PCET4-Fluid-Mechanics-Module-2-Properties-of-Fluids.pptx

2.1 Compressible and Incompressible
Fluids
 INCOMPRESSIBLE FLUIDS
• A fluid is considered incompressible if its density remains
nearly constant regardless of pressure or temperature
changes.
• Assumption: Most liquids are treated as incompressible
since their density variations are negligible under normal
conditions.
• Applications:
o Hydraulics (water in pipelines, pumps, and turbines)
o Aerodynamics at low speeds (air at speeds less than
Mach 0.3)

Fluids
• For incompressible fluids, density ρ (rho) is constant. The
main governing equations include:
1. Continuity Equation (Conservation of Mass)
where: A = cross-sectional area (m2
)
V = velocity (m/s)
𝑨𝟏𝑽 𝟏=𝑨𝟐𝑽 𝟐

Fluids
2. Bernoulli’s Equation (Energy Conservation for
Incompressible Flow)
where: P = pressure (Pa)
V = velocity (m/s)
ρ = fluid density (kg/m3
)
g = gravity (9.81 m/s2
)
h = height (m)

Fluids
 EXAMPLE: Water flows through a horizontal pipe that
narrows from 10 cm diameter to 5 cm diameter.
The pressure in the wider section is 200 kPa, and
the velocity is 3 m/s. Find:
1. The velocity in the narrow section.
2. The pressure in the narrow section.

Fluids
 COMPRESSIBLE FLUIDS
• A fluid is compressible if its density significantly changes
when pressure or temperature changes.
• Mach number (M) determines compressibility effects:
o M < 0.3 Incompressible
→
o M 0.3 Compressible effects become significant
≥ →
o M 1 Supersonic flow
≥ →

Fluids
• Applications:
o High-speed aerodynamics (airplanes, rockets, jet
engines)
o Gas dynamics (steam turbines, nozzles, shock waves)
o Compressors and gas flow systems

Fluids
• For compressible fluids, density varies with pressure and
temperature. The governing equations include:
1. Equation of State (Ideal Gas Law)
where: P = pressure (Pa)
ρ = density (kg/m3
)
R = specific gas constant (J/kg·K)
T = temperature (K)

Fluids
2. Speed of Sound in a Gas
where: a = speed of sound (m/s)
γ = ratio of specific heats (Cp/Cv)
R = specific gas constant (J/kg·K)
T = temperature (K)

Fluids
3. Mach Number: The ratio of an object's speed to the
speed of sound in the same medium. It helps determine
whether flow is incompressible or compressible.
where: M = Mach number
V = velocity of fluid (m/s)
a = speed of sound (m/s)

Fluids
3. Isentropic Flow Equations
These equations describe the relationship between
temperature, pressure, and density in an isentropic process.

Fluids
 EXAMPLE: Find the speed of sound in air at 300 K where γ is
1.4 and specific gas constant is 287 J/kg·K.
 EXAMPLE: A jet is flying at 500 m/s. If the speed of sound in
the surrounding air is 340 m/s, determine
the Mach number.
 EXAMPLE: A gas is contained in a closed cylinder at a
pressure of 150kPa, temperature of 350K,
and occupies a volume of 0.5m³. If the gas is
oxygen (O₂) with a specific gas constant R=259.8 J/kg·K,
determine the mass of the gas inside the cylinder.

Fluids
 EXAMPLE: Air flows through a converging-diverging nozzle
under isentropic conditions. The stagnation
pressure is 500 kPa, and the pressure at a certain
point in the nozzle is 200 kPa. Assuming air
behaves as an ideal gas with γ=1.4, find:
1. The temperature ratio T1/T2
2. The density ratio ρ1/ρ2

2.1 Differential and Integral form of the
Fluid Dynamic Equation
 Fluid dynamics is governed by fundamental equations that
describe how fluid velocity, pressure, and other properties
change in space and time. These equations can be written in two
forms:
1. Integral Form – Describes the behavior of the entire fluid
system as a whole.
2. Differential Form – Describes fluid properties at an
infinitesimally small point.

 INTEGRAL vs. DIFFERENTIAL FORMS

1. Integral Form of Fluid Dynamic Equations
The integral form applies to a control volume (a fixed region in
space) and is derived from fundamental conservation laws:
1.1. Conservation of Mass (Continuity Equation - Integral Form)
The Conservation of Mass states that mass cannot be
created or destroyed in a fluid system. This principle leads
to the Continuity Equation, which expresses that the mass
flow rate of a fluid remains constant along a streamline in
steady flow.

The integral form of the Continuity Equation is used for a
control volume approach, analyzing fluid flow through an
enclosed region. It is given as:
where: ρ = fluid density (kg/m3
) CV = control volume
V = velocity vector (m/s) CS = control surface
dA = elemental area on the control surface

• First term: Rate of change of mass inside the control
volume.
• Second term: Net mass flux (mass flow rate) across the
control surface.
• If the fluid is incompressible or steady-state flow (d/dt),
the equation simplifies to:
which means the mass flow rate in = mass flow rate
out.

• EXAMPLE: Air enters a converging-diverging nozzle with a
velocity of 150m/s at a section where the cross-
sectional area is A1 = 0.02m² and the air density is ρ1 =
1.2kg/m³.
At the throat of the nozzle, the area reduces to A2 =
0.01m², and the density of air changes to ρ2 = 0.9kg/m³ due
to compressibility effects.
Find the velocity of air at the throat V2
.

1.2. Momentum Equation (Newton’s Second Law - Integral Form)
The Momentum Equation in fluid mechanics is derived
from Newton’s Second Law, which states that the net force
acting on a fluid control volume equals the rate of change
of momentum of the fluid.
where: ΣF = sum of forces acting on the control volume
(pressure,
gravity, shear, etc.)
• The change in momentum inside the control volume +
net flux of momentum out of the control surface = Total

1.2. Momentum Equation (Newton’s Second Law - Integral Form)
In steady flow (d/dt), the equation simplifies to:
This equation is used to analyze the force exerted by fluids
in motion, such as in jets, nozzles, turbines, and pipe bends.

• EXAMPLE: A water jet with a velocity of 20 m/s and a mass
flow rate of 10 kg/s strikes a flat vertical plate
perpendicularly and is brought to rest. Find the force
exerted by the jet on the plate.

• EXAMPLE: Water flows through a pipe bend at steady state.
The inlet diameter is 0.3m, and the velocity at the
inlet is 5 m/s. The outlet diameter is 0.2m, and the
velocity at the outlet is 9m/s.
The bend turns the water 90° upward while
maintaining the same pressure at the inlet and
outlet.
The density of water is 1000 kg/m³.
Find the horizontal and vertical forces that the bend
exerts on the fluid.

1.3. Energy Equation (First Law of Thermodynamics - Integral
Form)
where: e = total energy per unit mass; (internal energy,
kinetic energy, potential energy)
=
Q = heat transfer rate (W)
W = work done by the system (W)
• Rate of energy change inside the control volume + net
energy flux through the surface = Heat & Work input/output.

1.3. Energy Equation (First Law of Thermodynamics - Integral
Form)
For a steady-state system (no change in stored energy), the
equation simplifies to:
This equation is useful in analyzing:
• Pumps and turbines (where work is done)
• Heat exchangers (where heat is added or removed)
• Pipes and nozzles (where kinetic energy and pressure
change)

• EXAMPLE: A pump moves water from one reservoir to
another at a rate of 50 kg/s.
The water enters at 10 m and exits at 50 m
elevation.
The inlet velocity is 3 m/s, and the outlet velocity is
5 m/s.
The pressure at both inlet and outlet is 100 kPa.
Find the power required by the pump assuming no
heat transfer and neglecting friction losses.

2. Differential Form of Fluid Dynamic Equations
The differential form describes the flow at every infinitesimal
point and is obtained using differential calculus.
2.1. Continuity Equation (Differential Form)
The Continuity Equation represents the principle of
conservation of mass in fluid mechanics. The differential
form describes how mass is conserved at a point within a
fluid flow field.

For a compressible fluid, the differential form of the
continuity equation is:
where: ρ = fluid density (kg/m³)
V= (u,v,w) = velocity vector in Cartesian coordinates
∂t/ ρ= local rate of change of density
∂
∇⋅(ρV) = divergence of mass flux (how mass flows
into or out of a small volume)

For incompressible flow (ρ constant), the equation
simplifies to:
This means that in an incompressible flow, the net
volume flow rate entering or leaving a small control
volume is zero.

• EXAMPLE: A steady, two-dimensional, incompressible flow
has velocity components:
u=3x+2y, v= 2x+y
−
Check if the continuity equation is satisfied.

2.2. Navier-Stokes Equation (Momentum Equation - Differential
Form)
The Navier-Stokes equations describe the motion of
viscous fluid substances and are derived from Newton’s
Second Law (Force = Mass × Acceleration) applied to fluid
motion.
The differential form of the Navier-Stokes equation for
an incompressible, Newtonian fluid is:

Form)
where: ρ = fluid density (kg/m³)
V = (u,v,w) = velocity vector (m/s)
P = pressure (Pa)
μ = dynamic viscosity (Pa·s)
∇2
V = Laplacian of velocity (viscous diffusion term)
ρg = body force per unit volume (e.g., gravity)

Form)
Breakdown of terms:
Unsteady term ( V/ t) describes changes in velocity
∂ ∂ →
over time.
Convective term V V accounts for fluid inertia.
⋅∇ →
Pressure term P force due to pressure gradients.
−∇ →
Viscous term μ∇2
V accounts for internal friction due to
→
viscosity.
Body force term ρg external forces like gravity.
→

Form)
For inviscid flow (μ=0):
(Simplifies to Euler’s Equation.)

Form)
where: = kinetic energy
q = heat source term
The total energy per unit mass changes due to work,
heat transfer, and viscous dissipation.

PCET4-Fluid-Mechanics-Module-2-Properties-of-Fluids.pptx

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