Poiseuille’s Formula

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 –

1. The flow of the liquid is steady and parallel to the axis of the tube.
2. The pressure is constant over any cross-section of the tube.
3. The liquid velocity is zero at the walls of the tube and increases towards the axis of the tube.
4. 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 –

1. Proportional to the terminal velocity of the body i.e. ( F \propto v )
2. Proportional to the size or area of approach of the body i.e. ( F \propto r )
3. 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.

1. A dashpot is used for damping of vibrations and shock in machines, vehicles etc.
2. In dashpot cylinder, a viscous liquid or hydraulic oil is filled in the cylinder. This liquid is called dashpot oil.
3. 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.