FLUID DYNAMICS
BERNOULLI’S EQUATION
MOMENTUM EQUATION
Prepaid by :(130640106110) Shah Ashish
(130640106111) Shah Nirali
(130640106112) Shah Nishita
(130640106113) Shah Rushi
(130640106114) Shah Smit
(130640106115) Sheikh Amir
(130640106117) Sheth Binal
FLUID DYNAMICS
• In physics, fluid dynamics is a subdiscipline
of
fluid
mechanics
that
deals
with fluid flow—the natural science
of fluids (liquids and gases) in motion.
• It has several subdisciplines itself, including
aerodynamics (the study of air and other
gases in motion) and hydrodynamics (the
study of liquids in motion).
BERNOULLI’S EQUATION
It states,
“in a steady , ideal flow of an incompressible
fluid, the total energy at any point of the fluid is
constant.”
the total energy consists of potential energy, kinetic
energy and pressure energy.
BERNOULLI’S EQUATION
proof :we know that,
density (w) = W/V
=> W = w*V
=> W = w*a1*dl1
=> a1*dl1 = W/w
Similarly , W =w*a2*dl2
=> a2*dl2 = W/w
=> a1*dl1 = a2*dl2 =W/w
BERNOULLI’S EQUATION
We know that work done by pressure ,
= force * displacement
= p1*a1*dl1
Similarly, work done by pressure for other side
= - p2*a2*dl2
So,
total work done by pressure,
= p1*a1*dl1 – p2*a2*dl2
=p1*a1*dl1 – p2*a2* dl1
=a1*dl1(p1-p2)
= W/w (p1-p2) ....... Pressure energy
BERNOULLI’S EQUATION
Loss of potential energy,
= WZ1- WZ2
=W (Z1 – Z2)
Gain in kinetic energy ,
= W( V2^2/2g-v1^2/2g)
=W/2g (V2^2 – V1^2)
We know that ,
Loss of potential energy + work done by pressure .
= gain in kinetic energy
BERNOULLI’S EQUATION
So,
W (Z1-Z2) + W/w(p1-p2) = W/2g(V2^2-V1^2)
(Z1-Z2)+P1/w-p2/w=v2^2/2g-v1^2/2g
Z1 + V1^2/2g + p1/w = Z2 + v2^2/2g + p2/w
........... Bernoulli’s equation .
MOMENTUM EQUATION
The momentum equation is a statement of Newton's Second
Law and relates the sum of the forces acting on an element of
fluid to its acceleration or rate of change of momentum. You
will probably recognise the equation F = ma which is used in
the analysis of solid mechanics to relate applied force to
acceleration. In fluid mechanics it is not clear what mass of
moving fluid we should use so we use a different form of the
equation.
Newton's 2nd Law can be written:
• The Rate of change of momentum of a body is equal to the
resultant force acting on the body, and takes place in the direction
of the force.
• To determine the rate of change of momentum for a fluid we will
consider a streamtube as we did for the Bernoulli equation,
• We start by assuming that we have steady flow which is nonuniform flowing in a stream tube.
MOMENTUM EQUATION
MOMENTUM EQUATION
• In time a volume of the fluid moves from the inlet a
distance , so the volume entering the streamtube in
the time is
• this has mass,
• and momentum
• Similarly, at the exit, we can obtain an expression for
the momentum leaving the steamtube:
•
We can now calculate the force exerted by the fluid
using Newton's 2nd Law. The force is equal to the
rate of change of momentum
MOMENTUM EQUATION
• We know from continuity that , and if we have a fluid
of constant density, i.e. , then we can write
•
For an alternative derivation of the same expression,
as we know from conservation of mass in a stream
tube that
• we can write
MOMENTUM EQUATION
• The rate at which momentum leaves face 1 is
• The rate at which momentum enters face 2
is
• Thus the rate at which momentum changes
across the stream tube is
• i.e.
• This force is acting in the direction of the
flow of the fluid.
MOMENTUM EQUATION
• The force in the x-direction
• And the force in the y-direction
MOMENTUM EQUATION
And the force in the y-direction
MOMENTUM EQUATION
• We then find the resultant force by combining
these vectorially:
• And the angle which this force acts at is given
by
MOMENTUM EQUATION
1. Fluid Mechanics by G.J. RAJPARA
2. Bernoulli’s Equation by R.P.
RETHALIYA