**This article is formulated according to the 10th chapter in the NCERT Syllabus of 11th Std Physics OR 1st PUC Physics.**

## Fluids

The materials that can flow are called fluids. Liquids and gases are collectively known as fluids. Unlike a solid, a fluid has no definite shape of its own.

## Hydrostatics (Fluids statics)

The study of fluids at rest is known as hydrostatics.

## Hydrodynamics (Fluids dynamics)

The study of fluids in motion is termed hydrodynamics.

## Density

Density of a substance is defined as the mass per unit volume of the substance.

𝝆 = 𝑴/V

SI unit of density is 𝑘𝑔𝑚^{−3} and dimensions are 𝑀𝐿^{−3}

## Relative density

It is defined as the ratio of the density of the substance to the density of water.

𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒/𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑎𝑡𝑒𝑟

Relative density has no unit. It is a pure number.

**Note:** Density of water at 4°𝐶 is maximum and is equal to 1000𝑘𝑔𝑚^{−3} Pressure: The pressure is defined as the magnitude of the force acting perpendicular to the surface of an object per unit area of the object.

𝑷 = 𝑭/𝑨

SI unit of pressure is 𝑁𝑚^{−2} or 𝑝𝑎𝑠𝑐𝑎𝑙(𝑃𝑎). Dimensions of pressure is 𝑀𝐿^{−1}𝑇^{−2}

**Note:** Pressure is a scalar quantity because hydrostatic pressure is transmitted equally in all directions, when force is applied, which shows that a definite direction is not associated with pressure.

## Measurement of pressure

The normal force exerted by the fluid at a point may be measured and the arrangement is as shown. It consists of an evacuated chamber with a spring that is calibrated to measure the force acting on the piston. This device is placed at a point inside the fluid. The inward force exerted by the fluid on the piston is balanced by the outward spring force and is thereby measured.

## Pascal’s Law

“The pressure in a fluid at rest is same at all points if they are at the same height.”

**Pascal’s Explanation:** Consider an element 𝐴𝐵𝐶-𝐷𝐸𝐻 in the form of a right-angled prism in the fluid. As the element is very small, every part of it is located at the same height from the liquid surface. Then the effect of gravity is same at all these points. Let 𝐹_{𝑎}, 𝐹_{𝑏} and 𝐹_{𝑐} be the normal forces exerted by the fluid on the faces 𝐵𝐸𝐻𝐶, 𝐴𝐷𝐻𝐶 and 𝐴𝐷𝐸𝐵 respectively. Let 𝐴_{𝑎}, 𝐴_{𝑏} and 𝐴_{𝑐} be the area of the faces 𝐵𝐸𝐻𝐶, 𝐴𝐷𝐻𝐶 and 𝐴𝐷𝐸𝐵 respectively.

Since the element 𝐴𝐵𝐶-𝐷𝐸𝐻 is in equilibrium, net force acting on that element should be zero.

𝐹_{𝑐} = 𝐹_{𝑏} sin 𝜃

𝐹_{𝑎} = 𝐹_{𝑏} cos 𝜃

By geometry,

𝐴_{𝑐} = 𝐴_{𝑏} sin 𝜃

𝐴_{𝑎} = 𝐴_{𝑏} cos 𝜃

Pressure on 𝐴𝐷𝐸𝐵 = 𝐹_{𝑐}/𝐴_{𝑐} = 𝐹_{𝑏} sin 𝜃/𝐴_{𝑏} sin 𝜃 = 𝐹_{𝑏}/𝐴_{𝑏} − − − (1)

Pressure on 𝐵𝐸𝐻𝐶 = 𝐹_{𝑎}/𝐴_{𝑎} = 𝐹_{𝑏} 𝑐𝑜𝑠 𝜃/𝐴_{𝑏} 𝑐𝑜𝑠 𝜃 = 𝐹_{𝑏}/𝐴_{𝑏} − − − (2)

Pressure on 𝐴𝐷𝐻𝐶 = 𝐹_{𝑏}/𝐴_{𝑏} − − − (3)

The above equations says, 𝐹_{𝑐}/𝐴_{𝑐} = 𝐹_{𝑎}/𝐴_{𝑎} = 𝐹_{𝑏}/𝐴_{𝑏} implies that 𝑃_{𝑎} = 𝑃_{𝑏} = 𝑃_{𝑐}

Hence, pressure exerted is same in all directions in a fluid at rest.

## Expression for pressure

Consider a vessel containing a liquid of density 𝜌, which is in equilibrium. Consider a cylindrical element of fluid having area of base 𝐴 and height ℎ. As the fluid is in equilibrium, the net force acting on it is zero.

𝐹_{1} + 𝑚𝑔 − 𝐹_{2} = 0

But 𝑃 = 𝐹/𝐴 therefore 𝐹_{1} = 𝑃_{1}𝐴 and 𝐹_{2} = 𝑃_{2}𝐴

𝑃_{1}𝐴 + 𝑚𝑔 − 𝑃_{2}𝐴 = 0

𝑃_{2}𝐴 − 𝑃_{1}𝐴 = 𝑚𝑔

(𝑃_{2} − 𝑃_{1})𝐴 = 𝑚𝑔

(𝑃_{2} − 𝑃_{1})𝐴 = (𝐴ℎ𝜌)𝑔

(𝑃_{2} − 𝑃_{1}) = 𝜌𝑔ℎ

If the 𝑝𝑜𝑖𝑛𝑡 1 is in the figure is shifted to the top of the fluid, which is open to atmosphere, 𝑃_{1 }may be replaced by atmospheric pressure 𝑃_{𝑎} and 𝑃_{2} by 𝑃 then,

𝑃 − 𝑃_{𝑎} = 𝜌𝑔ℎ

𝑷 = 𝑷_{𝒂} + 𝝆𝒈𝒉

The above equation tells us that pressure depends on height, so that the pressure in a fluid at rest is same at all points, if they are at same height.

## Gauge Pressure

The pressure 𝑃 at depth ℎ from the surface of the fluid is greater than the atmospheric pressure by an amount 𝜌𝑔ℎ. The excess pressure at depth ℎ is called Gauge pressure. Gauge pressure is the difference of the actual pressure and the atmospheric pressure.

## Atmospheric pressure

The pressure of the atmosphere at any point is equal to the weight of a column of air of unit cross-sectional area extending from that point to the top of the atmosphere. At sea level it is 1.013 × 10^{5}𝑃𝑎 and

1 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 1.013 × 10^{5}𝑃𝑎

## Hydrostatic paradox

Consider three vessels 𝐴, 𝐵 and 𝐶 of different shapes. They are connected at the bottom by a horizontal pipe. On filling with water, the level in the vessel is same. Though they hold different amount of water, any how the pressure of the water at the bottom is same. This result is known as hydrostatic paradox.

## Measurement of atmospheric pressure

## Mercury Barometer

Torricelli invented a mercury barometer to measure the atmospheric pressure. It consists of a long glass tube closed at one end and filled with mercury and inverted into a trough of mercury as shown. The space in the tube above the mercury column is almost empty and can be neglected. This space is called Torricelli space. Now the pressure at 𝐴 = pressure at 𝐵, which are at same level. But pressure at 𝐴 = atmosphere pressure, 𝑃_{𝑎}

𝑃_{𝑎} = pressure at 𝐵

𝑷_{𝒂} = 𝝆𝒈𝒉 where 𝜌 is density of mercury.

**Note:**

(1) At sea level, the mercury column in the barometer is found to have a height of 76𝑐𝑚. The pressure equivalent to this column is 1 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (1 𝑎𝑡𝑚).

(2) A common way of stating pressure is in terms of 𝑐𝑚 or 𝑚𝑚 𝑜𝑓 𝐻𝑔.

(3) A pressure equivalent to 1𝑚𝑚 is called a 𝑡𝑜𝑟𝑟.

1𝑚𝑚 𝑜𝑓 𝐻𝑔 = 1 𝑡𝑜𝑟𝑟 = 133𝑃_{𝑎}

(4) 𝑚𝑚 𝑜𝑓 𝐻𝑔 and 𝑡𝑜𝑟𝑟 are used in medicine and physiology. In meteorology, a common unit is the 𝑏𝑎𝑟 and 𝑚𝑖𝑙𝑙𝑖 𝑏𝑎𝑟. (1𝑏𝑎𝑟 = 10^{5}𝑃𝑎)

## Open tube monometer

It is a useful instrument for measuring pressure differences. It consists of a U-tube containing a low-density liquid for measuring small pressure differences or a high-density liquid for large pressure differences. One end of the tube is open to the atmosphere and the other end is connected to the system whose pressure is to be measured. The pressure 𝑃 at 𝐴 is equal to the pressure at point 𝐵. The gauge pressure is 𝑃 − 𝑃_{𝑎 }= 𝜌𝑔ℎ which is proportional to the height of the liquid.

## Pascal’s law for transmission of fluid pressure

Whenever external pressure is applied to any part of the fluid contained in a vessel, it is transmitted undiminished and equally in all directions.

## Applications of the law

Hydraulic machines work based on Pascal’s law for the transmission of fluid pressure. In these devices, fluids are used for transmitting pressure.

## Hydraulic machines

The devices which work on the Pascal’s law are known as hydraulic machines. Ex: Hydraulic lift, hydraulic brakes etc.

## Hydraulic lift

It consists of a horizontal fluid filled container. Both the ends of the container are fitted with two cylinders having pistons of different area of cross-section as shown in the figure.

**Explanation:** Let the force of magnitude 𝐹_{1} be applied to a small piston of surface area 𝐴_{1. }This generates a pressure,

𝑃 = 𝐹_{1}/𝐴_{1}

This pressure is transmitted undiminished through the fluid to a larger piston of surface area 𝐴_{2}

Hence,

𝑃 = 𝐹_{1}/𝐴_{1} = 𝐹_{2}/𝐴_{2}

𝑭_{𝟐} = 𝑭_{𝟏}(𝑨_{𝟐}/𝑨_{𝟏})

As 𝐴_{2} > 𝐴_{1}, 𝐹_{2} > 𝐹_{1}

This shows that a small force applied on the smaller piston appears as a large force on the larger piston.

## Hydraulic brakes

When we apply a little force on the pedal with our foot, the master piston 𝑃 moves inside the master cylinder and the pressure caused is transmitted through the brake oil to act on a piston of large area (𝑃_{1} and 𝑃_{2}). A large force acts on the piston and is pushed down expanding the brake shoes (𝑆_{1} and 𝑆_{2}) against the brake lining and retard the motion of the wheel.

## Archimedes’ principle

“When a body is immersed completely or partially in a liquid it appears to lose a part of its weight and this apparent loss of weight is equal to the weight of the liquid displaced by the body.”

## Fluid dynamics

Fluid dynamics deals with fluid flow – the science of flow of fluids.

## Types of flow

The flow of fluids is divided into two types, namely.

**(i) Streamline (Steady) flow**

**(ii) Turbulent flow**

## Streamline (Steady) flow

If fluid flows such that the velocity of its particles at a given point remains constant with time, then the fluid is said to have a streamlined flow.

**Explanation:** Consider a liquid passing through a tube as shown. If the velocity of the flow is small, all the particles which come to 𝐴 will have the same speed and will move in the same direction. As the particle goes from 𝐴 and 𝐵 its speed and direction may change, but all the particles at 𝐵 will have the same speed, also if one particle through 𝐴 has gone through 𝐵 then all the particles passing through 𝐴 go through 𝐵.

## Streamline

The path followed by the particle of a fluid in a streamlined flow or steady flow is called streamline.

## Properties of streamlines

(i) The tangent at any point on the line of flow gives the direction of flow.

(ii) The streamline may curve and bend, but they cannot cross each other.

## Equation of continuity

Consider the streamlined flow of a fluid of density 𝜌, through a pipe 𝐴𝐵 of the non-uniform cross-section. Let 𝑣_{1} be the velocity of the liquid entering at 𝐴 of the cross-sectional area 𝑎_{1} normal to the surface. Let 𝑣_{2} be the velocity with which it flows out at 𝐵 where the area of cross-section 𝑎_{2} normal to the surface.

Mass of fluid entering at 𝐴 per second = 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜌(𝑎_{1}𝑣_{1})

Mass of fluid entering at 𝐵 per second = 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜌(𝑎_{2}𝑣_{2})

Since the flow is steady, The mass of the fluid entering per second is equal to the mass of the fluid flowing out per second.

𝜌𝑎_{1}𝑣_{1} = 𝜌𝑎_{2}𝑣_{2}

𝑎_{1}𝑣_{1} = 𝑎_{2}𝑣_{2}

This equation is called the equation of continuity and it is the statement of the law of conservation of mass in the flow of incompressible fluids. In general, 𝒂𝒗 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

**Note:** The product 𝑎𝑣 gives the volume flux or flow rate and remains constant throughout the pipe of flow. Thus, at narrower portions where the streamlines are closely packed, velocity increases and vice versa.

## Turbulent flow

When the speed of flow exceeds a limiting value called critical velocity, the orderly motion of the fluid is lost and it acquires an unsteady motion called turbulent motion. **Ex:** Floods, hurricanes, whirlpools, etc.

## Differences between streamlined flow and turbulent flow

Streamline flow | Turbulent flow |

It is a regular and orderly flow | It is an irregular and disorderly flow |

The lines of flow are parallel to each other | The lines of flow are not parallel to each other |

The velocity of the flow is less than the critical velocity | The velocity of the flow is greater than the critical velocity |

Different particles cross a given point with the same velocity | Different particles cross a given point with different velocity |

## Bernoulli’s principle

“For streamline flow of an ideal (non-viscous, incompressible) fluid, the sum of pressure, the kinetic energy per unit volume and potential energy per unit volume remains a constant.”

𝑷 + (𝟏/𝟐)𝝆𝒗^{𝟐} + 𝝆𝒈𝒉 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

**Proof:** Consider an incompressible and non-viscous fluid flowing through a pipe 𝐵𝐸 of varying cross-sectional area. Let the flow is streamlined. Let 𝑃1 and 𝑃2 be the pressure and 𝑎1 and 𝑎2 be the area of the cross-section at 𝐵 and 𝐷 respectively. Let 𝑣1 be the speed of the fluid at 𝐵 and 𝑣_{2} at 𝐷. In a small interval of time ∆𝑡, the fluid at 𝐵 moves through a distance 𝑣_{1}∆𝑡 to 𝐶. At the same time the fluid, initially at 𝐷 moves through a distance 𝑣_{2}∆𝑡 to 𝐸. The work done on the fluid in the region 𝐵𝐶 is, 𝑊_{1} = 𝑓𝑜𝑟𝑐𝑒 × 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡

𝑊_{1 }= 𝑃_{1}𝑎_{1 }× 𝑣_{1}∆𝑡

𝑊1 = 𝑃_{1}𝑎_{1}𝑣_{1}∆∆𝑡

𝑊_{1} = 𝑃_{1}∆𝑉

Work done on the fluid in the region 𝐷𝐸 against the pressure is

𝑊_{2} = 𝑃_{2}𝑎_{2}𝑣_{2}∆𝑡

Since the flow is streamline, 𝑎_{1}𝑣_{1}∆𝑡 = 𝑎_{2}𝑣_{2}∆𝑡

𝑊_{2} = 𝑃_{2}∆𝑉

The net work done on the fluid is, 𝑊 = 𝑊_{1} − 𝑊_{2}

𝑊 = (𝑃_{1} − 𝑃_{2})∆𝑉

A part of this work done is utilised to change the kinetic energy ane remaining is utilised in changing the gravitational potential energy. If 𝜌 is the density of the fluid, then the mass of the fluid passing through the pipe in time ∆𝑡 is,

∆𝑚 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑣𝑜𝑙𝑢𝑚𝑒

∆𝑚 = 𝜌 × 𝑎_{1}𝑣_{1}∆𝑡 = 𝜌 × 𝑎_{2}𝑣_{2}∆𝑡

∆𝑚 = 𝜌∆𝑉

Change in kinetic energy is, ∆𝐾 = (1/2)(∆𝑚)(𝑣_{2}^{2} − 𝑣_{1}^{2})

∆𝐾 = (1/2)𝜌∆𝑉(𝑣_{2}^{2} − 𝑣_{1}^{2})

Change in potential energy is,

∆𝑈 = ∆𝑚𝑔(ℎ_{2} − ℎ_{1})

∆𝑈 = 𝜌∆𝑉𝑔(ℎ_{2} − ℎ_{1})

Applying work-energy theorem for this volume of fluid, we get,

𝑊 = ∆𝑈 + ∆𝐾

(𝑃_{1} − 𝑃_{2})∆𝑉 = 𝜌∆𝑉𝑔(ℎ_{2} − ℎ_{1}) + (1/2)𝜌∆𝑉(𝑣_{2}^{2} − 𝑣_{1}^{2})

Dividing each term by ∆𝑉, (𝑃_{1} − 𝑃_{2}) = 𝜌𝑔(ℎ_{2} − ℎ_{1}) + (1/2)𝜌(𝑣_{2}^{2} − 𝑣_{1}^{2})

On rearranging, 𝑃_{1} + 𝜌𝑔ℎ_{1} + (1/2)𝜌𝑣_{1}^{2} = 𝑃_{2} + 𝜌𝑔ℎ_{2} + (1/2)𝜌𝑣_{2}^{2}

In general, 𝑷 + 𝝆𝒈𝒉 + (𝟏/𝟐)𝝆𝒗^{𝟐} = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

## Limitations of Bernoulli’s theorem

1) While proving Bernoulli’s theorem, we have assumed that no energy is lost due to friction, but in practice due to the viscosity of the fluid some energy is lost. The kinetic energy lost gets converted to heat. Thus, Bernoulli’s equation applies only to non-viscous fluids.

2) Bernoulli’s theorem cannot be applied to compressible fluids, as the elastic energy of the fluid is not taken into consideration.

3) Bernoulli’s equation does not hold good for non-steady or turbulent flow, as the velocity and pressure may vary with time and position.

## Speed of Efflux

The word efflux means fluid out flow.

## Torricelli’s law

“Torricelli discovered that, the expression for speed of efflux from an open tank is similar to that of a freely falling body.”

**Proof: **Consider a tank containing a liquid of density 𝜌 with a small hole in its side at a height 𝑦_{1} from the bottom of the tank. Let 𝑦_{2} be the height of the free surface of the liquid from the bottom of the tank.

Let 𝑃 be the pressure of air above the free surface of the liquid. From the equation of continuity,

𝑎_{1}𝑣_{1} = 𝑎_{2}𝑣_{2}

𝒗_{𝟐} = (𝒂_{𝟏}/𝒂_{𝟐})𝒗_{𝟏}

Where 𝑎_{1} → Cross-sectional area of the hole

𝑎_{2} → Cross-sectional area of the tank

𝑣_{1} →Velocity of fluid coming out of the hole

𝑣_{2} →Velocity of fluid at the top surface of the liquid

Since 𝑎_{2} ≫ 𝑎_{1}, top layer of the liquid is approximately at rest. i.e. 𝑣_{2} = 0.

Also the pressure of the fluid at the hole 𝑃_{1} is same as that of the atmospheric pressure 𝑃_{𝑎}. Applying Bernoulli’s equation to point 1 and 2,

𝑃_{𝑎} + 𝜌𝑔𝑦_{1} + (1/2)𝜌𝑣_{1}^{2} = 𝑃 + 𝜌𝑔𝑦_{2}

(1/2)𝜌𝑣_{1}^{2} = 𝑃 − 𝑃_{𝑎} + 𝜌𝑔𝑦_{2} − 𝜌𝑔𝑦_{1}

(1/2)𝜌𝑣_{1}^{2} = 𝑃 − 𝑃_{𝑎}_{ }+ 𝜌𝑔(𝑦_{2} − 𝑦_{1})

𝑣_{1}^{2} = (2/𝜌)(𝑃 − 𝑃_{𝑎} + 𝜌𝑔ℎ) (since 𝑦_{2} − 𝑦_{1 }= ℎ)

𝑣_{1}^{2} = [2(𝑃 − 𝑃_{𝑎})/𝜌] + 2𝑔ℎ

𝑣_{1} = √[(2(𝑃 − 𝑃_{𝑎})/𝜌) + 2𝑔ℎ]

If the tank is open to atmosphere, then, 𝑃 = 𝑃_{𝑎}, 𝒗_{𝟏} = √𝟐𝒈𝒉. This is same as the speed of a freely falling body. This equation is known as Torricelli’s law.

## Applications of Bernoulli’s principle

## Venturi meter

Venturi meter is used to measure the speed of an incompressible fluid in a pipe. It consists of a tube with normal cross-sectional area 𝑎_{1} with a constriction of 𝑎_{2} at the middle of the tube. One arm of 𝑈-tube monometer is connected to the point where the area of cross-section is 𝑎_{1} and the other arm is connected at the constriction.

## Measurement of speed

Let 𝜌 be the density of the fluid flowing in the pipe and 𝜌_{𝑚} be the density of the monometer liquid. Let 𝑣_{1} and 𝑣_{2 }be the velocities of the fluid at broader region and constriction respectively. Then from equation of continuity,

𝑎_{1}𝑣_{1} = 𝑎_{2}𝑣_{2}

Speed at the constriction, 𝑣_{2} = (𝑎_{1}/𝑎_{2})𝑣_{1}

Using Bernoulli’s equation, assuming the flow is horizontal,

𝑃_{1} + (1/2)𝜌𝑣_{1}^{2} = 𝑃_{2} + (1/2)𝜌𝑣_{2}^{2}

𝑃_{1} + (1/2)𝜌𝑣_{1}^{2} = 𝑃_{2} + (1/2)𝜌[(𝑎_{1}/𝑎_{2})𝑣_{1}]^{2}

𝑃_{1} + (1/2)𝜌𝑣_{1}^{2} = 𝑃_{2} + (1/2)𝜌𝑣_{1}^{2}[𝑎_{1}/𝑎_{2}]^{2}

𝑃_{1} − 𝑃_{2} = (1/2)𝜌𝑣_{1}^{2}[(𝑎_{1}/𝑎_{2}) − 1]^{2}

Due to this pressure difference the fluid level in the 𝑈-tube connected at the constriction, rise above that in the other arm. The difference in the heights ℎ is the direct measure of pressure difference.

𝑃_{1} − 𝑃_{2} = 𝜌𝑚𝑔ℎ = (1/2)𝜌𝑣_{1}^{2}[(𝑎_{1}/𝑎_{2} ) − 1]^{2}

Speed of the fluid at wide neck is,

𝑣_{1} = √[2𝜌_{𝑚}𝑔ℎ/𝜌[(𝑎_{1}/𝑎_{2}) − 1]^{2}]

𝒗_{𝟏} = √(𝟐𝝆_{𝒎}𝒈𝒉/𝝆)[(𝒂_{𝟏}/𝒂_{𝟐}) − 𝟏]^{−}^{𝟏}^{/}^{𝟐}

By measuring the value of ℎ speed of the fluid can be calculated.

## Filter pumps (Aspirators)

When a fluid passes through a region at a large speed, the pressure in that region decreases. This fact is used in this device. The air in the tube 𝐴 is pushed using a piston. As the air passes through the constriction 𝐵 its speed is considerably increases and consequently pressure drops. Thus, the liquid rises from the vessel and is sprayed with the expelled air. Bunsen burner, atomiser and sprayers work on the same principle.

## Carburettor

The function of the carburettor is to deliver the rightly proportioned mixture of petrol vapour and air to the cylinder of a petrol engine. This also works based on Bernoulli’s principle.

## Uplift of an aircraft

The shape of the wings of aircraft is specially designed so that the velocity of the layers of air on its upper surface is more than that on the lower surface. According to Bernoulli’s principle where velocity of the fluid is high, the pressure is low and vice-versa. So, the pressure 𝑃_{1 }is low at the upper surface of the wing and pressure 𝑃_{2} is high at the lower surface of the wing. This difference in pressure causes an upwards thrust called dynamic lift on the wings of the aircraft.

## Swing bowling

When swing bowlers in cricket deliver the ball, the ball changes its plane of motion in air. This kind of deflection from the plane of projection can be explained on the basis of Bernoulli’s principle.

## Blood flow and heart attack

Bernoulli’s principle helps in explaining blood flow in artery. The artery may get constricted due to the accumulation of plaque on its inner walls. In order to drive the blood through this artery a greater demand is placed on the activity of the heart. The speed of the flow of the blood in this region is which lowers the pressure. The heart exerts further to open this artery and forces the blood through. As the blood rushes through the opening, the internal pressure once again drops due to same reasons leading to a repeat collapse. This may result in heart attack.

## Viscosity

The property of a liquid by virtue of which it opposes relative motion between its different layers is called viscosity.

**Explanation: **When the liquid flows on a horizontal surface, the velocities of different layers of the liquid will be different and there is a relative motion between successive layers of the liquid. In the absence of an external force, the faster layer tends to accelerate the slower one and the slower one tends to retard the faster one.

## Viscous force

In case of a liquid having relative motion between the layers internal forces are developed, which retard the relative motion. These retarding forces are called viscous force.

**Note:**

(1) The viscous force does not operate as long as the liquid is at rest. They come to play only when there is a relative motion between its layers.

(2) Greater viscosity favours streamline flow whereas lower viscosity causes turbulent motion.

## Co-efficient of viscosity (𝜼)

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

**Explanation:** Consider a liquid enclosed between two glass plates as shown. The bottom plate is fixed, and top plate is moved with a constant velocity, 𝑣 with respect to the fixed plate. The fluid in contact with the surface has same velocity as the surface. Therefore, the layer of the liquid in contact with the top surface moves with a velocity, 𝑣 and the layer in contact with the fixed surface remain at rest. The velocities of the layers increase uniformly from bottom to the top. Due to this motion, a portion of liquid initially having the shape ABCD takes the shape AEFD after a short interval of time Δ𝑡. If the distance between the plates is 𝑙 and plate at the top moves through a distance Δ𝑥 in time Δ𝑡 then, 𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑠𝑡𝑟𝑎𝑖𝑛 = Δ𝑥/𝑙. The strain in a flowing liquid increases continuously with time. Therefore, in case of liquids the stress is directly proportional to the rate of change of strain. 𝑆𝑡𝑟𝑎𝑖𝑛 𝑟𝑎𝑡𝑒 = Δ𝑥/𝑙Δ𝑡 = 𝑣/𝑙. The co − efficient of viscosity, 𝜼 = (𝑭/𝑨)/(𝒗/𝒍) = 𝑭𝒍/𝒗𝑨. The SI unit of viscosity is 𝑁𝑠𝑚^{−2}. It can be expressed also in 𝑝𝑎𝑠𝑐𝑎𝑙 𝑠𝑒𝑐𝑜𝑛𝑑. The dimensions are 𝑀𝐿^{−1}𝑇^{−1}.

## Temperature dependence of viscosity

1) As the temperature of the liquid increases, the distance between the molecule increases. Hence the magnitude of cohesion force decreases and the viscosity decreases.

2) When the temperature of the gas increases, the change of momentum and number of collisions also increases and hence the co-efficient of viscosity increases.

## Viscous drag or drag force

When an object moves relative to a fluid, the fluid exerts a friction like retarding force on the object. This force is called viscous drag or drag force. Viscous drag is due to the viscosity of the fluid.

**This chapter will continue in Part-2 which will be the next post.**