Bernoulli's Principle
MECHANICAL PROPERTIES OF FLUIDS
1. Bernoulli's equation is a general expression that relates the
pressure difference between two points in a pipe to both
velocity changes (kinetic energy change) and elevation
(height) changes (potential energy change).
2. The Swiss Physicist Daniel Bernoulli developed this
relationship in 1738.
3. This is considered to be the qualitative behavior that lowers
the pressure in the regions with high velocities. This is termed
as the Bernoulli effect.
For a streamline fluid flow, the sum of the pressure(P),
the kinetic energy per unit volume (ρv^2/2) and the
potential energy per unit volume(ρgh) remains
constant.
Bernoulli's
P+ρv^2/2+ρgh
Equation
K.E/Vol. = mv^2/2v = 1/2ρv^2
P.E/Vol. = mgh/v = (m/v)gh = ρgh
DERIVATION - ASSUMPTIONS
Fluid flow through a pipe of varying width.
Pipe is located at changing heights.
Fluid is incompressible.
Flow is laminar.
No energy is lost due to friction : applicable only to non-
viscous fluids.
W2
{
t
R S W1 = F1 x disp.
= P1A1 x V1 t
W2 = F2 x disp.
W1 = P2A2 x V2 t
A2
{
t Net Work Done = W1-W2
P Q h2 = P1A1V1 t - P2A2V2 t
-A1 j h1
Therefore, the work done on the fluid is given as:
dW = F1dx1 – F2dx2
dW = p1A1dx1 – p2A2dx2
dW = p1dV – p2dV = (p1 – p2)dV.
We know that the work done on the fluid was due to conservation of gravitational force and change
in kinetic energy. The change in kinetic energy of the fluid is given as:
dK=1/2m2v22−1/2m1v21=1/2ρdV(v22−v21)
The change in potential energy is given as:
dU = mgy2 – mgy1 = ρdVg(y2 – y1).
Therefore, the energy equation is given as:
dW = dK + dU
(p1 – p2)dV = 1/2ρdV(v22−v21) + ρdVg(y2 – y1)
(p1 – p2) = 1/2ρ(v22−v21) + ρg(y2 – y1)
Rearranging the above equation, we get,
p1+1/2ρv21+ρgy1=p2+1/2ρv22+ρgy2.
This is Bernoulli’s equation.
PRINCIPLE OF CONTINUITY
If the fluid is in streamline flow and is in-compressible then we
can say that mass of fluid passing through different cross
sections are equal.
From the above situation, we can say the mass of liquid inside the
container remains the same.
The rate of mass entering = Rate of mass leaving
The rate of mass entering = ρA1V1Δt--------------------------(1)
The rate of mass entering = ρA2V2Δt--------------------------(2)
Using the above equations,
ρA1V1=ρA2V2
This equation is known as the Principle of continuity.
SPEED OF EFFLUX: TORRICELLI'S LAW
Suppose we need to calculate the speed of efflux for the following
setup.
Using Bernoulli’s equation at point 1 and point 2,
p+1/2ρv21+ρgh=p0+1/2ρv22v22=v2/+2p−p0/ρ+2gh
Generally, A2 is much smaller than A1; in this case, v12 is very
much smaller than v22 and can be neglected. We then find,
v22=2p−p0ρ+2gh
Assuming A2<<A1,
We get, v2=√2gh.
Hence, the velocity of efflux is √2gh
VENTURI METER
It is a device that is based on Bernoulli’s theorem and is used
for measuring the rate of flow of liquid through the pipes.
Using Bernoulli’s theorem, Venturi meter formula is given as:
BLOOD FLOW AND HEART ATTACK
In persons suffering with advanced heart condition, the artery gets constricted
due to the accumulation of plaque on its inner walls. In order to drive the blood
through this constriction a greater activity of the heart is required. The speed of
the flow of the blood in this region is raised which lowers the pressure inside, and
the artery may collapse due to this external pressure.The heart exerts further
pressure to open this artery and forces the blood through. As the blood rushes
through the opening, the internal pressure once again drops leading to a repeat
collapse. This may result in heart attack.
DYNAMIC LIFT AND MAGNUS EFFECT
Dynamic lift is the force which acts on a body such as an airplane wing hydro fall or spinning
ball by virtue of its motion through a fluid.
For example, during the game of cricket, tennis, baseball, or golf, we have noticed that a
spinning ball deviates from its parabolic trajectory during its motion in the air.
1. Ball moving without spin
2. Ball moving with spin
The Magnus effect is an observable phenomenon that is commonly associated with a spinning
object moving through the air or a fluid.
In the case of a ball spinning through the air, the turning ball drags some of the air around
with it. A spinning object moving through a fluid departs from its straight path because of
pressure differences that develop in the fluid as a result of velocity changes induced by the
spinning body.
1. Aerofoil or lift on aircraft wing.
BERNOULLI’S PRINCIPLE EXAMPLE
Q1. Calculate the pressure in the hose whose absolute pressure is 1.01 x 105 N.m-2 if the speed
of the water in hose increases from 1.96 m.s-1 to 25.5 m.s-1. Assume that the flow is
frictionless and density 103 kg.m-3
Ans: Given,
Pressure at point 2, p2 = 1.01 × 105 N.m-2
Density of the fluid, ρ = 103 kg.m-3
Velocity of the fluid at point 1, V1 = 1.96 m.s-1
Velocity of the fluid at point 2, V2 = 25.5 m.s-1
From Bernoulli’s principle for p1,
p1=p212ρv22−12ρv21=p2+12ρ(v22−v21)
Substituting the values in above equation, we get
p1=(1.01×105)+12(103)[(25.5)2−(1.96)2]
p1 = 4.24 × 105 N.m-2
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