This document discusses Euler's equation in fluid mechanics. It provides background on the history of understanding fluid motion, defines key terms like pressure and fluid pressure. It then defines Euler's equation, which relates velocity, pressure and density of a moving fluid based on Newton's second law of motion. Bernoulli's equation is derived from integrating Euler's equation, relating pressure, velocity and fluid height. Applications of these equations in understanding bird flight and airplane wing design are discussed. The document provides detailed definitions and derivations of these important fluid mechanics equations.

2. Euler’s Equation in
Fluid Mechanics
TITTLE

3. Contents:
 Euler's equation
 Pressure and its types
 Bernoulli's Equation
 Derivation of equation

4. History
For thousands of years, human beings vainly
sought to fly "like a bird," not realizing that this is
literally impossible, due to differences in
physiognomy between birds and homo sapiens.
No man has ever been born (or ever will be)
who possesses enough strength in his chest
that he could flap a set of attached wings and lift
his body off the ground. Yet the bird's physical
structure proved highly useful to designers of
practical flying machines.

5. Only in 1853 did Sir George Cayley (1773-
1857) incorporate the avian airfoil to create
history's first workable (though engine-less)
flying machine, a glider. Much, much older
than Cayley's glider, however, was the first
manmade flying machine built "according to
Bernoulli's & Euler Equation"—only it first

6. Motion of fluid energykinetic
Pressure in fluid energyPotential
KE + PE is constant

7. Types of Pressure
Air Pressure
Air exerts pressure because it has mass.
Because the force of gravity pulls down on the
mass of air, the air has weight.
Weight of the air is the force that produces air
pressure or atmospheric pressure.

8. Fluid Pressure
Fluid is a material that can easily flow. All
Liquids and Gases are fluids.
In a fluid, all of the forces exerted by the
individual particles combine to make up the
pressure exerted by the fluid.
Balanced Pressure
When you hold out your hand, you are
holding up air. 1,000 Newtons of air, which
is about the same weight as that of a large
washing machine!

9. Definition
The Euler's equation for steady flow of an
ideal fluid along a streamline is a relation
between the velocity, pressure and
density of a moving fluid. It is based on
the Newton's Second Law of Motion. The
integration of the equation gives
Bernoulli's equation in the form of energy
per unit weight of the following fluid.

10. What was Bernoulli said?
Bernoulli's principle states that an
increase in the speed of a fluid occurs
simultaneously with a decrease in
pressure or a decrease in the fluid's
potential energy.The principle is named
after Daniel Bernoulli who published it in
his book Hydrodynamica in 1738

11. Newton's Second Law of Motion
The acceleration of an object as produced by a
net force is directly proportional to the magnitude
of the net force, in the same direction as the net
force, and inversely proportional to the mass of
the object.
This verbal statement can be expressed in
equation form as follows:
a = Fnet / m
The above equation is often rearranged to a more
familiar form as shown below.
Fnet = m • a

12. It is based on the following
assumptions:
 The fluid is non-viscous (i,e., the
frictional losses are zero).
 The fluid is homogeneous and
incompressible (i.e., mass density
of the fluid is constant).

13.  The flow is continuous, steady and
along the streamline.
 The velocity of the flow is uniform
over the section.
 No energy or force (except gravity
and pressure forces) is involved in
the flow.

14. Derivation Of Equation
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.

16. 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

17. We know that the external forces tending to accelerate the fluid element in the direction
of the streamline
(1)
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,
( 2)
Mass of the fluid element =

18. We see that the acceleration of the fluid element
(3)
Now, as per Newton's second law of motion, we know that Force = Mass *Acceleration
Dividing both sides by
or,
(4)
This is the required Euler's equation for motion as in the form of a differential
equation.

19. Integrating the above equation,
or in other words,
which proves the Bernoulli's equation.

20. ApplicationsApplications
 What allowsWhat allows
birds andbirds and
planes to fly.planes to fly.
 The secretThe secret
behind flight isbehind flight is
‘under the‘under the
wings.’wings.’

22. AIRFOIL
On top: greater air speed and
less air pressure
On bottom: less air speed and
more air pressure

23. DRAGDRAG
THRUSTTHRUST
LIFTLIFT
GRAVITYGRAVITY