**Viscosity**

It is the property of the fluid due to which an opposing force comes into action whenever there is a relative motion between its different layers.

The backward dragging force is called viscous drag or viscous force which acts tangentially on the layers of the fluid in motion and always try to destroy its motion.

Generally liquids like water, alcohol etc. are less viscous than thick liquids like coal tar, blood glycerine.

According to Newton, the force of viscosity depends on

1) F ∝ A (Area of the layers in contact

2) F ∝ dv/dx (Velocity gradient b/w the two layers)

F ∝ Adv/dx

F = – ηAdv/dx

where η is the coefficient of viscosity of the fluid. It depends on the nature of the liquid and gives the measure of viscosity. The -ve sign shows that the viscous force acts in the opposite direction of the motion of the liquid.

Definition of coefficient of viscosity (η)

η = F/A(dv/dx)

η = Fdx/Adv

If A =1

dv/dx = 1

then F =η

Hence coefficient of viscosity of a liquid is defined as the tangential force required to maintain a unit velocity gradient between its two parallel layers each of unit area.

• SI Unit of η is Nsm^{-2} or Pa s or Poiseuille or decapoise.

• CGS Unit of η is dyne s cm^{-2} or Poise.

• Dimensional Formula is [ML^{-1}T^{-1}]

Effect of Temperature on the viscosity

The viscosity of the liquid decreases with temperature while it increases in the case of gases.

## Stoke’s Law

According to Stokes law, the backward viscous force acting on a small spherical body of radius **r** moving with uniform velocity **v** through a fluid of viscosity **η **is given by **F=6πηrv**This formula can be derived from the dimensional method.

**Terminal Velocity**

The maximum constant velocity acquired by a body falling through a viscous medium is called its terminal velocity.

**Expression for Terminal Velocityv=2 r ^{2} (ρ-σ)g/9η**

**Numerical Problem : • Eight raindrops of radius 1 mm each falling down with terminal velocity 5 cm/s coalesce to form a bigger drop. Find the terminal velocity of the bigger drop. [20 cm/s]**

**Types of liquid flow :1) Streamline Flow (Steady Flow) – **It is a type of liquid flow in which each particle of the liquid follow exactly the path of its preceding particle and has the same velocity as that of its preceding particle while crossing through that point.

**Streamline**– The path taken by a liquid particle under steady flow is called streamline.

**2) Turbulent Flow – **~~When the rate of flow of liquid is large, the flow becomes turbulent. In this type of flow, the velocity of the liquid at any point varies rapidly and randomly with time. The circular motion of liquid (called eddies) also generated. ~~~~For Example – An obstacle placed in the path of a fast moving liquid causes turbulence. ~~When a liquid moves with a velocity greater than its, critical velocity, the motion of the particles of the liquid becomes disorderly or irregular. such a flow is called turbulent flow.

3**) Laminar Flow – **It is a steady flow of the liquid in which liquid moves in layer i.e. each layer slides over the other without mixing with each other. **Critical Velocity**

It is the velocity of the liquid flow up to which its flow streamlines and above which its flow becomes turbulent.

v_{c} = kη/ρr

Where

k = constant

η =coefficient of the viscosity of the liquid

ρ = density of the liquid

r = radius of the tube**Reynolds Number** (R_{e} or N_{R})

It is a dimensionless number whose value decides the nature of the flow of a liquid through a pipe.

Re = ρDv_{c}/η

where

ρ = density of the liquid

D = Diameter of the tube through which liquid is flowing

v_{c} = critical velocity of the liquid

η =coefficient of the viscosity of the liquid

• For R_{e}<1000 – The flow is streamlined or laminar.

• For R_{e}> 2000 – The flow is turbulent.

• 1000<R_{e}<2000 – The flow of the liquid is unstable changing from streamline to turbulent flow.**Note:** Renold number is a pure number that’s why it don’t have any units and dimensions.**Equation of continuity**

av = constant or

a_{1}v_{1}=a_{2}v_{2}

**Numerical Problem• Water flows through a horizontal pipe of radius 2 cm at a speed of 5 m/s. Estimate the radius of the nozzle of the pipe, if we want to set up the speed of the water coming out to be 20 m/s. [1 cm]Ideal Fluid**

An ideal fluid is the one which is non-viscous, incompressible and its flow is steady (streamline) and irrotational.

**Bernoulli’s Theorem**

It states that when a non-viscous and incompressible fluid flows in streamline flow through a pipe of varying cross-sectional area, then at every point of its flow the sum of pressure energy, kinetic energy and potential energy per unit volume remains constant.

Mathematically

P + ½ ρv

^{2}+ ρgh = constant

This theorem is based on the principle of conservation of energy.