Important Viscosity & Poiseuille’s Formula Formulas for JEE and NEET

Viscosity

• The property of a fluid by virtue of which an internal frictional force acts between its different layers which opposes their relative motion is called viscosity.
• These internal frictional force is called viscous force.

Viscosity

• Viscous forces are inter-molecular forces acting between the molecules of different layers of liquid moving with different velocities.

• where, (dv/dx) = rate of change of velocity with distance called velocity gradient, A = area of cross-section and = coefficient of viscosity.
• SI unit of η is Nsm^-2 or pascal-second or decapoise. Its dimensional formula is [ML^-1T^-1].
• The knowledge of the coefficient of viscosity of different oils and its variation with temperature helps us to select a suitable lubricant for a given machine.
• Viscosity is due to transport of momentum. The value of viscosity (and compressibility) for ideal liquid is zero.
• The viscosity of air and of some liquids are utilised for damping the given parts of some instruments.
• The knowledge of viscosity of some organic liquids is used in determining the molecular weight and shape of large organic moleculars like proteins and cellulose.

Variation of Viscosity

The viscosity of liquids decreases with increase in temperature

where, η₀ and η_t is are coefficient of viscosities at 0°C and t°C, α and β are constants.

The viscosity of gases increases with increase in temperatures as

η ∝ √T

The viscosity of liquids increases with increase in pressure but the viscosity of water decreases with increase in pressure.

The viscosity of gases do not changes with pressure.

Poiseuille’s Formula

The rate of flow (v) of liquid through a horizontal pipe for steady flow is given by

where, p = pressure difference across the two ends of the tube. r = radius of the tube, n = coefficient of viscosity and 1 = length of the tube.

The Rate of Flow of Liquid

Rate of flow of liquid through a tube is given by

v = (P/R)

where, R = (8 ηl/πr⁴), called liquid resistance and p = liquid pressure.

Stoke's Law

When a sphere moves through a thick liquid, like honey, the resistance it faces depends on how fast it's moving, how big the sphere is, and how thick the liquid is. This resistance is called Stokes' drag. 

Imagine stirring honey with a spoon—the harder and faster you stir (velocity), the bigger the spoon (radius), and the thicker the honey (viscosity), the more resistance you feel. 

Illustration of Stoke's Law

This resistance force is calculated using a formula that takes into account the viscosity of the liquid, the size of the sphere, and how fast it's moving.

• where F is the frictional force — known as Stokes' drag — acting on the interface between the fluid and the particle
• η is the viscosity
• R is the radius of the spherical object
• V is the flow velocity