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6. Bernoullis Equation
It relates the variables describing the steady laminer of liquid. It is based on energy conservation.
Assumptions
The fluid is incompressible, nonviscous, non rotational and streamline flow.
Mass of the fluid entering from side S
dm_{1} = p A_{1} dX_{1} = p dV_{1}
The work done in this displacement dx_{1} at point S is
Wp_{1} = F_{1}dx_{1} = P_{1}A_{1}dx_{1}
Wp_{1} = P_{1}dV_{1} {∵ A_{1}dx_{1} = dV_{1}}
At the same time the amount of fluid moves out of the tube at point T is dm_{2} = pdV_{2}
According to equation of coutniuity
The work done in the displacement of dm_{2} mass at point T
Wp_{2}  P_{2}dV_{2}
Now applying work energy theorem.
h = Gravitational head
= volume head
6.1 Application of Bernoullis principle
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 and 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 figure, then relative to the ball air will be moving from right to left.
(A) (B) (C)
The resultant velocity of air above the ball will be (V rw) while below it (V rw) (shown figure). So in accordance with Bernoulli's principle pressure above the ball will be less than below it. Due to this difference of pressure an upward force will act on the ball and hence the ball will deviate from its usual path OA_{0} and will hit the ground at A_{1} following the path OA_{1} (figure shown) i.e., if a ball is thrown with back spin, the pitch will curve less sharply prolonging the flight.
Similarly if the spin is clockwise, i.e., the ball is thrown with topspin, the force due to pressure difference will act in the direction of gravity and so the pitch will curve more sharply shortening the flight.
Furthermore, if the ball is spinning about a vertical axis, the curving will be sideays as shown in figure. producing the so called out swing or in swing.
Action of Atomiser : The action of aspirator, carburettor, paintgun, scentspray or insectsprayer is based on Bernoulli's principle. In all these by means of motion of a piston P in a cylinder C high speed air is passed over a 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 : This is also based on Bernouilli's principle. The wings of the aeroplane are having tapering as shown in figure. Due to this specific shape of wings when the aeroplane runs, air passes at higher speed over it as compared to its lower surface. This difference of air speeds above and below the wings, in accordance with Bernoulli's principle, creates a pressure difference, due to which an upward force called ' dynamic lift' ( = pressure difference × area of wing) acts on the plane. If this force becomes greater than the weight of the plane, the plane will rise up.
Ex.15 If pressure and velocity at point A is P_{1} and V_{1} respectively & at point B is P_{2}, V_{2} is the figure as shown. Comment on P_{1} and P_{2}.
Sol. From equation of continuity A_{1}V_{1} = A_{2}V_{2}
here A_{1} > A_{2}
⇒ V_{1} < V_{2} ....(1)
from Bernoulli's equation. We can write
Torricielli's Law of Efflux (Velocity of efflux)
Crosssectional Area of hole at A is greater than B If water is come in tank with velocity v_{A} and going out side with velocity
on applying Bernoulli theorem at A and B
Range (R)
Let us find the range R on the ground.
Considering the vertical motion of the liquid.
(H h) = or
Now, considering the horizontal motion,
R = vt or or
From the expression of R, following conclusions can be drawn,
(i) R_{h} = R_{H h }
as and
This can be shown as in Figure
(ii) R is maximum at H/2 and R_{max} = H.
Proof : R^{2 }= 4 (Hh – h^{2})
For R to be maximum.
or H  2h = 0 or h = H/2
That is, R is maximum at h=H/2
and
Proved
Ex.16 A cylindrical dark 1 m in radius rests on a platform 5 m high. Initially the tank is filled with water up to a height of 5 m. A plug whose area is 10^{–4} m^{2} is removed from an orifice on the side of the tank at the bottom Calculate
(a) initial speed with which the water flows from the orifice
(b) initial speed with which the water strikes the ground and
(c) time taken to empty the tank to half its original volume
(d) Does the time to be emptied the tank depend upon the height of stand.
Sol. The situation is shown in figure
(a) As speed of flow is given by
(b) As initial vertical velocity of water is zero, so its vertical velocity when it hits the ground
So the initial speed with which water strikes the ground.
(c) When the height of water level above the hole is y, velocity of flow will be and so rate of flow
Which on integration improper limits gives
= 9.2 × 103s – ~ 2.5 h
(d) No, as expression of t is independent of height of stand.
6.1 Venturimeter
Figure shows a venturimeter used to measure flow speed in a pipe of non  uniform crosssection. We apply Bernoulli's equation to the wide (point 1) and narrow (point 2) parts of the pipe, with h_{1} = h_{2}
From the continuity equation
Substituting and rearranging, we get
Because A_{1} is greater than A_{2} , v_{2} is greater than v_{1} and hence the pressure P_{2} is less than P_{1}. A net force to the right acceleration the fluid as it enters the narrow part of the tube (called throat) and a net force to the left slows as it leaves. The pressure difference is also equal to ρgh, where h is the difference in liquid level in the two tubes. Substituting in Eq. (1), we get
6.2 Pivot Tube
It is a device used to measure flow velocity of fluid. It is a U shaped tube which can be inserted in a tube or in the fluid flowing space as shown in figure shown. In the U tube a liquid which is immiscible with the fluid is filled upto a level C and the short opening M is placed in the fluid flowing space against the flow so that few of the fluid particles entered into the tube and exert a pressure on the liquid in limb A of U tube. Due to this the liquid level changes as shown in the figure show.
At end B fluid is freely flowing, which exert approximately negligible pressure on this liquid. The pressure difference at ends A and B can be given by measuring the liquid level difference h as
It is a gas, then
P_{A}  P_{B} = ρhg
It if the liquid of density , then
P_{A}  P_{B} = h(ρ  ρ_{g})g
Now if we apply Bernoulli's equation at the ends A and B we'll have
Now by using equations, we can evaluate the velocity v, with which the fluid is flowing.
Note : Pivot tube is also used to measure velocity of aeroplanes with respect to wind. It can be mounted at the top surface of the plain and hence the velocity of wind can be measured with respect to the plane.
6.3 SIPHON
It is a pipe used to drain liquid at a lower height but the pipe initially rises and then comes down
let velocity of outflow is v and the pipe is of uniform crosssection A. Applying Bernoulli's Equation between P (top of tank) and R (Opening of pipe) we get
⇒
here velocity is considered zero at P since area of tank is very large compared to the area of pipe
Naturally for siphon to work
h_{1 }> 0
Now as area of pipe is constant so by equation of continuity
as A_{v} = constant so velocity of flow inside siphon is also constant between Q and R
(P + pgh + 1/2 pv^{2})_{Q} = (P + pgh + 1/2 pv^{2})R ⇒ P_{Q} + pgh_{2} = P_{0}  pgh_{1} (v is same)
⇒ P_{Q} = P_{0}  Pg (h_{1} + h_{2}) as P_{Q} = 0 ⇒ P_{0} > pg (h_{1} + h_{2}) means (h_{1} + h_{2}) should not be more than P/pg for siphon to work
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127 videos464 docs210 tests
