## What is Poiseuille’s formula?

*Poiseuille’s formula gives the discharge of a viscous fluid from a capillary tube.*

Consider a horizontal *capillary tube* of length ( l ) and diameter of cross section ( d = 2 \ r ) . A liquid of coefficient of viscosity ( \eta ) is flowing through the capillary. The tube is under a pressure difference of ( p ) across its ends.

Then, discharge from the tube is given by Poiseulli’s formula –

Q = \left ( \frac {V}{t} \right ) = \left ( \frac {\pi \ p \ r^4}{8 \ \eta \ l} \right )

= \left ( \frac {\pi \ p \ d^4}{128 \ \eta \ l} \right )

*This formula is called Poiseulle’s formula to find viscosity of a liquid.*

Assumptions used in the derivation of Poiseuille’s formula are –

- The flow of the liquid is steady and parallel to the axis of the tube.
- The
*pressure*is constant over any cross-section of the tube. - The liquid
*velocity*is zero at the walls of the tube and increases towards the axis of the tube. - The tube is held horizontal so that
*gravity*does not influence the flow of liquid.

## Stokes Law

*Stoke’s law gives a expression for viscous forces acting on a body when it moves within a viscous liquid.*

According to Stokes’ law, the viscous forces acting on a moving body is –

- Proportional to the terminal velocity of the body i.e. ( F \propto v )
- Proportional to the size or area of approach of the body i.e. ( F \propto r )
- Proportional to the
*coefficient of viscosity*of the liquid or fluid i.e. ( F \propto \eta )

Therefore, viscous force acting on a small spherical body of radius ( r ) , moving with *uniform velocity* ( v ) through a fluid of viscosity ( \eta ) will be –

F = 6 \ \pi \ \eta \ r \ v

### Stokes formula

Consider that a spherical body of radius ( r ) is dropped in a chamber of viscous liquid of viscosity ( \eta ) .

- When the sphere is dropped in liquid, it will attain its terminal velocity ( v ) depending on the viscosity of the fluid. In such a case, the various forces acting on the sphere are shown in figure.

- Various forces acting on the sphere are –

(1) Weight of the sphere acting vertically downwards.

W = \left ( \frac {4}{3} \pi r^3 \rho g \right )

Here ( \rho ) is the *density* of material of sphere.

(2) Upward thrust called *buoyant force* equal to the weight of the liquid displaced.

U = \left ( \frac {4}{3} \pi r^3 \sigma g \right )

Here ( \sigma ) is the *density* of liquid.

(3) *Viscous force* ( F ) , acting in the upward direction.

Therefore, net force acting on the body will be –

F_{net} = W - ( U + F )

Now, according to Stokes’ law –

F = 6 \ \pi \ \eta \ r \ v

Thus, the force of viscosity increases as the velocity of the body increases. A stage will reach when the weight of the body becomes just equal to the sum of the *buoyant force* and the viscous force. Thus, ( W = U + F )

*At this stage, net force on the body becomes zero and the body begins to fall with a uniform maximum velocity which is called***terminal velocity.**

When the body attains the terminal velocity, then –

U + F = W

So, \quad \frac {4}{3} \pi r^3 \sigma g + 6 \pi \eta r v = \frac {4}{3} \pi r^3 \rho g

Or, \quad 6 \pi \eta r v = \frac {4}{3} \pi r^3 ( \rho - \sigma ) g

Hence, \quad \eta = \frac {2 \ r^2}{9 \ v} ( \rho - \sigma )

*This expression is termed as Stoke’s formula to find viscosity of a liquid.*

### Dashpot formula

*A dashpot is a piston cylinder arrangement in which a heavy viscous liquid is used for damping purpose.*

- A dashpot is used for damping of vibrations and shock in machines, vehicles etc.
- In dashpot cylinder, a viscous liquid or hydraulic oil is filled in the cylinder. This liquid is called
.*dashpot oil* - Piston moves in the dashpot cylinder against the viscous forces offered by dashpot oil.

Consider a dashpot in which –

- ( W ) is the weight of the piston along with external disturbing force of vibration.
- ( D ) is the diameter of the piston.
- ( l ) is length of the piston.
- ( t ) is the clearance between the piston diameter and the dashpot cylinder bore.
- ( v ) is the terminal velocity reached by the piston in the dashpot.

Then, viscosity is given by –

\eta = \left ( \frac {4 W t^3}{3 \pi D^3 l v } \right )

*This equation is known as dashpot formula to find viscosity of a liquid.*