This document outlines a presentation on fluid mechanics concepts including the momentum equation, Bernoulli's theorem, and applications. It discusses the momentum equation, force exerted by fluid flow on pipe bends using a 2D flow equation, Euler's equation of motion, Bernoulli's theorem, and applications like Pitot tubes, Venturimeters, and orifice meters. It also provides an elementary introduction to notches and weirs.

1. In partial fulfillment of the subject
Fluid Mechanics
Submitted by:
Sagar Damani 130120119041 (MECH:4-A{A3})
Sanket Chopde 130120119039
2141906
GANDHINAGAR INSTITUTE OF TECHNOLOGY
PRESENTATION
ON
The Energy Equation and its Application

2. Chapter Outline
Momentum equation
Force exerted by the fluid flow on pipe bend(2-D
Dimensional flow equation)
Euler’s equation of motion
Bernoulli’s theorm
Applications of Bernoulli’s theorm
 Pitot tube
 Venturimeter
Orifice meter
Elementary introduction of Notches and Weirs

3. THE MOMENTUM EQUATION
• It is based on the law of conservation of momentum or on
the momentum principle,
Which states,
• “the net force acting on a fluid mass is equal to the
change in momentum of flow per unit time in that
direction”.
The force acting on a fluid mass’m’is given by the
Newton’s second law of motion.
F=m*a

4. Conti…
Now, a=dv/dt
F=m(dv/dt)
=d(mv)/dt
Equation is known as the momentum principle.
Equation can also be written as
F.dt=d(mv)
Which is known as the impulse-momentum equation and states
that the impulse of a force F acting on a fluid o mass m in a short
interval of time dt is equal to the change of momentum d(mv) in
the direction of force.

5. Conti…

6. Force exerted by a flowing fluid on a pipe-bend
Consider two sections(1) and(2),as shown in fig.
Let v1=vel.of flow at section(1)
p1=pressure intensity at section(1)
A1=area of cross-section of pipe at section(1)
V2,P2,A2=corresponding values at section (2).

7. Conti…
Let Fx and Fy be the components of the forces exerted by the flowing fluid on
the bend in x and y respectively.Then the force exerted by the bend on the fluid
in the direction of x and y will be Fx and Fy but in opp.direction.
Net force acting on fluid in the x direction=Rate of change of momentum in x
direction.
P1A1-P2A2 cosθ-Fx=M.dv
=density.Q(V2 cosθ-V1)
Fx= density.Q(V2 cosθ-V1)+P1A1-P2A2 cosθ
Similarly the momentum equation in Y-direction gives
0-P2A2 sinθ-Fy=density.Q(-V2 sinθ-0)
Fy=density.Q(-V2 sinθ)-P2A2 sinθ
Now the resultant force (Fr) acting on the bend and angle made by the F with X
=

8. Euler’s Equation Of Motion
Let us consider a steady flow of an ideal fluid along a
streamline and small element AB of the flowing fluid
as shown in figure.

9. Conti…
Let,
•dA = Cross-sectional area of the fluid element
•ds = Length of the fluid element
•dW = Weight of the fluid element
•P = Pressure on the element at A
•P+dP = Pressure on the element at B
•v = velocity of the fluid element
We know that the external forces tending to accelerate the
fluid element in the direction of the streamline

10. We also know that the weight of the fluid element,
From the geometry of the figure, we find that the component
of the weight of the fluid element in the direction of flow,

11. Mass of the fluid element =
We see that the acceleration of the fluid elementt

12. Now, as per Newton's second law of motion, we know
that Force = Mass *Acceleration
Dividing both sides by
,
Or
This is the required Euler’s equation of motion for a fluid

13. BERNOULLI’S THEOREMBERNOULLI’S THEOREM
Bernoulli’s theorem which is also known as Bernoulli’s
principle, states that an increase in the speed of moving
air or a flowing fluid is accompanied by a decrease in the
air or fluid’s pressure or sum of the kinetic (velocity
head), pressure(static head) and Potential energy of the
fluid at any point remains constant, provided that the
flow is steady, irrotational, and frictionless and the fluid
is incompressible.
The Bernoulli equation is an
approximate equation that is valid
only in in viscid regions of flow where net
viscous forces are negligibly small
compared to inertial, gravitational, or
pressure forces. Such regions occur
outside of boundary layers and wakes.

14. BERNOULLI’S EQUATIONBERNOULLI’S EQUATION
If a section of pipe is as shown above,
then Bernoulli’s Equation can be written as;

15. BERNOULLI’S EQUATIONBERNOULLI’S EQUATION
Where (in SI units)
P= static pressure of fluid at the cross section;
ρ= density of the flowing fluid in;
g= acceleration due to gravity;
v= mean velocity of fluid flow at the cross section in;
h= elevation head of the center of the cross section with
respect to a datum.

16. Limitations on the Use of the Bernoulli
Equation
Steady flow and incompressible flow
No heat transfer into and out of the fluid
Constant internal energy (constant
temperature)
Assumptions made in Bernoulli’s equation:
 Ideal fluid
Stream lined flow
Irrotational flow
The gravity force and pressure force are only
considered

27. NOTCHES AND WEIRS
Notches
It is defined as a device which measures the flow rate of a
liquid through a small channel or tank.
It is an opening in the side of a tank or a small channel in such
a way that the surface of liquid in the tank or small channel is
below the top edge of opening.

28. Weirs
It is a concrete or masonary structure which is placed
in an open channel over which the flow occurs.

29. THANK YOU.