the viscosity of air is
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Solution: Click For PDF Version We will use two results one from surface tension and the other from viscosity for solving this problem. i. The gauge pressure
where
ii. According to Poiseuille’s law of viscous flow of fluid of density
As the air from inside the bubble is flowing out through the pipe, its radius
will change with time. Let
where
As the air flows out of the tube the rate of mass flux will be
Note,
With the above two results we can obtain a differential equation for
Integrating this equation, we find the variation of r as a function of time. It is
We solve for the constant of integration c by using the boundary condition that at
We thus obtain the relation
Data of the problem is
Substituting these values, we find that the time taken by the bubble radius to fall to 21.6 mm is 3617 s.
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inside a
soap bubble of radius r is
is the
surface tension of the soap solution.
in a pipe of radius R, length L, driven by pressur difference
is
be the
radius of the soap bubble at time t. Mass of air inside the bubble at
time t is
is the
density of air.
, as the
radius of the bubble will decrease with time when air flows out of it.
,
This
condition gives
