Bernoulli's Theorem :
We will learn that Bernoulli equation relates the speed of a fluid at a point, the pressure at that point and the height of that point above a reference level. It is just the application of work–energy theorem in the case of fluid flow. The analysis of the flow of a fluid becomes much simplified if we consider the fluid to be incompressible and nonviscous and that the flow is irrotational. Incompressibility means that the density of the fluid is same at all the points and remains constant as time passes.
This assumption is quite good for liquids and is valid in certain cases of flow of gases. Viscosity of a fluid is related to the internal friction when a layer of fluid slips over another layer. Mechanical energy is lost against such viscous forces. The assumption of a nonviscous fluid will mean that we are neglecting the effect of such internal friction. Irrotational flow means there is no net angular velocity of fluid particles. When you put some washing powder in a bucket containing water and mix it by rotating your hand in circular path along the wall of the bucket, the water comes into rotational motion. Quite often water flowing in rivers show small vortex formation where it goes in rotational motion about a centre a liquid possesses three kinds of energies.
It is the energy possessed by a liquid due to its velocity. So, kinetic energy possessed by a liquid of mass
m
moving with a speed
v
is
For liquids it is customary to talk about energy per unit mass or energy per unit volume, so we have
Potential Energy
It is the energy possessed by a liquid due to its position.
Consider a beaker filled to a height
h
with liquid of density
p
. The beaker is placed on a horizontal surface as shown in Figure.
Assuming zero potential energy level (ZPEL) to be assigned to the horizontal surface, the potential energy of the free surface of the liquid is
P
.
E
. =
mgh
It is the energy possessed by a liquid due to its pressure. Consider a beaker having frictionless piston attached to its base and filled with liquid of density
as shown in Figure.
∙
The piston will move backwards because of the force due to pressure given by
F
=
PA
. The area of the piston is assumed to be small, so that there is no variation of pressure on the cross-section of the piston. Work done (
W
) by this force to displace the piston backwards by
x
is
W
=
Fx
= (
PA
)
x
=
P
(
Ax
) =
PV
This work done is stone as pressure energy in the liquid, so we have
Pressure Energy =
PV
So, from above we see that pressure energy per unit volume is just equal to the pressure of the liquid.
-
Bernoulli’s equation relates the pressure, flow speed and height for flow of an ideal (incompressible and non-viscous) fluid. The pressure of a fluid depends on height as in the static situation and it also depends on the speed of flow.
-
The dependence of pressure on speed can be understood from the continuity equation. When an incompressible fluid flows along a tube with varying cross section, its speed must change, and so, an element of fluid must have an acceleration. If the tube is horizontal, the force that causes this acceleration has to be applied by the surrounding fluid. This means that the pressure must be different in regions of different cross section.
-
When a horizontal flow tube narrows and a fluid element speeds up, it must be moving towards a region of lower pressure in order to have a net forward force to accelerate it. If the elevation also changes, this causes additional pressure difference
Proof Of Bernoulli's Equation
Figure shows flow of an ideal fluid through a tube of varying cross section and height. The fluid at the left end exerts a force
P
1
A
1
, which moves the fluid through a distance Δ
x
1
and therefore does work
W
1
=
F
1
Δ
x
1
=
P
1
A
1
Δ
x
1
on the fluid. At the same time an equal volume of fluid at the upper right end is displaced through Δ
x
2
. So, the work done on this fluid volume is
W
1
=
F
1
Δ
x
1
=
P
1
A
1
Δ
x
1
∙
At right end of tube the work done on this fluid element is negative because the force on it is in the opposite direction to the displacement. The total work done on the entire fluid element shown is
W
=
W
1
+
W
2
=
P
1
A
1
Δ
x
1
–
P
2
A
2
Δ
x
2
Using equation of continuity,
A
1
Δ
x
1
=
A
2
Δ
x
2
= Δ
V
W
= (
P
1
–
P
2
)
Δ
V
The change in kinetic energy between the right and left end is
And the change in potential energy is
Δ
U
= Δ
mgh
2
– Δ
mgh
1
Using energy conservation,
Δ
W
= Δ
K
+ Δ
U
Dividing this equation by Δ
V
and substituting
we find
…(i)
Equation (i) is called Bernoulli's equation. This equation is, in fact, the conservation of energy.
Applications And Limitations
-
The flow is assumed to be steady, i.e., there is no change in pressure, velocity and density of the fluid at any point with respect to time.
-
However, in the problems of unsteady flow with gradually changing conditions, Bernoulli's equation can be applied without appreciable error. For example, a problem of emptying a large tank can be solved by applying Bernoulli's equation.
-
The flow is assumed to be incompressible. Since liquids are incompressible, Bernoulli's equation can be applied to all liquids. However, it can be applied to the problems of gas flow when there is little variation in pressure, velocity and temperature so that density of gas can be assumed to be constant.
-
The flow is assumed to be irrotational. The irrotationality means the net angular momentum at any point in the fluid flow is zero.
-
The fluid is assumed to be ideal, i.e., the energy loss due to friction is assumed to be absent.
Magnus effect
Magnus effect :
When a spinning ball is thrown, it deviates from its usual path in flight. This effect is called Magnus effect and plays an important role in tennis, cricket, soccer, etc., as by applying appropriate spin the moving ball can be made to curve in any desired direction. If a ball is moving from left to right and also spinning about a horizontal axis perpendicular to the direction of motion as shown in Fig., then relative to the ball air will be moving from right to left.
Attraction between two closely parallel moving boats
Attraction between two closely parallel moving boats (or buses):
When two boats or buses move side by side in the same direction, the water (or air) in the region between them moves faster than that on the remote sides. Consequently in accordance with Bernoulli's principle the pressure between them is reduced and hence due to pressure difference they are pulled towards each other creating the so-called attraction.
Blowing off roofs by wind storms
Blowing off roofs by wind storms :
During tornado or hurricane, when a high-speed wind blows over a straw or tin roof, it creates a low pressure (
p
) in accordance will Bernoulli's principle. However, the pressure below the roof (i.e., inside the room) is still atmospheric (=
P
0
). So due to this difference of pressure the roof is lifted up and is then blown off by the wind.
Action of atomiser
Action of atomiser :
The action of aspirator, carburettor, paint-gun, scent-spray or insect-sprayer is based on Bernoulli's principle. In all these, by means of motion of piston
P
in cylinder
C
, high speed air is passed over tube
T
dipped in liquid
L
to be sprayed. High speed air creates low pressure over the tube due to which liquid (paint, scent, insecticide or petrol) rises in it and is then blown off in very small droplets with expelled air.
Working of aeroplane
∙T
his is also based on Bernoulli's principle. The wings of the aeroplane are having tapering as shown. Due to this specific shape of wings when the aeroplane runs, air passes at higher
⇒
speed over it as compared at its lower surface.
