Name: Surface Tension and Its Causes
Uploaded: 2019-07-19
Duration: 3 min 11 s
Description: Surface tension is caused by attraction of particles in the surface layer by the bulk of liquid .Here we wiss study about the causes and variation in surface tension. Let's watch.

Question

Derive Poiseuille&rsquo;s formula for the volume of a liquid flowing per second through a pipe under streamlined flow

Accepted Answer

Consider a liquid flowing steadily through a horizontal capillary tube. Let $v = (\frac{v}{T})$ be the volume of the liquid flowing out per second through a capillary tube. It depends on coefficient of viscosity (η) of the liquid, radius of the tube (r), and the pressure gradient $\frac{P}{l}$ . Then
$v α η^{a} r^{b} {(\frac{P}{l})}^{c} v = k η^{a} r^{b} {(\frac{P}{l})}^{c} . . . . . (i)$
Where k is a dimensionless constant.
There fore $[v] = \frac{V o l u m e}{T i m e} = [L^{3} T^{- 1}] : [\frac{d P}{d x}] = \frac{P r e s s u r e}{D i s \tan c e} = [M L^{- 2} T^{- 2}] [η] = [M L^{- 1} T^{- 1}] a n d [r] = [L]$
Substituting in equation (i)
$[L^{3} T^{- 1}] = [M L^{- 1} T^{- 1}]^{a} [L]^{b} [M L^{- 2} T^{- 2}]^{c} M^{0} L^{3} T^{- 1} = M^{q + c} L^{- a + b - 2 c} T^{- a - 2 c}$
So, equating the powers of M, L and T on both sides, we get
$a + c = 0$ , $- a + b - 2 c = 3$ , and $- a - 2 c = - 1$
We have three unknowns a, b and c. We have three equations, on solving, we get $a = - 1, b = 4 a n d c = 1$
Therefore, equation (i) becomes,
$v = k η^{- 1} r^{4} {(\frac{P}{l})}^{1}$
Experimentally, the value of k is shown to be $\frac{π}{8}$ , we have
$v = \frac{π r^{4} P}{8 η l}$
The above equation is known as Poiseuille’s equation for the flow of liquid through a narrow tube or a capillary tube. This relation holds good for the fluids whose velocities are lesser than the critical velocity (vC).

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