# Fluid Mechanics

PART #1

## Fluid Mechanics Syllabus

1.Fluid Properties
2.Fluid Pressure
3.Fluid Statics
4.Fluid Kinematics
5.Fluid Dynamics
6.Boundary Layer

## Fluid Mechanics

➢ Fluid Mechanics is the science that deals with behavior of fluids at rest (fluid statics) or in motion (fluid dynamics) and the interaction of fluids with solids or other fluids at the boundaries

### What is Fluid

➢ A fluid is a substance which deforms continuously when subjected to external shearing forces.Gases And Liquids both are fluids

## Properties of Fluid

### Mass Density

➢ The density of a substance is the quantity of matter contained in unit volume of the substance. ➢ Density of water is 997 kg/m3 , approximately 1000 kg/m3

➢ Density of air is 1.225 kg/m3

➢ Density of mercury is 13600 kg/m

### Relative Density/Specific gravity

➢ The ratio of the weight of a unit volume of some substance to the weight of an equal volume of a standard substance(water), measured under standard pressure and temperature, is called specific gravity .Relative Density/Specific gravity

➢ In other words Relative density, or specific gravity, is the ratio of the density (mass of a unit volume) of a substance to the density of a given standard substance(water). ### Specific weight

➢ The specific weight, also known as the unit weight, is the weight per unit volume of a material. ### Bulk Modulus

➢ It is the property of that fluid that represents the variation of density with pressure at constant temperature. Mathematically, it is represented as : ### Viscosity / Dynamic Viscosity(µ)

➢ When a fluid moves relative to a solid or when two fluids move relative to each other internal resistance of a fluid to motion is called as viscosity.

➢ Cause of viscosity is cohesion.

➢ The dynamic viscosity has the dimension ML-1T-1 and the unit of kg/m.s (or, N.s/m2 or Pa.s)

➢ A common unit of dynamic viscosity is poise which is equivalent to 0.1 Pa.s.

### Kinematic Viscosity (ν)

The ratio of dynamic viscosity to density.Kinematic Viscosity (ν) ➢ It has got the dimension of L2 T-1 and unit of stoke.

➢ 1 stoke = 0.0001 m2 /s

## Variation of Viscosity with Temperature

### For Liquids

➢ The viscosity of liquids decreases with the increase of temperature. This is due to reason that the viscous forces in a fluid are due to cohesive forces and molecular momentum transfer.

➢ In liquids the cohesive forces predominates so with the increase in temperature, the cohesive forces decreases with the result of decreasing viscosity. ### For Gases

➢ In the case of gases the cohesive force are small and molecular momentum transfer predominates. With the increase in temperature, molecular momentum transfer increases and hence viscosity increases ## Variation of Viscosity with Pressure

### For Liquids

➢ Dynamic viscosity is only function of temperature and also liquids are considered approximately incompressible, so there is no effect on viscosity due to pressure variation. ### For Gases

➢ Dynamic viscosity is only function of temperature but gases are compressible fluids, so there is no effect on dynamic viscosity due to pressure variation but kinematic viscosity decreases with increase in pressure. ## Newton’s Law of Viscosity

➢ When two layers of a fluid, a distance ‘dy' apart move one over the other at different velocities say u and u + du as shown, the viscosity together with relative velocity causes a shear stress acting between the fluid layers.

➢ This shear stress is proportional to the rate of change of velocity with respect to y. ## ### Ideal Fluid

➢ A fluid, which is incompressible and is having no viscosity, is known as an ideal fluid.

➢ Ideal fluid is only an imaginary fluid as all the fluids, which exist, have some viscosity

### Real Fluid

➢ A fluid, which possesses viscosity, is known areal fluid.

➢ All the fluids: in actual practice, are real fluids.

### Newtonian Fluid

➢ A real fluid, in which the shear stress is directly, proportional to the rate of shear strain (or velocity gradient), is known as a Newtonian fluid.

➢ Viscosity remains constant with flow.

➢ Example water, air, petrol, Diesel, kerosene etc. ### Non-Newtonian Fluid

➢ A real fluid, in which shear stress is not proportional to the rate of shear strain (or velocity gradient), known as a Non-Newtonian fluid. ➢ There are various types of non Newtonian fluids

### Dilatant Fluid

➢ If value of n > 1

➢ So by comparing we get apparent viscosity :

➢ So as flow increases apparent viscosity also increases,means resistance to flow increases.

➢ Such fluids also called shear thickening fluids.

➢ Examples are honey, butter, rise starch etc.

### Pseudo Plastic Fluids

➢ If 0 < n < 1

➢ So by comparing we get apparent viscosity :

➢ So as flow increases apparent viscosity decreases, means resistance to flow decreases.

➢ Such fluids also called shear thinning fluids.

➢ Examples are lipstick, blood, milk, syrup, nail polish,ketchup etc.  ### Ideal Bingham

➢ Fluids which require certain minimum shear stress to start a flow and once flow is started it behaves like Newtonian fluids.

➢ Viscosity remains constant with flow.

➢ Example tooth paste, mud, jam, sewage sludge etc. ### Thixotropy Fluid

➢ This type of fluids have time depended viscosity.

➢ They have properties like Bingham fluids but their viscosity is not constant. ➢ There are two types of non Thixotropy Fluid.

### Grade 1 Thixotropy or Thixotropy Fluid

➢ If value of n < 1

➢ So by comparing we get apparent viscosity :

➢ Viscosity of Grade 1 Thixotropy fluid decreases with time.

➢  Examples are Paints and printer ink ## Surface Tension

➢ Surface tension is defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two immiscible liquids such that the contact surface behave like a membrane under tension

➢ Surface tension is created due to the unbalanced cohesive forces acting on the liquid molecules at the fluid surface.

➢ Molecules in the interior of the fluid mass are surrounded by molecules that are attracted to each other equally. However, molecules along the surface are subjected to a net force toward the interior.

➢ A tensile force may be considered to be acting in the plane of the surface along any line in the surface. The intensity of the molecular attraction per unit length along any line in the surface is called the surface tension.

➢ The SI unit is N/m.

➢ For equilibrium : ➢ Fluid droplet have spherical form due to surface tension.

## Surface Energy

➢ Surface energy can be defined as the work per unit area done by the force that creates the new surface.

➢ Work done by force. ➢ Here A = increase in surface area.

## Surface Tension on Liquid Droplet

➢ Consider a small spherical droplet of a liquid of radius ‘R'. On the entire surface of the droplet,

the tensile force due to surface tension will be acting.

➢ Let σ = surface tension of the liquid

➢ ΔP= Pressure intensity inside the droplet (in excess of the outside pressure intensity) ## Surface Tension on Bubble

➢ Consider a small spherical bubble of a liquid of radius ‘R'. On the entire surface of the bubble, the tensile force due to surface tension will be acting.

➢ Let σ = surface tension of the liquid

➢ ΔP= Pressure intensity inside the bubble (in excess of the outside pressure intensity) ## Surface Interactions with Solids

➢ When a liquid droplet comes with contact with a solid surface, due to surface tension it makes a certain angle with solid surface, called contact angle.

➢ Different Conditions :

## Capillarity

➢ Capillarity is defined as a phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of liquid when the tube is heldvertically in the liquid.

➢ The rise of liquid surface is known as capillary rise while the fall of the liquid surface is known as capillary depression.

➢ If adhesive force is less then cohesive force : Capillary fall

➢ If adhesive force is more then cohesive force : Capillary rise

➢ Applying force balance for the element : ➢ Here θ= contact angle

p= liquid density

## PRESSURE OF A FLUID

➢ All fluid molecules will be in constant and random motion called “Brownian motion”, due to which fluid at rest in a vessel, does exerts force on all the walls of the vessel, with which it is in contact.

➢ This force caused a pressure called fluid pressure. ➢ Unit of the pressure is Pascal and 1 Pascal = 1 N/m2

## Pascal’s Law

➢ It states that the pressure or the intensity of the pressure at a point in a static fluid is equal in all directions.

➢ Let us consider a elemental area dA :

➢ Forces acting on it in all directions are Fx, Fy and Fz.

➢ According to pascal’s law : ## HYDROSTATICS LAW

➢ Hydrostatic law defines the variation of pressure with respect to the elevation of fluid particles under static condition and states that “as the depth from a fluid surface level increases the pressure increases”.

➢ When an object pushed in a fluid to its bottom, it compresses the fluid, so there arise an increase in pressure force around it and is termed as “Hydrostatic force.”

➢ Let us consider a static fluid element as shown in figure.

➢ Under equilibrium of the element, the resultant of forces in any direction is zero and resolving the forces in the direction along the central axis gives.  ➢ This equation gives the variation of pressure in a static fluid in any direction.

➢ If the cylindrical fluid element is vertical, then the angle of orientation θ= 0 ➢ If depth h from the free surface of the fluid and pressure at surface is P0 then :  ➢ If the cylindrical fluid element is horizontal, then the angle of orientation θ= 90 ➢ It means in a static fluid pressure remains constant at all the points which are lies at same depth or height. It is the other statement of the hydrostatic law. ## Pressure Measurement

➢ Pressure measurement can be done by either absolute basic or with respect to atmospheric pressure.

Absolute Pressure = Gauge Pressure + Atmospheric pressure

Vacuum Pressure = Atmospheric pressure - Absolute Pressure

➢ Vacuum pressure also called negative gauge pressure.

## SOME QUESTION FROM THESE TOPICS

1.Surface tension is caused by the force of ____ at the free surface

(A) Cohesion

(D) None of these

2.Rain drop are spherical because of

(A) Viscosity

(B) Air resistance

(C) Surface tension forces

(D) Atmospheric pressure

3.The surface tension of mercury at normal temperature compared to that of water is

(A) More

(B) Less

(C) Same

(D) More or less depending on size of glass tube.

4.If the surface tension of water air interface is 0.073 N/m, the gauge pressure inside a rain drop of 1 mm diameter will be.

(A) 0.146 N/m2

(B) 73 N/m2

(C) 146 N/m2

(D) 292 N/m2

5.Property of a fluid by which molecules of different kinds of fluids are attracted to each other is called _____ .

(B) Cohesion

(C) Viscosity

(D) Compressibility

6.One stoke is equal to

(A) 1 cm2/sec

(B) 1 m2/sec

(C) 1 mm2/sec

(D) 10 m2/sec

7.Newton’s law of viscosity is a relationship between?

(A) Rate of shear strain & temperature

(B) Shear stress & rate of shear strain

(C) pressure, velocity & temperature

(D) Shear stress & velocity

8..The viscosity of a gas

(A) Decreases with increases in temperature

(B) Increases with increases in temperature

(C) Is independent of temperature

(D) In independent of pressure for very high pressure intensities.

9.Choose the correct relationship

(A) Specific gravity = gravity × density

(B) Dynamic viscosity = kinematic viscosity × density

(C) Gravity = specific gravity × density

(D) Kinematic viscosity = dynamic viscosity × density

10.Select the wrong statement

(A) Viscosity of a fluid is that property which determines the amount if its resistance to a shearing force

(B) Viscosity is due primarily to interaction between fluid molecules

(C) Viscosity of liquids decreases with increases in temperature

(D) Viscosity of liquids is appreciably affected by change in pressure

11.____ is not a property of fluid.

(A) Mass density

(B) Area

(C) Specific volume

(D) Specific weight

12.The specific gravity of liquid is 2.5 assuming g = 9.8 m/s2 , The mass of 1 Litre of the liquid

(A) 1 kg

(B) 9.8 kg

(C) 24.5 kg

(D) 2.5 kg

13.The shear stress developed in lubricating oil, of viscosity 9.81 poise, filled between two parallel plates 1 cm apart and moving with relative velocity of 2 m/s is:

(a) 20 N/m2

(b) 196.2 N/ m2

(c) 29.62 N/ m2

(d) 40 N/ m2

14.An oil of specific gravity 0.9 has viscosity of 0.28 Strokes at 380C. What will be its viscosity in Ns/m2?

(a) 0.2520 (b) 0.0311 (c) 0.0252 (d) 0.0206

15.Decrease in temperature, in general, results in

(a) An increase in viscosities of both gases and liquids

(b) A decrease in the viscosities of both liquids and gases

(c) An increase in the viscosity of liquids and a decrease in that of gases

(d) A decrease in the viscosity of liquids and an increase in that of gases

16.Newton’s law of viscosity is a relationship between?(A) Rate of shear strain & temperature

(B) Shear stress & rate of shear strain

(C) pressure, velocity & temperature

(D) Shear stress & velocity

17.The viscosity of a gas

(A) Decreases with increases in temperature

(B) Increases with increases in temperature

(C) Is independent of temperature

(D) In independent of pressure for very high pressure intensities.

18.Choose the correct relationship

(A) Specific gravity = gravity × density

(B) Dynamic viscosity = kinematic viscosity × density

(C) Gravity = specific gravity × density

(D) Kinematic viscosity = dynamic viscosity × density

19.Select the wrong statement

(A) Viscosity of a fluid is that property which determines the amount if its resistance to a shearing force

(B) Viscosity is due primarily to interaction between fluid molecules

(C) Viscosity of liquids decreases with increases in temperature

(D) Viscosity of liquids is appreciably affected by change in pressure

20.____ is not a property of fluid.

(A) Mass density

(B) Area

(C) Specific volume

(D) Specific weight

21.The specific gravity of liquid is 2.5 assuming g = 9.8 m/s2 , The mass of 1 Litre of the liquid

(A) 1 kg

(B) 9.8 kg

(C) 24.5 kg

(D) 2.5 kg

22.The shear stress developed in lubricating oil, of viscosity 9.81 poise, filled between two parallel plates 1 cm apart and moving with relative velocity of 2 m/s is:

(a) 20 N/m2

(b) 196.2 N/ m2

(c) 29.62 N/ m2

(d) 40 N/ m2

23.An oil of specific gravity 0.9 has viscosity of 0.28 Strokes at 380C. What will be its viscosity in Ns/m2?

(a) 0.2520 (b) 0.0311 (c) 0.0252 (d) 0.0206

24.Decrease in temperature, in general, results in

(a) An increase in viscosities of both gases and liquids

(b) A decrease in the viscosities of both liquids and gases

(c) An increase in the viscosity of liquids and a decrease in that of gases

(d) A decrease in the viscosity of liquids and an increase in that of gases