Mechanical Properties of fluid 1

S. U. Hiswankar PGT(Physics)

Jawahar Navodaya Vidyalaya, Wardha

Fluid:

A fluid (both liquid and gases) is a substance that can flow.

Liquid in equilibrium:

O

S

U

T

W

θ

R

The reaction R can resolve in two components

1) Tangential component OT = R cos θ

2) Normal component OW = R sin θ

component OT = R cos θ = 0

As R ≠ 0 hence cos θ = 0 or θ = 90⁰

i.e. liquid always exerts force perpendicular to the surface of the container at every point.

Measurement of pressure :

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Pascal’s Law:

A change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.

For equilibrium of fluid element we have

F_b sin θ = F_c

F_b cos θ = F_a

From geometry of figure we have

A_b sin θ = A_c

A_b cos θ = A_a

From above equations we get

or

P_a = P_b = P_c

or

Hence pressure exerted in all direction is same

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1) Hydraulic lift :

Hydraulic lift is used to lift heavy objects.

Pressure exerted on the liquid

Force on larger piston is

F = P x A

Since A>a, therefore, F > f

Hence by making the ratio A/a large, very heavy load (like cars and trucks) can be lifted.

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Applications of Pascal’s Law :

2) Hydraulic brakes :

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Variation of liquid pressure with depth :

a) Force due to liquid pressure at the top

F₁ = P₁A, (↓)

b) Force due to liquid pressure at the bottom

F₂ = P₂A, (↑)

c) Weight of the cylinder acting downward

W = Mass x g (↓)

W = Volume x density x g

W = Ahρg

As the liquid cylinder is in equilibrium

Net downward force = Net upward force

F₁ + W = F₂

F₂ – F₁ = W

P₂A – P₁A = Ahρg

P₂ – P₁ = hρg

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Buoyant force :

The upward force acting on a body immersed in the fluid is called upthrust or buoyant force and the phenomenon is called buoyancy.

Archimedes principle:

It states that when a body is partially or wholly immersed in a fluid, it experiences an upward thrust equal to the weight of the fluid displaced by it & its upthrust act through the centre of gravity of the displaced fluid.

Apparent weight of immersed body :

Apparent weight = Actual weight – Buoyant force

W_app = Vσg - Vρg

W_app = Vσg(1 – ρ/σ)

W_app = W (1 – ρ/σ)

Law of flotation:

It states that a body will float in a liquid if the weight of the liquid displaced by the immersed part of the body is equal to or grater then the weight of the body.

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c

Viscosity :

Viscosity is the property of fluid by virtue of which an internal force of friction comes into play when fluid is in motion and which opposes the relative motion between its different layers.

Fixed surface

F

V = 0

V = maximum

Coefficient of viscosity:

M

v

c

N

v + dv

x

X + dx

Fixed surface

V = 0

According to Newton, a force of viscosity F acting tangentially between two layers

F α A

{contd….

η is coefficient of viscosity

F

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

Note :

1. Negative sign shows that the viscous force acts in a direction opposite to the direction of motion of the liquid.

2)

Dimensions of η

[η] = [M¹ L^-1 T^-1]

3) SI unit of η is N s /m² or Kg /m s or decapoise or poiseuille.

4) CGS unit of η is dyne s /cm² or g /cm s or poise

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Effect of temperature on viscosity :

1. When liquid is heated, the kinetic energy of its molecules increases and the intermolecular attractions become weaker. Hence the viscosity of a liquid decreases with the increase in its temperature.

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1. Viscosity of gases is due to the diffusion of molecules form one moving layers to another. But the rate of diffusion of gas is directly proportional to the square root of the absolute temperature, so viscosity of gas increases with temperature.

Poiseuille’s formula :

The volume of a liquid flowing out per second through a horizontal capillary tube of length ℓ and radius r, under a pressure difference p applied across its end is given by

=

Stokes’ Law :

The retarding (backward) viscous force acting on a small spherical ball of radius r moving with uniform velocity v through fluid of viscosity η is given by.

F = 6πηrv

The viscous force F acting on a sphere moving through fluid may depends on

1. coefficient of viscosity η of the fluid
2. radius r of the spherical body
3. velocity v of the body

i.e.

F = k η^a r^b v^c

[M¹ L¹ T^-2 ] = [M¹ L^-1 T^-1 ]^a [L¹ ]^b [L¹ T^-1 ]^c

[M¹ L¹ T^-2 ] = [M¹ L^-1 T^-1 ]^a [L¹ ]^b [L¹ T^-1 ]^c

[M¹ L¹ T^-2 ] = [ M ^a L^{- a + b + c} T^{- a-c} ]

On solving

a = 1, b = 1, c = 1

F = kηrv

F = 6πηrv

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Terminal velocity :

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

W

U

F

As body falls various forces acting on body are

ρ

σ

i) Weight of the body acting vertically downwards

W = mg = 4/3 π r³ ρ.g

ii) Upward thrust equal to the weight of the liquid displaced

U = 4/3 π r³ σ.g

iii) Viscous force acting upward

F = 6π η r v

When the body attend terminal velocity

U + F = W

i.e.

When the body attend terminal velocity v_t

4/3 π r³ σ.g + 6π η r v_t = 4/3 π r³ ρ.g

6π η r v_t = 4/3 π r³ (ρ – σ).g

Note: if ρ < σ , the terminal velocity is -ve ex air bubble rises through fluid

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Streamline flow :

When a liquid flows in such a way that each particle of the liquid passing a given point moves along the same path and has the same velocity as its predecessor, the flow is called streamline flow or steady flow.

Streamline :

A streamline may be defined as the path, the tangent to which at any point gives the direction of the flow of liquid at that point.

Properties of streamline flow:

1. No two streamline can cross each other
2. The tangent at any point on the streamline gives the direction of velocity of fluid particle at that point.
3. Grater the number of streamlines passing normally through a section of the fluid, larger is the fluid velocity at that point.
4. Fluid velocity remains constant at any point of the streamline, but it may be different at different points of the same streamline.

Turbulent flow :

When the liquid velocity exceeds a certain limiting value, called critical velocity, the liquid flow becomes zig-zag. The path and velocity of liquid changes continuously, haphazardly. This flow is called turbulent flow. It is accompanied by random irregular, local circular currents called vertices.

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Tube of flow :

Tube of flow is a bundle of streamlines having the same velocity of fluid element over any cross-section perpendicular to the direction of flow.

Laminar flow :

In a steady flow, liquid flows in the form of layer sliding past one another without getting mixed, called laminar flow.

Critical velocity :

The critical velocity of a liquid is that limiting value of its velocity of flow up to which the flow is streamlined and above which the flow becomes turbulent.

velocity is

1. Directly proportional to the coefficient of viscosity of liquid (η)
2. Inversely proportional to the density of liquid (ρ)
3. Inversely proportional to the diameter of the tube (D)

where, Reynolds number.

Note :

1. If R_e < 1000 the flow is laminar
2. If R_e > 2000 the flow is turbulent
3. If 1000 < R_e < 2000 the flow is unstable

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Equation of continuity : a₁v₁ = a₂v₂

Bernoulli’s Principle:

It states that sum of pressure energy, kinetic energy and potential energy per unit volume of an incompressible, non-viscous fluid in a streamlined, irrotational flow remains constant along a streamline.

m = volume x density

m = area of cross-section x length x density

m = a₁ v₁ ∆t ρ = a₂ v₂ ∆t ρ

a₁ v₁ = a₂ v₂ …… (*)

change in K.E. of fluid

= K.E. at B – K.E. at A

= ½ m (v₂^{2 -} v₁²)

= ½ a₁ v₁ ∆t ρ (v₂^{2 -} v₁²)

change in P.E. of fluid

= P.E. at B – P.E. at A

= mg (h₂- h₁)

= a₁ v₁ ∆t ρ g (h₂- h₁)

Net work done (F.S) on the fluid

= work done on fluid at A – Work done on fluid at B

= P₁ a₁ x v₁ ∆t – P₂ a₂ x v₂ ∆t

= a₁ v₁ ∆t (P₁– P₂ )

According to law of conservation of energy

Net work done on fluid = Change in KE – change in PE

a₁ v₁∆t (P₁–P₂ )=½ a₁v₁∆t ρ(v₂² - v₁²) - a₁v₁∆t ρ g(h₂-h₁)

Dividing both side by a₁ v₁ ∆t, we get

(P₁–P₂ ) = ½ ρ(v₂² - v₁²) - ρ g (h₂-h₁)

P₁ + ½ ρv₁² + ρ g h₁ = P₂ + ½ ρv₂² + ρ g h₂

P + ½ ρv² + ρ g h = constant

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Speed of efflux (outflow): Torricelli’s Law

A₁v₁ = A₂v₂

v₂ = A₂/A₁ .v₁

P_a + ½ ρ v₁² + ρgy₁ = P + ρ g y₂

½ ρ v₁² = ρg( y₂ – y₁ ) + (P - P_a )

½ ρ v₁² = ρgh+ (P - P_a )

Note :

1. If P >> Pa, the term 2gh may be ignored

2) If tank is open to atmosphere is P = Pa, v₁ = √2gh

This is speed of freely falling body known as Torricelli’s law

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Applications of Bernoulli’s Principle :

it is a device used to measure the rate of flow of a liquid through a pipe.

1) Venturimeter :

It consists of a horizontal tube having wider opening of cross-section a₁ and a narrower neck of cross-section a₂. These two regions of the horizontal tube are connected to a manometer, connecting a liquid of density ρ’.

Let the liquid velocities be v₁ and v₂ at the wider and narrower end then according to principle of continuity.

A₁v₁ = A₂v₂

or

If P₁ and P₂ be the pressure at wider and narrower portion, ρ is density of fluid, then according to Bernoulli’s equation

P₁ + ½ ρv₁² = P₂ + ½ ρv₂²

or

P₁ – P₂ = ½ ρ( v₂² - v₁² )

……. (1)

{ eqn (1)

If h is the height difference in arms of manometer

P₁ – P₂ = h ρ’ g

Volume of liquid flowing out per second

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2) Automizer

3) Aerofoil :

Aerofoil is the name given to solid object shaped to provide an upward vertical force as it moves horizontally through the air

The difference in pressure provides an upward lift called dynamic lift.

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4) Magnus effect :