- Some liquids like water Kerosene flow very easily while some liquids like honey, glycerine do not flow easily.
- The property by virtue of which a liquid or a fluid opposes the relative motion between two adjacent layers is called viscosity.
- The force which opposes the relative motion between two adjacent layers in a fluid is called viscous force for viscous drag or internal friction. The force of viscosity in case of liquids is analogous to the friction force in case of solids.

** Practical Examples of Viscosity:**

- The body of fishes, the hull of ships, torpedoes, and submarines is such that the opposition to their motion in the liquid (water) is minimum. The shapes of bullets, arrows, racing cars, jet planes, bombs are such that opposition to their motion in air i.e. viscous resistance is minimum. This design work is called streamlining of the shapes and shapes themselves are called streamlined shapes.
- The quality of fountain ink depends on the viscosity of the ink.
- Blood circulation in our body depends on the viscosity of blood.
- Lubricating oils used in heavy machines is more viscous than that used in light machines. Thus car engine requires lubricating oil with higher viscosity than lubricating oil required for the watch.

**Types of Flow:**

**Streamline Flow:**

- When liquid flows slowly so that its velocity is less than a certain value called critical velocity of flow or the streamline velocity of flow, then the flow of liquid is called streamline flow. Streamline flow is also called as laminar flow.
- e.g. Flow of water in canal

**Characteristics of Streamline Flow:**- For streamline flow, the velocity of flow of liquid is less than or equal to critical velocity.
- In streamline flow, the velocity at any given point of the liquid always remains constant.
- In streamline flow liquid can be assumed to be made up of distinct layers sliding over each other such that each layer has constant velocity.
- In streamline flow molecules in each layer move forward in the same layer only.
- For streamline flow velocity gradient is constant.

**Turbulent flow:**

- When liquid flows fast so that its velocity is greater than a certain value called critical velocity of flow or the streamline velocity of flow, then the flow of liquid is called turbulent flow. Turbulent flow is also called as nonlaminar flow.
- e. g. Flow of water in river

**Characteristics of Turbulent Flow:**- For turbulent flow, the velocity of flow of liquid is greater than critical velocity.
- In turbulent flow, the velocity at any given point of the liquid does not remain constant.
- In turbulent flow, the liquid does not move forward in form of layers.
- In turbulent flow molecules of liquid can move in any layer.
- For turbulent flow velocity gradient is not constant.

#### Origin of Viscosity:

- Consider the motion of fluid over a horizontal surface such that its velocity of flow is slow then we can observe that the velocity of the layer which is contact with the surface is zero.
- Velocities of other layers increase with an increase in the distance of the layer from the solid surface. If we consider a particular layer we can see that the layer immediately above it moves faster than the layer immediately below it.
- Thus upper layer will try to move lower layer with higher velocity and vice versa. In fact, the two layers together try to destroy relative motion as if some backward drag were acting tangentially between the two layers. This backward drag opposing the flow of liquid is called viscosity.

- The velocity of any layer of liquid depends on its distance from fixed layer. Let v and v + dv be the velocities of two layers situated at a distance of x and x + dx from the fixed layer
- Then the ratio dv/dx is called the velocity gradient.
- Velocity gradient of flow is defined as the rate of change of velocity with respect to the distance from the fixed layer.

**Newton’s Law of Viscosity:**

**Statement:**

- Viscous force (F) acting on any layer of liquid is directly proportional to
- the area of the layer (A) and
- the velocity gradient (dv/dx)

**Explanation:**

By Newton’s law of viscosity

F ∝ A ….(1)

F ∝ dv/dx ……..(2)

from (1) & (2)

F ∝ A . dv/dx

F = η . A . dv/dx

- This formula is known as Newton’s formula of viscosity. Where η (eta) is called coefficient of viscosity. The value of η depends upon temperature & the nature of the liquid.

**Coefficient of Viscosity:**

By Newton’s formula

F = η . A . dv/dx

let A = 1 unit and dv/dx = 1 unit

then, F = η i.e. η = F

- The coefficient of a viscosity of a liquid is defined as the tangential viscous force acting on the unit area of liquid lair per unit velocity gradient.
- SI Unit of η is N.s/m
^{2}. CGS unit of η is poise - 1 poise = 0.1 N.s/ m
^{2 }and 1 centipoise = 10^{– 2 }poise

**Relation between S.I. and c.g.s. units of Coefficient of Viscosity:**

- The S.I. unit of viscosity is Ns / m
^{2}and c.g.s unit is dyne-s / cm^{2}i.e. poise.

Now, 1 N = 10 ^{5} dyne and 1m = 100 cm

1 Ns / m^{2 }= 10 ^{5} dyne × s / (100 cm)^{2 }= 10 ^{5} dyne × s / 10^{4} cm^{2 }= 10 dyne × s / cm^{2}

**Dimension of Coefficient of Viscosity (η****) :**

By Newton’s formula,

Hence dimensions of coefficient of viscosity are [L^{-1}M^{1}T^{-1}]

** Stokes’ Law:**

**Statement:**

- The viscous drag on a spherical body moving with sufficiently small velocity through a viscous medium or the medium moves past the spherical body with streamline flow is directly proportional to
- the radius ‘r’ of the sphere,
- the relative velocity ‘v’ of the sphere through the medium. And
- the coefficient of viscosity ‘η’ of the medium.

**Mathematical Expression of Stokes’ Law:**

F = 6 π η r v

** Derivation of Stokes’ law Using Dimensional Analysis:**

- Let (F) be the viscous force acting on a spherical body such that it depends on the radius of spherical body (r) terminal velocity (v) and coefficient of viscosity (η).

F ∝ r ^{x} ….(1)

F ∝ v ^{y} ….(2)

F ∝ h ^{z} ….(3)

From (1), (2) & (3)

F ∝ r ^{x} v ^{y} η ^{z}

∴ F = k r ^{x} v ^{y} η ^{z} ……….(4)

∴ [F] = [r]^{x} [v]^{y} [η]^{z}

∴ [L^{1}M^{1}T^{-2}] = [L^{1}M^{0}T^{0}]^{x} [L^{1}M^{0}T^{-1}]^{y} [L^{-1}M^{1}T^{-1}]^{z}

∴ [L^{1}M^{1}T^{-2}] = [L^{x}M^{0}T^{0}] [L^{y}M^{0}T^{-y}] [L^{-z}M^{z}T^{-z}]

∴ [L^{1}M^{1}T^{-2}] = [L^{x+y-z}M^{z}T^{-y -z}]

Considering equality of two sides we have

x + y – z = -1, z = 1, -y – z = -2 i.e. y + z = 2

Substituting z = 1 in equation y + z = 2, we get y = 1

Substituting y = 1, z = 1 in equation x + y + z = -1, we get x = 1

Substituting x = 1, y = 1, and z = 1 in equation (4) we get

F = k r v η

Experimentally the value of k is found to be 6π

∴ F = 6π η r v

Thus Stoke’s law is proved.

**Terminal Velocity:**

- When a smooth rigid body is allowed to fall through a viscous fluid it carries with it the layers of the liquid in contact. This tends to produce a relative motion between the layers of fluid which is opposed by viscous force.
- The opposing force increases with the increases in the velocity of the body in case of small bodies the opposing force becomes equal to driving force and body start going down in fluid with constant velocity. This constant velocity is called terminal velocity.

**Expression for Terminal Velocity of a Spherical Body:**

- Consider a small sphere of density ρ and radius ‘r’ falls under gravity through a medium of density σ and coefficient viscosity η. As the sphere begins to fall down its velocity goes on increasing due to gravitational force acting on it. But as the velocity increases the opposing viscous force also increases. When the viscous force becomes equal and opposite to the gravitational force, the resultant force acting on the sphere becomes zero and sphere begin to fall with constant velocity it has already acquired. This velocity is known as terminal velocity.

Downward force = Weight of the body = mg = V ρ g

Upward force = viscous force + upthrust due to medium

∴ Upward force = Viscous force + weight of displaced medium

∴ Upward force = 6π η r v + V σ g

But, Downward force = Upward force

∴ V ρ g = 6π η r v + V σ g

∴ 6π η r v = V ρ g – V σ g

∴ 6π η r v = V (ρ – σ) g

This is an expression for terminal velocity of a sphere