Chapter – 13, Fluid Motion

The substance that can flow is called as fluid. For example, the liquids and gasses.

Volume of the fluid : 

The space occupied by the fluid is called as its volume. Its unit is liter, m3, cm3

Density of the fluid :

The mass of the fluid per unit volume is called as the density of the fluid.

Density = Mass / Volume

SI unit = kg / m3

CGS unit = gm / cm3

Specific gravity of of the substance :

It is defined as the density of the substance to the density of water at 4 0C. It is also termed as the relative density of the substance.

Specific Gravity = ( Density of the substance ) / ( density of thewater at 4 0C ) = ( weight of the body in air)  / ( weight of equal volume of water)  = ( weight of the body in air)  / ( loss of wight of body in water ) = ( wight of body in air )  / upthrust

Pressure Exerted By Liquid Column :

Let h = Height of the Liquid column

ρ = density of the liquid

P = pressure of the liquid column

g = Acceleration due to gravity.

m = mass of the liquid = volume of the liquid column * density of the liquid = area * height * ρ 

Now, P = Force / Area = ( Mass * acceleration ) / area = mg / Area = ( area * h * ρ * g ) / area = ρgh

P = ρgh 

This is the required pressure exerted by a liquid column.

Conclusion : 

1 ) Pressure exerted by liquid at rest is proportional to the height of the liquid and density of the liquid column.

2 ) Absolute pressure :

Let P0 = atmospheric pressure

P = P0ρgh

It is the total pressure or absolute pressure at the bottom of the liquid column.

3 ) Guage pressure :

P – P0 = ρgh

The difference of Absolute pressure and atmospheric pressure is termed as the guage pressure.

Pascal’s Law : ( Absence of g ) 

It states that ” the pressure of the fluid in equilibrium is the same everywhere if the effect of gravity can be neglected. ”

Pascal’s Law : ( In the presence of g ) 

It states that ” Pressure applied to any part of an enclosed fluid at rest is transmitted undiminished to every portion of the fluid and the walls of the vessel. ”

Proof : 

Let us consider a vessel containing fluid in equilibrium.

Let us assume that there is a volume element in the form of cylindrical column of fluid of height h. Three different forces are acting on the fluid column which are given below.

F1 = P01A = vertically downward force on the top face of the column

F2 = P2A = Vertically upward force on the bottom face of the column

F3 = mg = weight of the fluid in the cylindrical column acting downward through center of the gravity of the cylindrical column

Where A = area of cross section of the cylindrical column

P1 = pressure at the top surface of the column

P2 = pressure at the bottom surface of the column

m = mass of the fluid column = volume of the fluid column * density = Aρh

g = acceleration due to gravity

ρ = density of the fluid

As the fluid in cylindrical column is in equilibrium

Net force = 0

⇒ F1 – F2 + F3 = 0

⇒ P1A – P2A + mg = 0

⇒ P2A = P1A + ( Aρgh ) = 0

⇒ P2 = P1ρgh

From above equation it is clear that if P is increased in any way, P at any depth must increase by exactly same amount. This is nothing but Pascal’s law in the presence of acceleration due to gravity.

 

For 1 Marks 

1 ) What is fluid ?

Ans : The substance that can flow, is called as Fluid.

2 ) Is gas considered as fluid ?

Ans : Yes, gas is considered as fluid.

3 ) Define specific gravity ?

Ans : The specific gravity is defined as the ratio of the density of the object to the density of the water at 4 degree centigrade.

4 ) Define density ?

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

5 ) Define pressure of fluid.

Ans : Pressure of fluid is defined as the force exerted by the fluid per unit area of the wall of the container.

6 ) What is the relation between the pressure of the fluid and the height of the fluid column ?

Ans : Pressure of the fluid is directly proportional to the height of the fluid column.

7 ) Write down the expression for the absolute pressure.

Ans : P = P0 + ρgh 

where P = absolute pressure

P0 = atmospheric pressure

ρ = density of the fluid 

h = height of the fluid

8 ) What do you mean by Guage pressure ?

Ans : The difference between the Absolute pressure and atmospheric pressure is called as Guage pressure.

P – P0ρgh

where P = absolute pressure

P0 = atmospheric pressure

ρ = density of the fluid 

h = height of the fluid

9 ) State Pascal’s law in the presence of gravity of the earth.

Ans : It sates that “Pressure applied to any part of an enclosed fluid at rest is transmitted undiminished to every portion of the fluid and the walls of the containing vessel”.

10 ) Which law is known as principle transmission of the fluid pressure ?

Ans : Pascal’s law is known as principle of transmission of the fluid pressure.

11 ) Is Pascal’s law independent principle ?

Ans : No, Pascal’s law is not independent principle.

12 ) Which law is applicable to the Hydraulic press ?

Ans : Pascal’s law is applicable to the Hydraulic press.

13 ) Give two examples of the application of Pascal’s law.

Ans : Hydraulic press and hydraulic lift are the two examples of the application of Pascal’s law.

14 ) What is the SI unit of pressure ?

Ans : The SI unit of pressure is pascal or N/m2

15 ) Define Buoyancy.

Ans : When a body is immersed wholly or partially in a fluid, it displaces the fluid and the displaced exerts an upward force. This tendency of exerting upward force on the body immersed by the displaced fluid is called as Buoyancy.

16 ) Define Upthrust.

Ans : When a body is immersed in a fluid wholly or partially, it displaces the fluid. The displaced fluid exerts an upward force. The upward force is called as Buoyant force or Upthrust.

17 ) What is the relation between the weight of the body immersed in a liquid and the weight of the liquid displaced ?

Ans : The weight of the body immersed in a liquid and the weight of the liquid displaced are equal to each other.

18 ) Define floating.

Ans : When the weight of the immersed portion of the body is equal to the displaced fluid, then the body floats in a fluid. This phenomenon is called as floating.

19 ) State  Archimedes Principle.

Ans : It states that ” If a body is wholly or partially immersed in a fluid, it experiences an upthrust equal to the weight of the fluid displaced and this upthrust acts through the center of the gravity of the displaced fluid”.

20 ) What is the condition for the stable equilibrium of a floating body ?

Ans :The condition for the stable equilibrium of a floating body is that the meta – center must be always above the center of gravity of the body.

21 ) What is the apparent weight of a body immersed completely in a liquid ?

Ans : The apparent weight of a body immersed completely in a liquid is zero.

22 ) Why density increases with fall of temperature ?

Ans : As the volume of the given mass decreases with fall of temperature, and density is inversely proportional to the density, therefore, the density increases with fall of temperature.

23 ) At what temperature the density of the water is maximum ?

Ans : At 40 C, density of the water is maximum.

24 ) What is the indicated by the gradual fall of atmospheric temperature ?

Ans : The gradual fall of atmospheric temperature indicates the rainfall.

25 ) What is the indicated by the sudden fall of atmospheric temperature ?

Ans : The sudden fall of atmospheric temperature indicates the storm.

26 ) Why water does not come out of a dropper unless its rubber head is pressed hard ?

Ans : If the pressure exerted on the rubber head is greater than the atmospheric pressure, then water drop comes out of a dropper.

27 ) Why two holes are made to empty an oil tin ?

Ans : To make the pressure inside the tin greater than the atmospheric pressure, two holes are made to empty an oil tin. If one hole is made then the pressure inside the tin would be less than the atmospheric pressure.

28 ) Define the term ” Thrust ” .

And : Thrust is a force applied violently in a specified direction.

29 ) Define Viscosity.

Ans : The property of a fluid by virtue of which the it opposes the relative motion between its different layer, is called as viscosity.

30 ) Define Viscous force.

Ans : The force that opposes the relative motion between the different layers of the fluid, is known as Viscous force.

31 ) Give one similarity between the viscosity of a fluid and solid friction.

Ans : Both arises due to inter molecular forces.

32 ) What is the coefficient of viscosity ?

Ans : Coefficient of viscosity of a liquid is defined as the viscous force acting tangentially per unit area of liquid layer having a unit velocity gradient in a direction perpendicular to the direction of flow of liquid.

33 ) What is the unit of coefficient of viscosity ?

Ans : In SI – system, the unit of coefficient of viscosity is decapoise while in CGS – system, its unit is poise.

34 ) Write down the mathematical form of Newton’s law of Viscous Flow.

Ans : F = ηA( dv / dx )

where F = viscous force

η = Coefficient of viscosity

A = Area of liquid layer

dv / dx = velocity gradient

35 ) What is the relation between the decapoise and poise ?

Ans : 1 decapoise = 10 poise.

36 ) Write down the mathematical form of Stoke’s law.

Ans : F = 6πηrv

where F = viscous force

η = coefficient of viscosity

r = radius of small spherical body in different fluid

v = velocity of the small spherical body in the fluid.

37 ) Define terminal velocity.

Ans : The constant maximum velocity through which a spherical body moves through a fluid, is called as terminal velocity.

38 ) What is the relation between the terminal velocity and radius of the spherical body passing through the fluid ?

Ans : The terminal velocity of the spherical body passing through the fluid is directly proportional to the square of the radius of the spherical body.

39 ) How the terminal velocity varies with the coefficient of viscosity ?

Ans : The terminal velocity varies inversely with the coefficient of viscosity.

40 ) Give an example of application of Stoke’s law .

Ans : By using Stoke’s law, we can determine the coefficient of viscosity of very viscous liquids like grease or pitch.

41 ) What is meant by streamline flow ?

Ans : In a fluid motion, if every particle follows the same path with the same velocity ( both in magnitude and direction ) as that of its preceding particle when crossing that point, this type of flow of fluid is termed as streamline flow.

42 ) Define streamline ?

Ans : In a fluid motion, if every particle follows the same path with the same velocity ( both in magnitude and direction ) as that of its preceding particle when crossing that point. This path is termed as streamline.

43 ) Define Tube of flow ?

Ans : A group of streamlines is called as Tube flow.

44 ) Define turbulent flow.

Ans : If a liquid flows with a velocity greater than its critical velocity, then its flow is said to be turbulent flow.

45 ) Define critical velocity.

Ans : The velocity of a liquid up to which its flow is streamlined and above which its flow is turbulent, is known as critical velocity.

46 ) Write down the mathematical form of critical velocity.

Ans : vc = NRη / ( ρD )

Where v = Critical velocity

N = Reynolds number

η = coefficient of viscosity

ρ = density of the liquid 

D = diameter of the liquid through which the fluid is flowing.

47 ) What do you mean by Reynolds Number ?

Ans : The pure number which determines the type of flow of liquid through a pipe, is termed as Reynolds number.

48 ) What is the physical significance of Reynolds Number ?

Ans : Reynolds number gives the ratio of inertia force per unit area to the viscous force per unit area.

49 ) What is the effect of increase of temperature on the viscosity of the liquid ?

Ans : If the temperature increases the viscosity of the liquid decreases and vice versa.

50 ) Define pressure energy.

Ans : The energy required to move the liquid against pressure without imparting any velocity to it, is called as pressure energy.

51 ) State Bernoulli’s theorem.

Ans : It states that ” For a streamline flow of an ideal fluid, the sum of the pressure energy, kinetic energy and potential energy is constant.”

52 ) Name two applications of Bernoulli’s theorem.

Ans : Venturimeter and pitot tube are the two instrument that work on the principle of application of Bernoulli’s theorem.

53 ) Define Poise.

Ans : The coefficient of viscosity of a liquid is said to be one poise if a tangential force of 1 dy-cm-2 of the surface is required to maintain a relative of 1 cm / s between two layers of the liquid 1 cm apart.

54 ) Why a rain drop falling freely does not acquire a high velocity ?

Ans : A rain drop falling freely does not acquire a high velocity because it acquires terminal velocity after having fallen through a certain height.

55 ) Why a hot liquid moves faster than the cold liquid?

Ans : The coefficient of viscosity of hot liquid is greater than that of cold liquid. Terminal velocity is inversely proportional to the coefficient of viscosity. Therefore, a hot liquid moves faster than the cold liquid.

56 ) What is the velocity head ?

Ans : v2 / 2g is termed as velocity head.

57 ) What is the pressure head?

Ans : P / 2 g is called as pressure head

58 ) Why do the clouds float in the sky ?

Ans : The clouds float in the sky because their terminal velocity is zero.

59 ) Why a spinning cricket ball in air does not follow the parabolic path ?

Ans : Due to Magnus effect, a spinning cricket ball in air does not follow the parabolic path.

60 ) Why fire fighters have a jet attached to the head of their water pipes ?

Ans : To increase the velocity of water flowing out from the pipe, fire fighters have a jet attached to the head of their water pipes.

61 ) Why the aeroplanes and cars are given a streamline shape ?

Ans : To reduce the backward drag of the atmosphere, the aeroplanes and cars are given a streamline shape.

62 ) Give one point of difference between the streamline and turbulent flow.

Ans : In streamline flow, the velocity of the fluid should be less than the critical velocity whereas in turbulent flow, the velocity should be greater than its critical velocity.

63 ) Give one point of difference between the viscosity and friction.

Ans : The viscosity depends on the area of layer whereas the friction depends on the area of the surface in contact.

64 ) Why two streamlines can not intersect each other ?

Ans : Because at the point of intersection two tangents can be drawn.

65 ) Mention one feature of ideal fluid.

Ans : Ideal fluid is non viscous.

66 ) Why a larger rain drop falls faster than the smaller rain drop ?

Ans : As terminal velocity of rain drop is directly proportional to the square of its radius, a larger rain drop falls faster than the smaller rain drop.

67 ) In which condition, the two systems of fluids are said to be dynamically similar ?

Ans : If Reynolds number is same for any two systems of fluids whether they have different coefficient of viscosity, diameter, critical velocity and density, the systems are said to dynamically similar.

68 ) What is represented by the crowding streamlines ?

Ans : Crowding streamlines represent the faster flow of the fluid.

69 ) Write down the dimension of coefficient of viscosity.

Ans : ML-1T-1

70 ) Indicate the terminal velocity in the velocity versus time graph.

Ans :

For 2  Or 3  Or 5 Marks

1 ) What is the cause of viscosity ?

Ans : Let us consider that there are two neighbouring liquids layer. Suppose that the upper layer is faster than the lower one. Lower layer tries to retard the motion of the upper layer. On the other hand, the upper layer would try to accelerate the lower one. As a result, a backward tangential force is set up. This force tends to destroy the relative motion between the two layers. This accounts for the viscous force of both the liquids and gasses.

2 ) Derive the expression for the coefficient of viscosity.

Ans : According to Newton’s law of viscous force, the viscous force F acts tangentially on the layer of the fluids is directly proportional to the area of the layer and the velocity gradient.

Therefore,

∝ A ( dv/dx ) 

F = η A( dv/dx )

η = F / { A ( dv/dx ) }

where η is the coefficient of viscosity, A is the area of the layer, and ( dv / dx ) is the velocity gradient.

The above equation is the expression for the coefficient of viscosity.

3 ) What is the relation between decapoise and poise ?

Ans : 1 decapoise = 1 N s m-2 = 105 dyne * 1 s *  (102 )-2 cm-2 = 10 dyne s cm-2 = 10 poise.

4 ) Differentiate between the viscosity and solid friction.

Ans :

Viscosity Solid Friction
1 ) It depends upon the area of the layer. 1 ) It is independent of the area of the surface in contact.
2 ) It depends upon the relative velocity of the layers. 2 ) It does not depend upon the relative velocity of the surface in contact.
3 ) The SI unit of coefficient of viscosity is decapoise. 3 ) The coefficient of friction is dimensionless.
4 ) The coefficient of viscosity is the viscous force per unit area of the layers having unit velocity gradient. 4 ) The coefficient of friction is the ratio of frictional force to the normal reaction.
5 ) The value of coefficient of viscosity may be greater than 1. 5 ) The value of coefficient of friction lies between 0 and 1.

5 ) List any two points of similarity between the viscosity and solid friction.

Ans : Two pints of similarities between the viscosity and solid friction are given below.

  1. Both arise due to the internal force.
  2. Both come into play due to the relative motion.

6 ) Show that P = ρgh where symbols have usual meaning.

Ans : 

Let h = Height of the Liquid column

ρ = density of the liquid

P = pressure of the liquid column

g = Acceleration due to gravity.

m = mass of the liquid = volume of the liquid column * density of the liquid = area * height * ρ 

Now, P = Force / Area = ( Mass * acceleration ) / area = mg / Area = ( area * h * ρ * g ) / area = ρgh

P = ρgh 

( Showed )

7 ) Consider two liquids of same mass but different densities. Find the density of the mixture of liquids.

Ans : Let mass of the liquids = m

density of the first liquid = ρ1

density of the second liquid = ρ2

volume of the first liquid ( v1 ) = m / ρ1

volume of the second liquid ( v2 ) = m / ρ2

density of the mixture ( ρ ) = total mass / total volume = ( m + m ) / ( v1 + v2 ) = 2m/ {( m / ρ1) + ( m / ρ2 ) } = 2 ρ1ρ / ( ρ1 + ρ2

Therefore, the density of the mixture = 2 ρ1ρ / ( ρ1 + ρ2

8 ) Consider two liquid of different volumes and densities. Find the density of the mixture.

Ans : Let

volume of the first liquid = v1

volume of the second liquid = v2

total volume of the mixture ( v ) = v1 + v2

density of the first liquid = ρ1

density of the second liquid = ρ2

mass of the first liquid ( m1 ) = v1ρ1

mass of the second volume ( m2 ) = v2ρ2

total mass of the mixture ( m )  = m1 + m2 = v1ρ1 + v2ρ2

density of the mixture = m / v = ( v1ρ1 + v2ρ2 ) / (  v1 + v2 )

Therefore, the density of the mixture = ( v1ρ1 + v2ρ2 ) / (  v1 + v2 )

9 ) Find the expression for the absolute pressure.

Ans :

Let h = Height of the Liquid column

ρ = density of the liquid

P’ = pressure of the liquid column

g = Acceleration due to gravity.

m = mass of the liquid = volume of the liquid column * density of the liquid = area * height * ρ 

Now, P’ = Force / Area = ( Mass * acceleration ) / area = mg / Area = ( area * h * ρ * g ) / area = ρgh

P’ = ρgh 

Absolute pressure = P = P0ρgh

P0 = atmospheric pressure

10 ) Explain the working principle of Hydraulic Brake.

Ans :

It works on the principle of Pascal’s law.When the brake pedal is pressed, the lever system operates. The piston of the master cylinder is pushed The pressure is transmitted to pistons connected to the brake shoes of the wheel cylinder in accordance with the Pascal’s law. The pistons push the brake shoes away from each other. The brake shoes press against the inner rim of the wheel and retard the motion of the wheel. Since the area of the piston of the wheel cylinder is greater than the area of the piston of the master cylinder, therefore the small force is applied to the brake pedal produces a large thrust on the wheels. In order to apply equal pressure to all the wheels, the master cylinder is connected to all the wheels of the vehicle through tubes.

11 ) Name the main component of the Hydraulic brake.

Ans : Master cylinder is the main component of the Hydraulic brake.

11 ) Draw the graph showing variation of atmospheric pressure with height.

Ans :

12 ) Prove that the upthrust equals the weight of the liquid displaced.

Ans

Consider a solid cylinder of height h and cross-sectional area A completely immersed in a fluid of density ρ. The horizontal thrusts on the cylinder balances each other because these are equal in magnitude and opposite in direction. But the thrusts on the two end faces of the cylinder do not balance each other.

h1 = depth of the top face of the cylinder from the free surface of the fluid.

h2 = depth of the bottom face of the cylinder from the free surface of the fluid.

P = atmospheric pressure

Total downward thrust on the top face of the cylinder = ( P + ρgh1 ) A

Total upward thrust on the bottom face of the cylinder = ( P + ρgh2 ) A

Resultant thrust = ( P + ρgh2 ) A – ( P + ρgh1 ) A = ρg ( h2 – h1 )A = ρghA 

hA = Volume of the cylinder = volume of the displaced cylinder

hAρ = Volume of the displaced fluid * density of the fluid = mass of the displaced fluid = m 

Therefore, Resultant thrust = Upthrust = mg = weight of the displaced fluid.

13 ) Name the forces acting on the body immersed completely in a liquid.

Ans : The weight of the body acting vertically downwards through the center of gravity of the body and the upward thrust acting upward through the center of gravity of the liquid displaced are the forces acting on the body immersed completely in a liquid.

14 ) What happen when immiscible liquids of different densities are put in a vessel? 

Ans : when immiscible liquids of different densities are put in a vessel, the liquid of lowest density settles at the top and the highest density settles at the bottom.

15 ) Why mercury is used in barometer ?

Ans : Mercury is used in barometer due to its large density.

16 ) What height of water column produces the same pressure as a 760 mm high column of Hg ?

Ans : height of mercury column ( h1 ) = 760 mm = 76 cm

density of the mercury ( ρ1 ) = 13.6 g/cc

height of the water column ( h2 ) = ??

density of the water ( ρ2 ) = 1 gm / cc 

The pressure is same in both the cases

ρ1gh1 = ρ2gh2 

h2ρ1gh1 / ρ1g = ρ1h1 / ρ2 = ( 13.6 * 76 ) / 1 =1033.6 cm = 10.34 m 

Therefore, the required height is 10.34.

17 ) Derive the Stoke’s law by dimension analysis.

Ans : Let

F = viscous force arise due to relative motion of the different layers of the fluid.

r = radius of the spherical body inside the fluid

v = velocity of the spherical body

η = coefficient of the viscosity of the fluid.

∝ ηa ————————— ( 1 ) 

∝ rb ——————————– ( 2 ) 

∝ vc —————————– ( 3 ) 

On combining the above equations

∝ ηarbvc

F = K ηarbvc

where K is proportionality is constant

By dimension analysis

[ MLT-2 ] =  [ ML-1T-1 ]a[ L ]b[ LT-1 ]c

[ MLT-2 ] = [ MaL-a + b – cT-a – c ]

a = 1 ——————- ( 4 )

-a + b + c = 1——————— ( 5 )

-a -c = -2———————– ( 6 )

putting a = 1 equation ( 6 ), we get

-a -c = -2

-1 -c = -2

c = 1

putting c = 1 and a = 1 in equation ( 5 ) we get

-a + b + c = 1

-1 + b +1 = 1

b = 1

Now, putting a = 1, b = 1 and c = 1 in the following equation

F = K ηarbvc

F = Kηrv

on experiment it is found that K = 6π

F = 6πηrv

This is nothing but stoke’s law.

18 ) Write any two assumptions regarding Stoke’s law.

Ans : Two assumptions regarding Stoke’s law are given below :

1 ) The viscous medium is homogeneous and of infinite extent.

2 ) The spherical body is perfectly rigid and smooth.

19 ) Explain the phenomenon ‘ the rise of bubbles from soda water” .

Ans : The terminal velocity attained by the gas bubbles is given by

v = { 2r2ρ – ρ’ ) g } / 9η 

where

r = radius of the gas bubbles

ρ = density of the the material of the gas bubbles

ρ’ = density of the soda water 

g = acceleration due to gravity

η = coefficient of the viscosity of the soda water.

Here, ρ < ρ’

Therefore, v is negative and the direction of the terminal velocity attained by the gas bubbles is upward direction and this accounts the rise of gas bubbles from soda water.

20 ) What are the properties of streamlines ?

Ans : The properties of the streamlines are given below:

1 ) The tangent drawn at any point on the streamlines give the direction of the velocity of velocity of the liquid at that point.

2 ) Two streamlines can not intersect each other . If two streamlines intersect each other then it would mean that at the point of intersection, two tangents can be drawn and these will give the two different direction of the velocity of the liquid at point and this is not possible.

3 ) At a particular point of the streamline, the velocity of the liquid is constant. However, at different points, the velocity will be different.

4 ) Crowding of streamlines represents a faster flow of the liquid and vice versa.

5 ) It may be straight or curve.

6 ) Streamlining ( to give some suitable shape to the flow of liquid ) reduces the friction or pressure exerted by the liquid.

21 ) Distinguish between the streamline flow and turbulent flow.

Ans :

Streamline flow Turbulent flow
1 ) The velocity is less than the critical velocity. 1 ) The velocity is greater than the critical velocity.
2 ) Viscosity plays a dominant role. 2 ) Density plays a dominant role.
3 ) The flow is regular and orderly. 3 ) The flow is irregular and disorderly.
4 ) Each and every particle flow the path and velocity of its preceding particle when it passes through that point. 4 ) The paths and velocities of the liquid particles change continuously and haphazardly.
5 ) In this type of flow, the energy needed to drive the liquid is used up only in overcoming the viscous drag between the layers. 5 ) In such a flow, most of the energy needed to drive the liquid is dissipated in setting up eddies and whirpools in it.

 

22 ) Establish the expression for the critical velocity by the method of dimension analysis.

Ans : Let the critical velocity is proportional to the coefficient of viscosity η , the density ρ and diameter D of the tube.

Let vc ∝ ηa ———————– ( 1 )

vc  ρb ——————————– ( 2 ) 

vc ∝ Dc ————————- ( 3 ) 

On combining these equations we get

vc ∝ ηa ρbDc

vc = Kηa ρbDc

where K is proportionality constant

By dimension analysis

[ LT-1 ] = [ η ]a[ ρ ]b[ D ]c 

[ LT-1 ] = [ ML-1T-1 ]a[ ML-3 ]b [ L ]c 

[ LT-1 ] = [ Ma + b L-a – 3b + c T-a  ]

a + b = 0 ———————- ( 1 )

-a -3b + c = 1 ————————- ( 2 )

-a = -1 —————————- ( 3 )

⇒ a = 1

Putting a = 1 in equation ( 1 ) we get

a + b = 0

⇒ b = -1

Putting a = 1 and b = -1 in equation ( 2 ) we get

-a -3b +c = 1

⇒ -1 -3 ( -1 ) + c = 1

⇒ -1 + 3 + c = 1

⇒ c = 1 – 2

⇒ c = -1

Putting the values of a, b and c in the following equation we get

vc = Kηa ρbDc

vc= K η1 ρ-1D-1

vc =  ( Kη ) / ( ρD ) 

On experiment it is found that K = NR = Reynolds number

vc = NRη / ( ρD )

23 ) What is physical significance of Reynolds number ? 

Ans : Let

a = cross sectional area of the tube through which fluid is flowing.

v = velocity of the fluid

ρ = density of the fluid 

η = coefficient of viscosity 

∴ The volume of the fluid flowing per unit time = av 

The mass of the fluid flowing per unit time = avρ

Inertia force = Time rate of change of momentum = ( mass * velocity ) / time = ( mass / time ) * velocity = mass of the fluid flowing per unit volume * velocity = avρ * v = av2ρ

Viscous force = ηav / D 

( here D is used as linear dimension )

Now,

inertia force / viscous force = (  av2ρ ) / (  ηav / D ) = vρD / η = NR

Hence, Reynolds number represents the ratio of inertia force to viscous force. It is the key number in all those problems on fluid motion in which inertia forces and viscous forces operate. If Reynolds number of two systems are same, then the two systems are said to be dynamically similar, even if the individual values of v, η and D are different.

24 ) Discuss the factors influencing the viscosity of a fluid. 

Ans :The factors influencing the viscosity of a fluid are in the following:

a ) Temperature : With the rise in temperature, the viscosity of the liquid decreases but in case of gasses, it increases with the increases in temperature.

b ) Pressure : Viscosity of the liquids increases with the increase in pressure except water. The viscosity of the gasses is independent of the pressure.

25 ) Write any two importance of viscosity. 

Ans : Two importance of viscosity are given below :

a ) The liquids of high viscosity are used as buffers at railway stations.

b ) The phenomenon of viscosity is used to damp the motion of some instruments.

26 ) Write properties of ideal fluid.

Ans : Two properties of ideal fluid are given below :

a ) It is non viscous and incompressible.

b ) It has no shear stress.

c ) It has constant density

d ) It does not show the phenomenon heat conduction.

27 ) State the Bernoulli’s theorem.

Ans : For the streamline flow of an ideal fluid, the sum of the pressure energy, kinetic energy and potential energy per unit mass is always constant.

28 ) Explain the working principle of venturimeter.

29 ) Why is the speed of innermost layers of a whirlwind alarmingly high ? Explain.

Ans : Let P = pressure of the outer layers

p = pressure of the inner layers

v = speed of the inner layer

V = speed of the outer layer

During the formation of the whirlwind, air is drawn into circular layers,

P > p

∴ The pressure energy of the outer layer > The pressure energy of the inner layer

According to Bernoulli’s theorem, the kinetic energy of the inner layer increases

Hence, speed of the outer layer < speed of the inner layer

Thus, the speed of the innermost layers of a whirlwind is alarmingly high.

30 ) Why wings of aeroplanes are rounded outwards while flattened inwards ? 

Ans : The special design of the wings of aeroplanes increases the velocity at the upper surface and decreases the velocity at the lower surface. Hence, the kinetic energy at the upper surface is greater than that of the lower surface. According to Bernoulli’s theorem, the pressure energy at the upper surface will be greater than that of the lower surface. Consequently, the pressure at lower surface will be greater than that of the upper surface of the wings. This difference of pressures provides the lift.

31 ) A person standing near a fast – moving train has a danger of falling towards the train. ——-why ? 

Ans : The fast moving train reduces the pressure between the person and the train. Due to larger pressure on the other side of the person, he might be pulled towards the train. So, a person standing near a fast moving train has a danger of falling towards the train.

32 ) Why deep waters run slow ? 

Ans : The velocity of flow of liquid is inversely proportional to the cross sectional area. The cross sectional area increases as we approach towards the bottom of a river or a canal. Therefore, the velocity of the water at the top is greater than that of the deep waters. Hence, the deep waters run slow.

33 ) Why flags flutter on a windy ? 

Ans : On a windy day, velocity of the wind increases while its pressure decreases. So, flags flutter on a windy.

34 ) If two ships happen to parallel and adjacent to each other, they experience a pull trying to bring them together.—————- why ? 

Ans : If two ships come close to each other then the pressure between the two ships is reduced. But on the other side of the ships, the pressure is high. So, due to the pressure difference, the two ships experience a pull trying to bring them together if they are parallel and adjacent to each other.

35 ) Why light roofs are blown off during wind storm ? 

Ans : During wind storm, the velocity of the wind on the top of the  roof is higher than that of the wind under the roof. The pressure on the top is low. But the pressure under the roof is nearly atmospheric pressure. Due to this pressure difference, there is upward thrust on the roof. That is why roofs are blown off during winds storm.

36 ) Why two streamlines can not intersect each other ? 

Ans : If two streamlines intersect each other, then at the point of intersection two tangents can be drawn and these tangents will give the two different direction of the fluid at the same time. This situation is impossible. Therefore, two streamlines can not intersect each other.

37 ) Why pressure of water decreases when it flows from broader pipe to a narrow pipe ? 

Ans : According to equation of continuity, we know that the velocity v is inversely proportional to the square of the radius ( R ) of the pipe through which the liquid flows.

i.e v ∝ 1 / R2

But p ∝ 1 / v 

where p is the pressure

Therefore, p ∝ 1 / R2

That is pressure decreases with the increase in cross sectional area.  Hence pressure of water decreases when it flows from broader pipe to a narrow pipe.

38 ) In case of an emergency, a vacuum brake is used to stop the train. How does this brake work ? 

Ans : Steam at high pressure is made to enter the cylinder of the vacuum brake. Due to high velocity pressure decreased in accordance with the Bernoulli’s theorem. Due to this decrease of pressure, the piston gets lifted. Consequently the brake gets lifted.

39 ) Why dust generally settle down in a closed room ? 

Ans : The dust particles are very tiny and may be considered as sphere. The dust particles acquire the terminal velocity after falling through some distance in air. As the terminal velocity is directly proportional to the square of the radius of the sphere. So the dust particles have very small terminal velocity. Therefore the dust settles down in a closed room generally.

40 ) On what factors does the terminal velocity of a spherical ball falling through a liquid depend ? 

Ans : The factors influencing the terminal velocity of a spherical body falling through a liquid depend are given below :

1 ) It is directly proportional to the square of spherical body.

2 ) It depends on the nature of the spherical body and the fluid through which the body is flowing.

3 ) It is inversely proportional to the viscosity of the fluid.

41 ) How does a falling body attains a constant speed ? 

Ans : When a body falls it experiences three different forces weight of the body acting downward, upthrust due to buoyancy and the viscous force acting upward. The resultant force of these three forces are responsible for the downward acceleration of the body. As the velocity of the body increase, the viscous force also increases. This continuous until a stage is reached when the weight of the body just balance the sum of the upthrust and the upward viscous force. Now there is no net force neither in upward direction nor in downward direction. Consequently, the body will move with the constant speed.

42 ) How many different forms of energy does a flowing liquid possess ? Name them. Which law gives the relation between above forms of energy? 

Ans : A flowing liquid possess three different forms of energy. They are pressure energy, kinetic energy and potential energy. Bernoulli’s principle gives the relation between above forms of energy.

 Numerical Problems

1 ) A 0.5 kg block of brass ( density : 8 * 103 kg / m3 ) is suspended from a string. What is the tension in the string if the block is completely immersed in water ? g = 10 m/sec2

2 ) A house is connected to a city water main that is 100 m above the house in altitude. If the city water pressure is 4 atm, what will be the water pressure at house ? g = 10 m / sec

3 ) A metal block of area 0.10 m2 is connected to a 0.01 kg mass via a string that passes over an ideal pulley ( considered massless and frictionless ) as in figure. A liquid with a film of thickness of 0.3 mm is placed between the block and the table when released, the block moves to the right with a constant speed of 0.085 m / sec. Find the coefficient of viscosity of the liquid.

4 ) A metal plate 0.02 m2 in a area is lying on a liquid layer of thickness 10-3 m and coefficient of viscosity 12 decapoise. Calculate the horizontal force required to move the plate with a speed of 0.025 m / sec .

5 ) A square plate of  0.1 m side moves parallel to another plate with a velocity of 0.1 m /sec , both the plates being immersed in a water. If the viscous force is 2 * 10-3 N and viscosity of water is 10-3 decapoise, what is their distance apart ?

6 ) The terminal velocity of a copper ball of a radius 2.0 mm falling through a tank of oil at 200 C is 6.5 cm / sec. Compute the viscosity of the oil at 200 C. Density of oil is 1.5 * 103 kg/m3 , the density of copper is 8.9 * 103 kg/m3 .

 


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