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UNIT 10

MECHANICAL PROPERTIES OF FLUIDS

NOTES- PART 1

Fluids Fluids are those substances which can flow when an external force is applied on it. Liquids and gases are fluids. Fluids do not have finite shape but takes the shape of the containing vessel, The total normal force exerted by liquid at rest on a given surface is called thrust of liquid. The SI unit of thrust is newton. In fluid mechanics the following properties of fluid would be considered

(i) When the fluid is at rest - hydrostatics (ii) When the fluid is in motion – hydrodynamics

HYDROSTATISTICS

1.1Pressure on a liquid at rest.

When an object is submerged in water / liquid it exerts a force on the surface of the object. This force is

always normal to the surface of the object and cannot be tangential. If the tangential component of

force exists then it causes the liquid to flow which violates the condition. This total normal force exerted

by the liquid at rest per unit area of the surface in contact with it is called thrust of the liquid on the

surface and the thrust exerted by a liquid at rest per unit area in contact with it is called pressure of

liquid or hydrostatic pressure.

Note: definitions to be recalled

1. Pressure , its mathematical relation, S.I unit and Dimensional formulae

2. Density and relative density

1.2Variation of pressure with depth

Consider a liquid of density ρ contained in a vessel in equilibrium of rest. Imagine a cylinder of liquid of

height h and cross sectional area a. Let the point 1 & 2 lie on the flat faces of the cylinder.

The mass of the cylinder will be

M = volume * density = Ahρ

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Let P1 and P2 be the pressure of liquid on both the faces.

The liquid cylinder is under the action of vertical forces

F1=P1A acting vertically downward

F2=P2A acting vertically upward

Weight Mg =Ahρg acting vertically downwards

Since the liquid as well as the imaginary cylinder is under equilibrium the net force on it must be zero.

F1+Mg= F2

P1 +hρg =P2

P2-P1 =hρg

Conclusions framed

If 1& 2 are at the same level then the difference in pressure will be zero which shows that the

pressure is same at all points inside the liquid lying at the same depth in a horizontal plane.

If the point 1 was on the surface of the liquid then P1 will be denoted by Pa the atmospheric

pressure. Then P2- Pa = hρg. This difference in pressure is termed as gauge pressure.

It also shows that pressure exerted in a liquid column is independent of the area of cross section

but depends only on the height of the liquids and density of the liquid.

1.3Hydrostatic paradox

It shows that liquid pressure at a point is independent of the quantity of liquid but depends upon depth

of the point below the liquid surface.

1.4 Pascal’s law

It states that if gravity effect is neglected, the pressure at every point of liquid in equilibrium of rest is

the same.

It also states that the increase in pressure at one point of the enclosed liquid in equilibrium of rest is

transmitted equally to all points of the liquid and also to the walls of the container, provided the effect

of gravity is neglected.

Experimental proof of Pascal’s law

Consider a spherical vessel having four cylindrical tubes fitted with an

airtight frictionless piston of area of cross section a, a/2, 2a, 3a.

The vessel is filled with an incompressible liquid so that no air gap is left

inside. Push the piston with a force F. The pressure developed P=F/a

It is seen that all other pistons will be pushed outwards. To keep the

piston at the original position the forces F/2 2F, 3F are to be applied. Which shows that the pressure is

transmitted equally to all parts of the liquid.

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Hydraulic lift

it is a device which works on the principle of Pascal’s law. It

consists of two cylinders with different area of cross section.

They are connected to each other via a pipe. Each cylinder is

provided with a air tight frictionless piston. Let A1 and A2 be the

cross sectional area of the pistons. Where A1‹‹ A2.the cylinders

are filled with an incompressible liquid.

Let a force F1 be applied on the piston of area of cross section A1.

The pressure exerted on the liquid P1=F1/A1.

According to the law the pressure is transmitted equally to the piston of area A2 then

F2=P1*A2= (F1/A1)*A2.

As A2›› A1 then F2››F1

This shows that the small force applied on the smaller piston will be appearing

as a very large force on the larger piston.

Hydraulic brakes

1.5 Atmospheric Pressure

The pressure exerted by the atmosphere on earth is atmospheric pressure. It is about 100000 N/m2. It is equivalent to a weight of 10 tones on 1 m2. At sea level, atmospheric pressure is equal to 76 cm of mercury column. Then, atmospheric pressure = hdg = 76 x 13.6 x 980 dyne/cm2

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[The atmospheric pressure does not crush our body because the pressure of the blood flowing through our circulatory system balance this pressure.] Atmospheric pressure is also measured in torr and bar.

1 torr = 1 mm of mercury column 1 bar = l05 Pa

Aneroid barometer is used to measure atmospheric pressure.

1.6 Buoyancy When a body is partially or fully immersed in a fluid an upward force acts on it, which is called buoyant force or simply buoyancy. The buoyant force acts at the centre of gravity of the liquid displaced] by the immersed part of the body and this point is called the centre buoyancy.

1.7 Archimedes’ Principle When a body is partially or fully immersed in a liquid, it loses some of its weight and it is equal to the weight of the liquid displaced by the immersed part of the body. If W is the observed weight of a body of density σ when it is fully immersed in a liquid of density p, then real weight of the body

W’= W/ ( 1 – p / σ) (derivation refer notes discussed)

1.8 Laws of Floatation A body will float in a liquid, if the weight of the body is equal to the weight of the liquid displaced by the immersed part of the body. If W is the weight of the body and w is the buoyant force, then (a) If W > w, then body will sink to the bottom of the liquid. (b) If W < w, then body will float partially submerged in the liquid. (c) If W = w, then body will float in liquid if its whole volume is just immersed in the liquid, The floating body will be in stable equilibrium if meta-centre (centre of buoyancy) lies vertically above the centre of gravity of the body. The floating body will be in unstable equilibrium if meta-centre (centre of buoyancy) lies vertically below the centre of gravity of the body. The floating body will be in neutral equilibrium if meta-centre (centre of buoyancy) coincides with the centre of gravity of the body.

HYDRODYNAMICS

2.1Fluid flow

1. Streamline Flow: The flow of liquid in which each of its particle follows the same path as

followed by the proceeding particles and has the same velocity in magnitude and direction as

that of the preceding particle then the flow is called streamline flow.

The flow can be straight or curved and the tangent to each point

shows the direction of flow.

No stream lines can ever cross each other

More the crowding of the stream lines more the velocity of the particles at that point.

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A liquid can maintain a stream line flow only if its velocity is less than the critical

velocity

Critical velocity:

Critical velocity is the speed and direction at which the flow of a liquid through a tube changes from

smooth, or "laminar," to turbulent. It depends on multiple variables such as co efficient of viscosity,

density of liquid, radius of the tube.

Vc = Kƞ/ρr.

For the flow to be streamline the value of critical velocity should be large as possible.

Reynolds number:

It is a pure number that characterizes the flow of the liquid through a tube as either laminar or

turbulent. It physically signifies the ration of the internal force per unit area to the viscous force per unit

area for a flowing liquid. The Reynolds number is a dimensionless variable, meaning it has no units

attached to it.

NR = ρDVC/ƞ

If NR is between 0 and 2000 the flow is streamline

If NR is between 2000 and 3000 the flow is unstable.

If NR is above 3000 the flow is turbulent

2. Laminar Flow: The steady flow of liquid over a horizontal surface in the form of layers of

different velocities, is called laminar flow.

3. Turbulent Flow: The flow of liquid with a velocity greater than its critical velocity is disordered

and called turbulent flow.

2.2Viscosity It is a property shown by a liquid in motion. It is the property by virtue of which an internal

frictional force comes into play when the fluid is in motion in the form of layers having relative

motion. It opposes the relative motion of the different layers.

It is also called as fluid friction or viscous drag because it opposes the relative motion of

the different layers of liquid.

As per the experimental situation discussed in class it is to be noted that in a flowing

liquid the stress is proportional to the rate of change of strain or strain rate.

Therefore

Strain rate = change in shear train/ time interval

= (Δx/l)/Δt

=v/l

Coefficient of viscosity (ƞ) is defined as the ratio of shearing stress to the

strain rate.

Ƞ = (F/A)/(v/l)

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The unit of co efficient of viscosity is poiseuille (Pl) or Pa-s

Thus co efficient of viscosity of a liquid is said to be 1Pl or 1Pa-s if 1 Newton tangential force is

required to maintain the velocity gradient of 1ms-1/m between two parallel layers of liquid each

of area 1 sq.m

Relative viscosity of liquid=viscosity of liquid/ viscosity of water

2.3 Poiseuilles formula When a liquid flows through a horizontal tube with a steady flow under some external pressure,

the liquid moves in the cylindrical layers coaxial with the axis of the tube. The velocity of the

cylindrical layer is maximum along the axis of the tube and it goes on decreasing towards the

walls of the tube.

The rate of flow (v) of liquid through a horizontal pipe for steady flow is given by where, p =

pressure difference across the two ends of the tube. r = radius of the tube, n = coefficient of

viscosity and 1 = length of the tube.

2.4 Stokes law:

When a small spherical body moves through a viscous medium at rest, the layers of the medium

touching the body are dragged along with it. But the layers of the medium away from the body are at

rest. This causes a relative motion between layers of the medium. As a result of this, a backward

dragging force comes into play, which opposes the motion of the body. This backward dragging force

increases with the increase in velocity of the moving body.

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If ρ ‹ σ then value of v is negative and the body will move up with a constant velocity. It is due to this

reason that gas bubbles rise up through soda water bottle.

NOTE: Viscosity varies with both temperature and pressure.

1. Viscosity of liquids decreases with increase in temperature.

2. Viscosity of gases increases with increase in temperature.

3. Viscosity of liquids except water increases with increase in pressure.

4. Viscosity of gases remains unchanged with variation in pressure.

2.5 Equation of continuity

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2.6 Energy of a liquid ( derivation, refer notes discussed)

2.7 Bernoulli’s theorem (derivation, refer notes discussed)

2.8 Application of Bernoulli’s theorem

Atomizer ( read text and refer lecture notes)

Dynamic lift {magnus effect} (read text and refer lecture notes)

Blood flow and heart attack

Venturimeter( derivation, refer notes discussed)

Torricelli’s theorem (derivation, refer notes discussed)