CE6451 Fluid Mechanics and Machinery

What is Fluid Mechanics? First, what is a fluid?

Three common states of matter are solid, liquid, and gas.

A fluid is either a liquid or a gas.

Fluid:A fluid is a substance which is capable of flowing. A fluid

may be classified in to Liquid, gas and Vapour.

Next, what is mechanics? Mechanics is essentially the application of the laws of

force and motion. Conventionally, it is divided into two

branches, statics and dynamics.

Fluid Mechanics:

It deals with the behavior of fluid under the condition of

rest and motion. It may be divided in to three parts: Statics,

Kinematics and dynamics.

UNIT I FLUID PROPERTIES AND FLOW CHARACTERISTICS

Units and dimensions‐ Properties of fluids‐ mass density, specific weight, specific

volume, specific gravity, viscosity, compressibility, vapor pressure, surface tension

and capillarity. Flow characteristics – concept of control volume ‐ application of

continuity equation, energy equation and momentum equation.

Fluid Properties: 1. Density

2. Specific Gravity

3. Viscosity

4. Surface Tension

5. Capillarity

6. Compressibility

7. Bulk modulus

8. Vapour pressure

Density:

Mass density or specific mass: ρ

The density of a fluid is defined as its mass per unit volume at

standard temperature and pressure.

fluidtheofVolumefluidtheofMass

=ρ

ρ=m/v

unit , kg/m3

For water, ρ = 1000 kg/m3

Specific Weight or weight density: wIt is defined as Weight per unit volume at

standard temperature and pressure

gfluidofVolume

fluidofMassfluidofVolumefluidofWeightw ×==

w = ρg ;Unit: N/m3 ; w for water=9.81 kN/m3

Specific Volume: vIt is defined as Volume per unit mass of the fluid.

v=V/m = 1/ρ

Unit, m3/kg1 m3 = 1000 litres

Specific Gravity : S

It is the ratio of the specific weight of the liquid to thespecific weight of the standard fluid.

waterpure theof weight Specificliquid theof weight specific

=avitySpecificGr

Weight Density of Liquid= S X Weight density of water= S X 1000 X 9.81 N/m3

Density of Liquid= S X density of water= S X 1000 kg/m3

Kinematic viscosity:It is defined as the ratio between the dynamic viscosity and mass density of fluid.

Units: m2/s ; 1 Stroke = 10-4 m2/s

Viscosity:Dynamic viscosity:

Viscosity is defined as the property of fluid which offers resistance to flow.

Unit : Ns/m2 ; 1 Poise = (1/10) Ns/m2 ; 1 Centipoise= (1/100) Poise 

,,dydu

dydu μττα =

ρμν =

Dynamic Viscosity

ForceArea

Shear stress ( τ)

Shear rate (du/dy)

Slope of lineμ =

dyduτμ =

Types of Fluids:

Shear stress

Shear rate du/dy

Ideal Solid

Ideal Fluid

Ideal Fluid:

A fluid, which is incompressible and is having no viscosity, is known as Idealfluid. It is only imaginary Fluid

Real Fluid:

A fluid, which is having viscosity, is known as Real fluid. All fluids are Realfluids.

Newtonian fluids:

These fluids follow Newton’s viscosity equation. For such fluids μ does notchange with rate of deformation. Exp. Water, Kerosene etc.,

Non Newtonian fluids:

There fluids which do not follow the linear relationship between the shearstress and the rate of deformation. Exp. Polymer solutions, Blood et.,

Compressibility:When a fluid is subjected to a pressure increase the

volume of the fluid decreases. It is known as compressibility. Itis reciprocal of bulk modulus.

Compressibility = 1/k

Bulk modulus:It is define as the ratio of compressive stress to volumetric

strain.

The negative sign indicates the volume decreases as pressureincreases.

Unit :‐

vdvdpk −=

2mN

Surface Tension of Liquids

The phenomenon of surface tension arises due to the two

kinds of intermolecular forces. Surface Tension is caused by

the force of cohesion at the free surface.

(i) Cohesion : The force of attraction between the molecules of

the same liquid. Surface tension is due to cohesion between

particles at the free surface.

(ii) Adhesion : The force of attraction between unlike molecules

of a solid boundary surface in contact with the liquid

Case I ‐Water droplet:Let, P= Pressure inside the droplet above outside pressureD= Diameter of the dropletσ = Surface tension of the liquid

i. Pressure force = and

ii. Surface tension force acting around the circumference =σx π d

Under equilibrium conditions these two forces will be equal and opposite,

2

4dP π

×

dp

ddP

σ

πσπ

44

2

=

×=×

Case II ‐ Soap bubble:

i. Pressure force =                   , andii. Surface tension force acting around the circumference =2(σXπd)

2

4dP π

×

( )

dp

ddP

σ

πσπ

8

24

2

=

×=×

( )

dp

ldlPσσ2

2

=

×=××

Case III‐ A liquid Jet:i. Pressure force = p x l x d , andii. Surface tension force acting around the circumference =σX 2l

Capillarity:

Capillarity is a phenomenon by which a liquid rises into a thin

glass tube above or below its general level. surface relative to the

adjacent level of the fluid is called capillarity.

This phenomenon is due to the combined effect of cohesion and

adhesion of liquid particles.

For water θ= 0

For mercury θ= 1400

Expression for Capillary Rise:

σ=Surface tension of the liquidθ= Angle of contact between the liquid and glass tube

ρ = Density of the liquid

The weight of the liquid of height h in the tube = (Area of the tube x h) x ρ x g  

ghd ×××= ρπ 2

4Vertical component of the surface tensile force

θπσθσ

coscos)(

××=××=

dnceCircumfere

Equating these two eq. θπσρπ cos4

2 ××=××× dghd

dggd

dh××

=××

××=

ρθσ

ρπθπσ cos4

4

cos2

dgh

××=ρ

σ4If θ is equal to zero

Expression for Capillary Fall:

σ=Surface tension of the liquidθ= Angle of contact between the liquid and glass tube

ρ = Density of the liquid

Intensity of the pressure at the depth = (Area of the tube x h) x ρ x g  

ghd ×××= ρπ 2

4Vertical component of the surface tensile force

θπσθσ

coscos)(

××=××=

dnceCircumfere

Equating these two eq. θπσρπ cos4

2 ××=××× dghd

gd

dh××

××=

ρπθπσ

2

4

cos

dgh

××=ρ

θσ cos4θ For mercury and glass tube is 138o

Vapour pressure:• All liquids have a tendency to evaporate when exposed to a

gaseous atmosphere. The vapour molecules exert a partial

pressure in the space above the liquid, known as vapour pressure.

Gas Laws:Boyle’s LawAt a given temperature for a given quantity of gas, the product

of the pressure and the volume is a constant.P1V1= P2V2

Ideal Gas Law:Ideal Gas Law relates pressure to Temp for a gas 

P = ρRTT = Absolute temperature in oK unitsR= Gas constant = 287 Joule / Kg‐oK

Universal Gas constant:

PV=mRT

Where,M=Mass of gas in kg;     V= Specific Volume.P=Absolute pressure      ;  T=Absolute temperature

Pressure in a Fluid

hgp ρ=

Pressure:Force per unit area: 1 N/m2 = Pascal = Pa

Standard Atmosphere P = 101.33 kPa = 101.3 kN/m2

=10.3 m of mg