fluid dynamics class presentation notes.

FLUID DYNAMICS
 A fluid in motion is subjected to several forces, which
results in the variation of the acceleration and the
energies involved in the flow of the fluid. The study of
the forces and energies that are involved in the fluid
flow is known as Dynamics of fluid flow.
 The various forces acting on a fluid mass may be
classified as:
1. Body or volume forces :- Ex. Weight, Centrifugal
force, magnetic force, Electromotive force etc.
2. Surface forces :- Ex. pressure force, shear or
tangential force etc.
3. Line forces :- Ex. surface tension.

FORCES INFLUENCING MOTION OF FLUID
According to Newton’s second law of motion, the net force Fx acting on a fluid
element in the direction of x is equal to mass m of the fluid element multiplied by
the acceleration ax in the x- direction.
F = ma…………(1)
various forces that may influence the motion are due to
1. Gravity force (Fg)
2. Pressure force (Fp)
3. Viscous force (Fv)
4. Turbulence force (Ft)
5. Surface tension force (Fs)
6. Compressibility force (Fc)

FORCES ACTING ON FLUID IN MOTION:
 The various forces that influence the motion of fluid are due to
gravity force(ρg), pressure force(Fp), viscous force(Fv),
turbulence(Ft) and compressibility(Fc).
 If a certain mass of fluid in motion is influenced by all the above
forces, then according to Newto’s second law of motion :-
The net force along x-direction is given as :-
Fx =M.ax = (Fg )x + ( Fp )x + ( Fv )x + ( Ft )x + ( Fc )x
i) if the net force due to compressibility(Fc) is negligible, the
resulting net force Fx = (Fg) x + (Fp) x + (Fv) x + (Ft) x and the
equation of motions are called Reynolds’s equations of motion.
ii) For flow where (Ft) is negligible, the resulting equations of
motion are known as Navier – Stokes equation.
iii) If the flow is assumed to be ideal, viscous force (Fv) is zero
and the equations of motion are known as Euler’s equation of
motion.

According to Newton’s second law of motion, the equation (1) can be written as-
ma = F
ma = Fg + Fp + Fv + Ft + Fs + Fc……….(2)
In most of the problems of the fluids in motion the surface tension forces and the
compressibility forces are not significant. Hence these forces may be neglected .
Therefore equation (2) reduces to
1. Reynold’s equation of motion-
ma = Fg + Fp + Fv + Ft……………(3)
It is useful in the analysis of the turbulent flows.
2. Naiver –Stoke’s equation-
for laminar or viscous flows the turbulent forces also become less then
ma = Fg + Fp + Fv………….(4)
it is useful in the analysis of viscous flow.
3. Euler’s equation of motion-
viscous forces are less significant and hence it may be neglected
ma = Fg + Fp…………(5)
it is useful in the analysis of fluid flow.

Euler's equation of motion
Euler's laws of motion are equations of motion which extend Newton's laws of
motion for point particle to rigid body motion.
Consider the flow of an ideal fluid along stream tube. Seperate out a small element of
fluid of cross sectional area dA , length dS from moving fluid.
F= Fg + Fp…….(i)
Now net pressure force in the direction of flow is
Fp = pdA – ( p +dp) dA = - dpdA……..(ii)
Component of weight of the fluid element in the direction of flow is
Fg = - ρg dA dS cos θ (cos θ = ds dz)
= - ρg dA ds ds dz = - ρg dA dZ….....(iii)

mass of fluid element (m)= ρdA dS
accleeration (a)= V dv/ds
Total force F = Fp + Fg
= - dp dA - ρg dA dZ ……….(iv)
F= ma= Fp + Fg
ρ dA dS . V dv/ds = - dpdA - ρ g dA dZ
Divide both sides by – ρdA…….
Final eq…..dp/ ρ + gdZ + vdv = 0

Bernaulli's equation
BERNOULLIS THEOREM- Bernoullis principle which states that for any mass of
flowing liquid, when there is a continuous connection between all the particles of
the flowing liquid the total energy remains the same at every section provided.
there is no addition or subtractions of energy. Total energy is constant.
Assumption of Bernoulli’s Theorem-
1. Velocity is uniform.
2. Flow is uniform.
3. Flow is steady.
4. Flow is irrational
5. The fluid is incompressible.
6. Flow is caused only by gravity force
7. There is no loss of energy in the flow.
8. Centrifugal force in the curve path is neglected.
9. Frictional, as well as viscous drag forces, are also neglected.
10. Liquid flow in a continuous stream.

ENERGY POSSESSED BY A FLUID PARTICLE –
The different kinds of energy possessed by a fluid particle are :
(i) Potential energy (Datum energy)-Potential Energy of a liquid in motion It is
the energy possessed by a liquid particle by virtue of its position.
(ii) Kinetic energy -Kinetic energy of a liquid in motion It is the energy possessed
by a liquid particle by virtue of its motion or velocity.
(iii) Pressure energy -Pressure energy of a liquid in motion It is the energy
possessed by a liquid particle by virtue of its pressure.
The total energy of a liquid particle in motion is the sum of its potential
energy,Kinetic energy and pressure energy.
p/ρg + V2/2g + Z = const.
Z = potential head or Datum head
V2/2g = velocity head or kinetic head
p/ρg = pressure head or static head

fluid dynamics class presentation notes.

LIMITATIONS OF BERNOULLI’S THEOREM
1. Equation is derived with the assumption that fluid is non viscous, no friction
losses, only gravity, pressure force exists and it is incompressible.
2. Velocity is uniform but in actual practice velocity varies (maximum at
centre) and gradually decreases towards the walls of the pipe.
3. External forces acting on the fluid also affect the fluid flow.
Application of Bernoulli’s theorem
1. Venturimeter
2. Orifice meter
3. Pitot tube
4. Rotameter
5. Airplane wings
6. Measuring blood pressure
7. Swimming

VENTURIMETER
Venturimeter is a flow measurement instrument. Here, a converging section of a
pipe is used to increase the flow velocity and a corresponding pressure drop from
which the flow rate of the fluid is deduced based on Bernoulli's equation.
The venturimeter is an apparatus used to find out the discharge of a liquid flowing
at any point along a pipe line when it is running full.
Three Part-
(i) Convergent cone (ii) Throat (iii) Dovergent cone

EXPRESSION FOR THE DISCHARGE THROUGH A VENTURIMETER-
Where-
h = manometric height difference
Qact. = actual discharge
A1 = area at section 1
A2 = area at section 2
Cd = Coefficient of discharge

Orifice meter
orifice meter is a device used for measuring flow rate, for reducing pressure or
for restricting flow.
orifice plate is a thin plate with a hole in it, which is usually placed in a pipe.
When a fluid (whether liquid or gaseous) passes through the orifice, its
pressure builds up slightly upstream of the orifice[1] but as the fluid is forced to
converge to pass through the hole, the velocity increases and the fluid pressure
decreases.

EXPRESSION FOR THE DISCHARGE THROUGH A ORIFICE METER-
Where-
h = manometric height difference
Qact. = actual discharge
D1 = diameter of pipe at section 1
D2 = diameter of pipe at section 2
Cd = Coefficient of discharge

Rotameter
A rotameter is a gauge for measuring fluid flow using a graduated glass tube
with an enclosed free float. Also known as variable area flow meters,
rotameters are used to measure liquid or gas volumetric flow rates as they pass
through the tapered tube of the rotameter.

Pitot tube
A pitot tube (pitot probe) measures fluid flow velocity. It was invented by
a French engineer, Henri Pitot, in the early 18th century,[1] and was modified to its
modern form in the mid-19th century by a French scientist, Henry Darcy.[2] It is
widely used to determine the airspeed of aircraft;[3] the water speed of boats; and
the flow velocity of liquids, air, and gases in industry.

Types of Kinetic Energy
1. Radiant energy
2. Fluid energy
3. Thermal energy
4. Sound energy
5. Electrical energy
6. Mechanical energy

Momentum correction factors
Momentum correction factor is defined as the ratio of momentum of the flow per
second based on actual velocity to the momentum of the flow per second based
on average velocity across a section.
For turbulent flow the momentum correction factor is slightly higher than one
near to 1.2 and for laminar, its value is 1.33.
Momentum correction factor will be displayed by the symbol (β).
Momentum correction factor, β =(Momentum per second based on actual
velocity) / (Momentum per second based on average velocity)

Kinetic energy correction factors
Kinetic energy correction factor is basically defined as the ratio of kinetic energy
of the flow per second based on actual velocity across a section to the kinetic
energy of the flow per second based on average velocity across the same
section.
Kinetic energy correction factor will be displayed by the symbol (α)
Kinetic Energy Factor value for a fully developed laminar pipe flow is around 2
Kinetic energy correction factor, α = (K.E. per second based on actual
velocity) / (K.E. per second based on average velocity)

fluid dynamics class presentation notes.

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