1

CHAPTER-11

FLUIDS MECHANICS AND ITS PROPERTIES

In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress, or

external force. Fluids are a phase of matter and include liquids, gases and plasmas. They

are substances with zero shear modulus, or, in simpler terms, substances which cannot resist any shear

force applied to them.

Equation of Continuity

(i) When an ideal fluid passes through a tube of non-uniform cross-section in stream line

motion. The product of area of cross-section and velocity of the liquid is constant.

av = constant

(ii) The velocity of the fluid is inversely proportional to the area of cross-section.

a

1v

(iii) 2211 vava , where 1v and 2v are the velocities of the fluid at the areas of cross-section 1a

and 2a respectively.

(iv) When the outlet of a hose pipe is closed partially. The water rushes out with greater

velocity.

(v) In the case of water falling from a hose hold tap, the stream becomes narrow as it flows

down.

(vi) Equation of continuity is in accordance with the law of conservation of mass

(avd is constant)

Pressure Energy

(i) The energy possessed by a fluid by virtue of its pressure is called the pressure energy.

(ii) It is equal to work done in keeping an elementary mass of a fluid at a point against the

pressure existing at that point.

(iii) Pressure energy of an element of volume V is PV. (Here P is the pressure).

Pressure energy per unit volume = V

Pressure energy per unit mass =

P

(Here the density)

(iv) Pressure energy has unit and dimensions are same as energy.

Potential Energy

Potential energy per unit volume of liquid = ghV

mgh

Volume

liquidofenergyPotential

2

Kinetic Energy

Kinetic energy per unit volume of liquid = 2v2

1

Bernoulli's Theorem

(i) A fluid in motion possesses three types of energy – Kinetic energy, potential energy and

pressure energy.

(ii) Bernoulli's theorem states that the sum of the pressure energy, kinetic energy and potential

energy at any point in a steady flow is constant.

(iii) If P is the pressure, is the density, v is the velocity of fluid and h is the height from the

horizontal level, then

ghv2

1P 2

= constant (unit mass)

ghv2

1P 2 constant (unit volume)

(iv)

hg

v

2

1

g

P 2

constant

Here g

P

is called pressure head,

g

v

2

1 2

is called velocity head, h is called gravitation head.

The sum of the pressure head, the velocity head and the gravitational head at any point in a

steady flow of a fluid is constant.

(v) ghv2

1P 2 constant (unit volume)

Here 2v2

1 is called dynamic pressure and (P + gh ) is called static pressure.

(vi) Bernoulli's theorem is in accordance with the law of conservation of energy.

(vii) In the case of horizontal pipe

2v2

1P = constant i.e., 2

22211 v

2

1Pv

2

1P

21

2221 vv

2

1PP

(viii) When a liquid flows through a horizontal tube of variable cross-section, at a point of narrow

cross-section, area is less and velocity is more (as per equation of continuity) and pressure is

less (as per Bernoullis theorem).

Application of Bernoulli's Theorem

(i) A body immersed in a fluid at rest experiences a resultant force acting vertically upwards

called buoyancy or static lift and when it is in motion also it experiences an upwards lift

which is called dynamic lift.

3

(ii) The dynamic lift experienced by a body, when it is in motion in air, is called aerodynamic lift.

(iii) Aeroplanes get the dynamic lift because of the shape of their wings. The upper surface of

the wing is made more curved than the lower surface; air flows with greater speed above

the wing ; pressure above the wing is less. The wing gets dynamic lift upwards.

Dynamic lift = Avv2

1APP 2

22112

where is the density of air, A is the area of the wing, 1v and 2v are the speeds of air

above and below the wing and 1P and 2P are pressure above and below the wing.

(iv) The plane of motion of a spinning ball gets changed due to an effect called magnus effect.

A and B are opposite sides of a ball spinning in anticlockwise direction.

In the plane of the figure which represents horizontal plane, at A the velocity of air and the

velocity due to air drag are in the same direction whereas at B, they are in opposite

direction. The resultant velocity at A is greater than that at B. The pressure of air at A is less

than that at B. A net force acts on the ball from B side to A side due to this pressure

difference. The plane of motion of the spinning ball gets changed due to this force.

This is called magnus effect which is in accordance with Bernoulli's theorem.

(v) When wind blows over a house with high speed, pressure there is less than the pressure

inside the house (which is nearly at the atmospheric pressure). Consequently the roof is

lifted up and blown away by the wind.

(vi) When air is blown between two suspended light balls, the velocity of air between the balls is

more. The pressure between the walls is less, the balls attract each other.

(vii) When air is blown fast under one of the pans of a counterpoised balance. The pressure

under that pan gets decreased, pressure difference is developed. That pan lowers down.

(viii) Pieces of paper, straw etc. are pulled towards a fast moving train or vehicle because of

pressure difference developed due to speed difference.

(ix) Atomiser, paintgun and Bunsen burner work based on Bernoulli's theorem.

Torricelli's Theorem (i) The efflux velocity of a liquid from an orifice is equal to the velocity acquired by a freely

falling body from a height which is equal to that of the level from the orifice, gh2v

(ii) This theorem can be deduced from Bernoulli's theorem.

(iii) Time taken by the efflux liquid to reach the ground is

g

)hH(2t

4

Where H is the height of the free surface of liquid from the bottom.

(iv) Horizontal range of the liquid fall

R = vt =

hHh2g

hH2gh2

(v) Horizontal Range (R) is maximum when the orifice is at the half the water height H.

2

Hh

(vi) Under that condition maximum range, Hh2Rmax

(vii) The efflux velocity is independent of the size of the orifice and density of the liquid.

(viii) The volume of the liquid that flows out from the orifice per second V = Av where A is the

area of cross-section of the orifice. = gh2A . The volume rate of flow V depend on both the

size of the orifice and efflux velocity (v).

(ix) A cylindrical vessel of area of cross-section A has a hole of area of cross-section 'a' in its

bottom.

Time taken for the water level to decrease from 1h to 2h as water flows out from the hole is

21 hhg

2

a

At

Buoyancy

(i) When a body is partly or fully immersed in a liquid, the pressure is more at the bottom of the

body than at the top. As a result a force acts on it vertically upwards. This force is called

force of buoyancy.

(ii) Force of buoyancy F = weight of the displaced liquid = V g .

(iii) When the body is fully immersed in a liquid, the volume of the liquid displaced is equal to

the volume (V) of the body.

Weight of the body = Vdg where d is the density of the body.

Real weight of the body = Vdg where d is the density of the body.

Apparent weight of the body = Real weight – Force of buoyancy = )d(Vg

Apparent loss of weight of the body = Real weight – apparent weight

= Force of buoyancy = weight of the displaced liquid This is nothing but Archimedes' Principle.

5

Laws of Floatation

(i) Weight of the floating body is equal to the weight of the displaced liquid.

If 1V and d are the volume and density of the floating body and 2V and are the volume

and density of the displaced liquid.

21 VdV

d

V

V

1

2

(ii) Centre of buoyancy and centre of gravity of the floating body lie on the same vertical line.

Specific Gravity

(i) Specific gravity of a body = waterofdensity

bodytheofdensity

Because density of water in CGS system is unity specific gravity of a body is numerically

equal to the density of the body in CGS (unit system.)

(ii) Specific gravity of a solid body = volumesameofwaterofweight

bodytheofweight

= 21

1

ww

w

waterinweightitsofloss

bodytheofweight

Where 21 wandw are the weights of the body in air and water respectively.

(iii) Specific gravity of a liquid = 21

31

ww

ww

waterinweightitsofloss

liquidinkerinstheofweightofloss

Where 21 w,w and 3w are the weights of the body in air, water and liquid respectively.

(iv) Specific gravity has neither units nor dimensions.

Floating Ice

(i) When ice block floating in water melts, the level of water in the vessel remains unchanged.

(ii) When ice block with cork floating in water melts, the level of water in the vessels remains

unchanged.

(iii) When ice block with lead shot or metal ball floating in water melts, the level of water in the

vessel decreases.

(iv) If a block of ice, floating in a liquid of density less than that of water, melts then the liquid

level falls.

If a block of ice, floating in a liquid of density greater than that of water, melts then the liquid level rises.

Surface Tension

Capillary rise h surface tension T and angle of contact ''

rdg2

3

rh

cosT

Work done = energy = Area Surface tension

Energy for film = 2 (Area Surface tension)

6

Absorbed energy when drops of radius R splits into a identical drops of radius r, is

= T1nnr4T1nR4 3/13/223/12

Excess pressure inside the soap bubble = r

T4

Excess pressure inside the liquid drop = r

T2

Pressure Difference between convex and concave side of a liquid surface

21 r

1

r

1Tp

When two drops of radii 21 r,r coalesce to form a new drop of radius R under isothermal

condition, then R = 22

21 rr

When a soap bubble of radius 1r and another of radius 2r are brought together the radius of

the common interface is 21 r

1

r

1

r

1 .

Shape of meniscus in the capillary tube will be

(i) Convex if AC F2F

(ii) Concave if AC F2F

(iii) Horizontal if AC F2F

Where AF force of adhesion and CF = force of cohesion

Viscous Force (Newton's Formula)

(i) M and N are two layers of a liquid in stream line flow. Their distances from the bottom

surface are x and (x + dx) and their velocities are v and (v + dv) respectively. A is the area of

the cross – section of the liquid. According to Newton,

viscous force (F) acting on any layer is

dx

dvAF

Here dx

dv is called the velocity gradient and is the coefficient of viscosity.

(ii) The velocity gradient is in the transverse direction. i.e., perpendicular to the direction of liquid flow.

(iii) The viscous force acts tangential to the liquid layer and in a direction opposite to the

direction of liquid flows.

7

Coefficient of Viscosity (i) It is the tangential force per unit area required to maintain unit velocity gradient normal to

the direction of flow.

)dx/dv(

A/F

(ii) It is the ratio of the tangential stress A

F to the velocity gradient

dx

dv.

(iii) It it also called coefficient of dynamic viscosity and its SI unit is Pa s (Pascal-second) or

decapoise 1 decapoise = 1 Ns 2m

Coefficient of Kinematic Viscosity

(i) It is the ratio of coefficient of viscosity () to the density () of the liquid.

i.e., Coefficient of kinematic viscosity =

(ii) SI unit is 12 sm . Practical unit is stoke, 1 stoke = 124 sm10 .

(iii) Areal velocity (area/time) and coefficient of kinematic viscosity have the same unit and

dimensions.

Poiseuille's Formula

(i) Coefficient of viscosity lV8

pr4

Here l is the length of the capillary tube, r is the radius of the capillary tube, p is the

pressure difference between the ends of the capillary tube, V is the volume of the liquid

collected per second through the capillary tube.

(ii) Because p = hdg where 'h' is the height of the free surface of the liquid form the axis of the

capillary tube and 'd' is the density of the liquid

So,

lV8

rhdg 4

(iii) This expression is applicable only when the liquid flows is steady and the pressure

difference (p) is just sufficient to overcome the viscous forces in the flowing liquid.

Poiseuille's formula can also be written as

R

p

r8

prV

4

l

where R = 4r

l

r8 is called fluid resistance.

Stocke's Formula

(i) The viscous force F acting on a spherical body of radius a moving with velocity v in a liquid

of coefficient of viscosity is called stoke's formula F = 6 .av

(ii) F i.e., viscous force is more in highly viscous liquids like glycerine and honey.

(iii) F a i.e., viscous force is more for larger bodies.

(iv) F v i.e., unlike mechanical frictionless frictional force, viscous force depends on the

velocity of the body.

8

Terminal Velocity

(i) When a spherical body travels down a vertical column of viscous liquid, force and

buoyancy force oppose its motion. But the resultant force is downwards in the initial state.

As the body travels down its velocity increases and hence viscous force (6 av) increases

whereas buoyancy force and gravitational force (weight) remain constant. Thus the net force

on the zero at this stage and so its travels down with the constant velocity called terminal

velocity.

(ii) If d is the density of the body and is the density of the liquid, then

Force of buoyancy ga3

F 3B

(upward)

Weight of the body dga3

4W 3 (downward)

viscous force vF av6 (upwards)

When the body attains terminal velocity ( Tv ).

WFF BV

BV FWF

)d(ga3

4av6 3

T

dga.

9

2v

2

T

(iii) 2T av i.e., Sphere s of larger size travel in a liquid with greater terminal velocity.

(iv) .,e.i1

vT

A sphere travels with less terminal velocity in highly viscous liquid like

glycerine.

(v) Rain drop falling from large height reach the ground with terminal velocity.

The larger the size of the rain drop, the greater is its terminal velocity.

(vi) If the density of the liquid () is greater than that of the body (d), the body rises up and

moves with terminal velocity. Air bubble rising up to the top of a water lake may acquire

terminal velocity, if the depth of the lake is sufficiently large.

(vii) Tv Depends on the densities of the body and the liquid.

At a depth h, the density of liquid n having bulk modulus K is given by

K

dgh1sn

Where s = density of liquid on its surface, d = average density of the liquid.

9

Reynold Number

(i) Number which determine the nature of flow of liquid through a pipe is called reynold

number and it is given by

r

DvN c

R

where density of liquid, D = diameter of a tube, cv = critical velocity of liquid flowing.

(ii) For stream line or laminar flow of liquid. RN varies from 0 to 2000.

(iii) In between value 2000 to 3000, flow of liquid changes from stream line to turbulent flow.

(iv) For turbulent flow, value of RN is above 3000.

In physics, elasticity is the ability of a body to resist a distorting influence and to return to its

original size and shape when that influence or force is removed. Solid objects will deform when

adequate forces are applied to them. If the material is elastic, the object will return to its initial shape and

size when these forces are removed.

Stress

Stress = A

F

Area

forcestoringRe .

The cause for elasticity is stress but not strain. Stress is the property possessed by elastic bodies but strain

is possessed by all bodies. stress and strain both have no specified direction but have different values in

different directions so these quantities is example of tensors.

Strain

1. Longitudinal strain = l

e

lengthInitial

Elongation

During the longitudinal strain, the angle between the load and cross-section of the wire is .90o

2. Shearing strain, tan = l

x

If is smaller, i.e., tan .

Then, l

x

3. Bulk strain = V

V

volumeInitial

volumeinChange

Hooke's law

Within the proportionally limit, stress is directly proportional to strain.

Strain

StressE where E = coefficient of elasticity.

10

Elastic Modulii

Elastic modulii is of three different types :

(a) Young's modulus (Y)

(b) Rigidity modulus ( )

(c) Bulk modulus (K)

Young's Modulus

Young' modulus (Y) = strainTensile

stressTensile

Y = Ae

lF

Where, F = Force applied, l = Initial length, A = Area of cross-section, e = elongation.

(i) The value of Young's modulus increases on mixing impurity.

(ii) A material with large value of 'Y' require a large force produce small change in its length.

(iii) The value of Young's modulus decreases with increase in temperature.

Rigidity Modulus

Rigidity modulus () = strainShear

stressTangential

xA

FA/F

l

F = Force applied, l = Initial length of the side, A = Area, x = transverse displacement.

Bulk Modulus

Bulk modulus (K) = strainBulk

stressBulk or

P

A

F;

)V(

pV

V

VA

F

Where, p = Increase in pressure, V = Initial volume, V = Change in volume.

Compressibility

Inverse of Bulk modulus is called compressibility.

Compressibility = K

1

11

Stress-Strain Graph Graph given below represents the typical stress-strain curve for a ductile metal.

Poisson's Constant ()

The ratio between lateral strain to longitudinal strain in called Poisson's constant.

b.e

l.b

strainalLongitudin

strainLateral

b = Initial thickness, b = Decrease in thickness, l = Initial length, e = Elongation

Relations between Elastic Constant Y, , K and

(i) )1(2

Y

(ii) )21(3

YK

(iii)

1

K3

1

Y

3

(iv) K62

2K3

Force constant ()

Force constant is the force required for unit elongation.

(i) = e

F where, e = Elongation produced.

(ii) If Y is the Young' modulus, l is the length and A is the area of cross-section of the then

= l

YA

Atomic Model of Elasticity Interatomic Force Constant

(i) If the distance between the atoms increases, then the distribution of charges become such that a

net attractive force acts between them and they are again brought to initial position.

(ii) Hook's law is obeyed for these atoms.

(iii) F = - 0r , where 0r interatomic distance, = interatomic force constant

(iv) The relation between Y, and 00 Yrisr

12

Work done in Stretching a Wire

It is given by

W = xF2

1ExtensionLoad

2

1

x = Elongation

or VolumeStrainStress2

1W

If the force acting on the wire is increased from 1F to 2F . Then within the elastic limit

x2

FFW 21

Energy Stored in Unit volume of the Wire

StrainStress2

1U

2)Strain(ulusmods'Young2

1U

Y

)Stress(

2

1U

2

Thermal Stress

(i) The thermal stress set up in the rod which is not free to expand or contract is given by,

stress in the rod = 12YA

F .

Y = Young's modulus, Linear coefficient of expansion and 12 = Temperature

difference.

(ii) Thermal force = F = YA 12

(iii) Two different rods of different materials are joined end to end and the composite rod is fixed

between the two supports. The temperature different i 12 . Then force in given by

22

2

11

1

12221211

YA

L

YA

LLL

F

(iv) If they have common, area of cross-sectional stress is given by,

2

2

1

1

12221211

Y

L

Y

L]L)(L[A

(v) If a gas is enclosed in a vessel of any rigid material, then the change in pressure of thermal stress

is

12KP

volume co-efficient of expansion, K = Bulk modulus of elasticity, 12 = Temperature

difference.

13

Shearing Angle

The angle of Shear () is given by

L

r

r = Radius of the wire, L = Length of the wire, = Angle of twisting.

The Depression of a Beam at its Centre

The depression at the centre of a beam is given by

Ybd4

MgL3

3

M = Suspended Mass, L = Length of the beam, b = Bread of the beam, Y = Young's modules and d =

Thickness of the beam.