Notes : Define Viscosity , Coefficient , Newton law, Units , Dimensions

Notes : Define Viscosity , Coefficient , Newton law, Units , Dimensions class 11 physics chapter 9 Mechanical Properties of fluids - Physics Kund

Introduction

Fluids (liquids and gases) can flow because their molecules are free to move. During flow, one layer of a fluid moves relative to another layer. Due to intermolecular forces, a resistive force develops between adjacent layers. This property of a fluid is called viscosity. It is one of the most important mechanical properties of fluids and explains why some liquids like water flow easily while others such as honey, glycerine, and engine oil flow slowly.

An ideal fluid is a hypothetical fluid that is incompressible and has zero viscosity. In reality, every fluid possesses some viscosity.

Viscosity

Viscosity is the property of a fluid by virtue of which it opposes the relative motion between its adjacent layers. It is also known as the internal friction of a fluid.

Whenever different layers of a fluid move with different velocities, a resisting force acts between them. This force opposes the relative motion of the layers and is called the viscous force or viscous drag.

Definition

Viscosity is the property of a fluid due to which an internal resisting force acts between its adjacent layers whenever there is relative motion between them.

No-Slip Condition

A fluid layer in contact with a solid surface always has the same velocity as that surface.

If the surface is at rest, the adjacent fluid layer also remains at rest.
If the surface moves with velocity v, the adjacent fluid layer also moves with velocity v.

This condition is known as the no-slip condition.

Velocity Distribution Between Parallel Plates

Consider a liquid enclosed between two large parallel plates separated by a distance l. The lower plate is fixed while the upper plate moves with a constant velocity v.

The fluid layer touching the lower plate has zero velocity.
The fluid layer touching the upper plate moves with velocity v.
The velocities of intermediate layers increase uniformly from the lower plate to the upper plate.

Thus, every layer has a different velocity, producing relative motion between adjacent layers.

Velocity Distribution in a Pipe

When a liquid flows through a horizontal pipe, the velocity of the liquid is not the same at every point.

Velocity is maximum along the central axis of the pipe.
Velocity gradually decreases towards the walls.
Velocity becomes zero at the walls because of the no-slip condition.

This variation of velocity across the pipe is called the velocity profile.

Shearing Stress

When the upper plate is pulled by a tangential force F, the liquid experiences a shearing force. The tangential force acting per unit area is called shearing stress.

$$ \text{Shearing Stress}=\frac{F}{A} $$

where

F = Tangential force applied
A = Area of the plate

Shear Strain

If the upper plate moves through a small distance $\Delta x$, the shape of the liquid changes.

Shear strain is defined as the ratio of the lateral displacement of the upper layer to the perpendicular distance between the layers.

$$ \text{Shear Strain}=\frac{\Delta x}{l} $$

where

$\Delta x$ = Horizontal displacement
l = Distance between the plates

Strain Rate

In a flowing fluid, shear strain changes continuously with time. Therefore, the rate of change of shear strain is called the strain rate.

$$ \text{Strain Rate}=\frac{\Delta x}{l\Delta t} $$

Since

$$ v=\frac{\Delta x}{\Delta t} $$

therefore,

$$ \text{Strain Rate}=\frac{v}{l} $$

Velocity Gradient

The velocity gradient is defined as the rate of change of velocity per unit perpendicular distance between two adjacent layers of a flowing fluid.

$$ \frac{dv}{dx} $$

A larger velocity gradient means a larger difference in velocity between adjacent layers and hence a greater viscous force.

Coefficient of Viscosity (NCERT Definition)

The coefficient of viscosity is defined as the ratio of shearing stress to strain rate.

$$ \eta=\frac{\text{Shearing Stress}}{\text{Strain Rate}} $$

Substituting the expressions for shearing stress and strain rate,

$$ \eta=\frac{\dfrac{F}{A}}{\dfrac{v}{l}} $$

$$ \eta=\frac{Fl}{Av} $$

where

$\eta$ = Coefficient of viscosity
F = Tangential force
A = Area of contact
v = Velocity of the upper plate
l = Distance between the plates

Newton's Law of Viscosity

According to Newton's law of viscosity, the viscous force acting between two adjacent layers of a fluid is directly proportional to the area of contact and the velocity gradient.

$$ F\propto A $$

$$ F\propto\frac{dv}{dx} $$

Combining the above relations,

$$ F\propto A\frac{dv}{dx} $$

Introducing the proportionality constant $\eta$,

$$ F=-\eta A\frac{dv}{dx} $$

The negative sign indicates that the viscous force always acts opposite to the direction of relative motion.

Definition of Coefficient of Viscosity

If the area of contact is unity and the velocity gradient is unity, then the viscous force acting between the layers is equal to the coefficient of viscosity.

$$ A=1,\qquad \frac{dv}{dx}=1 $$

Therefore,

$$ \eta=F $$

Definition: The coefficient of viscosity is defined as the viscous force acting per unit area between two adjacent layers of a fluid when the velocity gradient between them is unity.

Physical Significance of Coefficient of Viscosity

The coefficient of viscosity measures the resistance offered by a fluid to flow.

A fluid having a large value of viscosity flows slowly.
A fluid having a small value of viscosity flows easily.
Honey and glycerine have high viscosity.
Water has comparatively low viscosity.
Air has very low viscosity.

Units of Coefficient of Viscosity

SI Unit

The SI unit of coefficient of viscosity is Pascal-second (Pa·s).

$$ 1~\mathrm{Pa\cdot s}=1~\mathrm{N\,s\,m^{-2}} $$

The SI unit is also called Poiseuille (Pl).

CGS Unit

The CGS unit of coefficient of viscosity is Poise (P).

$$ 1~\mathrm{Poise}=1~\mathrm{dyne\,s\,cm^{-2}} $$

Relation Between SI and CGS Units

$$ 1~\mathrm{Pa\cdot s}=10~\mathrm{Poise} $$

Dimensional Formula

From Newton's law of viscosity,

$$ F=\eta A\frac{dv}{dx} $$

Therefore, the dimensional formula of coefficient of viscosity is

$$ [\eta]=[ML^{-1}T^{-1}] $$

Effect of Temperature on Viscosity

For Liquids

As the temperature of a liquid increases, its viscosity decreases because the intermolecular forces become weaker, allowing the layers of the liquid to move more freely.

$$ \eta\propto\frac{1}{\sqrt{T}} $$

For Gases

As the temperature of a gas increases, its viscosity increases because the molecules move faster and transfer momentum more effectively between adjacent layers.

$$ \eta\propto\sqrt{T} $$

Applications of Viscosity

Selection of suitable lubricants for machines.
Design and working of hydraulic brakes and shock absorbers.
Blood circulation through arteries and veins.
Manufacture of paints, varnishes, inks and cosmetics.
Damping unwanted vibrations in measuring instruments.
Transportation of petroleum products through pipelines.

Difference Between Viscosity and Solid Friction

Viscosity	Solid Friction
Acts between adjacent layers of a fluid.	Acts between two solid surfaces.
Depends on the velocity gradient.	Depends mainly on the normal reaction.
Acts throughout the fluid.	Acts only at the contact surface.
Depends upon the nature of the fluid and temperature.	Depends upon the nature of the surfaces in contact.

Important Formulae

$$ \text{Shearing Stress}=\frac{F}{A} $$

$$ \text{Shear Strain}=\frac{\Delta x}{l} $$

$$ \text{Strain Rate}=\frac{v}{l} $$

$$ \eta=\frac{\text{Shearing Stress}}{\text{Strain Rate}} $$

$$ \eta=\frac{\dfrac{F}{A}}{\dfrac{v}{l}} $$

$$ \eta=\frac{Fl}{Av} $$

$$ F=-\eta A\frac{dv}{dx} $$

$$ \frac{dv}{dx} $$

$$ [\eta]=[ML^{-1}T^{-1}] $$

$$ 1~\mathrm{Pa\cdot s}=10~\mathrm{Poise} $$

Coefficient of Viscosity of Some Common Fluids

Fluid	Temperature	Coefficient of Viscosity (Pa·s)
Air	20°C	0.018 × 10^-3
Water	0°C	1.8 × 10^-3
Water	20°C	1.0 × 10^-3
Water	100°C	0.3 × 10^-3
Blood	37°C	2.7 × 10^-3
Engine Oil	30°C	250 × 10^-3
Glycerine	20°C	1.5

Key Points for Revision

Viscosity is the internal friction of a fluid.
All real fluids possess viscosity, whereas an ideal fluid has zero viscosity.
The no-slip condition states that the fluid in contact with a solid surface has the same velocity as the surface.
Velocity is maximum at the centre of a pipe and zero at its walls.
The coefficient of viscosity is the ratio of shearing stress to strain rate.
Newton's law of viscosity is given by $$ F=-\eta A\frac{dv}{dx} $$
The viscosity of liquids decreases with increase in temperature, whereas the viscosity of gases increases.
The SI unit of viscosity is Pa·s (Poiseuille), and the CGS unit is Poise.

Frequently Asked Questions (FAQs)

1. What is viscosity?

Viscosity is the property of a fluid by virtue of which it opposes the relative motion between its adjacent layers. It is also called the internal friction of a fluid.

2. What is viscous force?

Viscous force is the internal resisting force acting between adjacent layers of a moving fluid.

3. What is the coefficient of viscosity?

The coefficient of viscosity is the ratio of shearing stress to strain rate. It is also defined as the viscous force acting per unit area when the velocity gradient is unity.

4. What is the SI unit of viscosity?

The SI unit is Pascal-second (Pa·s) or Poiseuille (Pl).

5. What is the CGS unit of viscosity?

The CGS unit of viscosity is Poise (P).

6. Why does the viscosity of liquids decrease with temperature?

On heating, intermolecular forces become weaker, allowing the liquid layers to move more freely.

7. Why does the viscosity of gases increase with temperature?

As temperature increases, gas molecules move faster and transfer momentum more effectively, increasing viscosity.

Multiple Choice Questions (MCQs)

Viscosity is also known as
- A. Surface tension
- B. Internal friction
- C. Elasticity
- D. Compressibility
Answer: B
The SI unit of viscosity is
- A. Newton
- B. Pascal
- C. Pascal-second
- D. Joule
Answer: C
The CGS unit of viscosity is
- A. Stoke
- B. Poise
- C. Dyne
- D. Pascal
Answer: B
The dimensional formula of viscosity is
- A. $$[MLT^{-2}]$$
- B. $$[ML^{-1}T^{-1}]$$
- C. $$[ML^2T^{-2}]$$
- D. $$[M^0LT]$$
Answer: B
The viscosity of liquids with increase in temperature
- A. Increases
- B. Decreases
- C. Remains constant
- D. Becomes zero
Answer: B
The viscosity of gases with increase in temperature
- A. Increases
- B. Decreases
- C. Remains constant
- D. Becomes zero
Answer: A
According to Newton's law of viscosity, the viscous force is
- A. $$F=\eta A$$
- B. $$F=-\eta A\frac{dv}{dx}$$
- C. $$F=\eta v$$
- D. $$F=\eta\frac{dx}{dv}$$
Answer: B
Velocity is maximum in a pipe at the
- A. Wall
- B. Centre
- C. Bottom
- D. Surface
Answer: B

True or False

All real fluids possess viscosity. (True)
An ideal fluid has zero viscosity. (True)
Viscosity is the internal friction of a fluid. (True)
The velocity of a fluid is maximum at the wall of a pipe. (False)
The viscosity of liquids decreases with increase in temperature. (True)
The viscosity of gases increases with increase in temperature. (True)
Poise is the SI unit of viscosity. (False)
Newton's law of viscosity applies to Newtonian fluids. (True)

Fill in the Blanks

Viscosity is also called internal friction.
An ideal fluid has zero viscosity.
The SI unit of viscosity is Pascal-second (Pa·s).
The CGS unit of viscosity is Poise.
The velocity of a liquid in a pipe is maximum at the centre.