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CRITICAL VELOCITY
Critical velocity is the velocity of liquid below up to which its flow is steady and above which its flow becomes turbulent.
When the velocity of the liquid flowing through a pipe is small, the flow is steady. As velocity is increased, a stage is reached when experiments show that for cylindrical pipes critical velocity is given by
 = Coefficient of viscosity of the liquid = Radius of pipe = Density of the liquid Derivation of expression for critical velocity We can derive the formula for critical velocity by dimensional analysis. The critical velocity vc of a liquid flowing through a pipe depends upon
| ||
(iii) Radius of the | pipe |
Equating the powers of M, L and T
By experiment the value of k = 1100
EQUATION OF CONTINUITY
(MASS FLOW RATE)
Consider the flowing of the fluid on the pipe PQ as shown on the figure below
Let
If 
are densities of fluid at P and Q
are densities of fluid at P and QMass flow rate
Is the mass of liquid (fluid) flowing through a tube per second.
Then, the rate of mass of fluid leaving at P and Q at the time interval 
be
be
SI unit of mass flow rate is kg/s
Applying the law of conservation of mass
This is the equation of continuity for the compressible fluid flow.
If the liquid is incompressible, the density of liquid P and Q will be equal.
Flow is steady. No liquid can cross the side, so the mass of liquid passing through section P is equal to the mass of liquid passing through section Q in one second.
Equation of continuity
States that “the mass of a liquid flowing through a pipe of varying cross – section is constant when density of the liquid does not change”.
AV = Constant.
ENERGY OF A LIQUID
A moving liquid can possess the following types of energies
- K.E. due to its position
- P.E. due to its position
- Pressure energy due to pressure of the liquid
- K.E of a liquid
(i) K.E of a liquid
Is the energy possessed by the liquid due to its motion
M – Mass liquid
V – Velocity of a liquid flow
(ii) P.E of a liquid
Is the energy possessed by the liquid due to its position
P.E of liquid of mass m at a height h is given by
P.E = 
*P.E per unit mass = m=1
m=1 P.E per unit mass = |
In figure below, is a wide tank containing a liquid
At its bottom there is a side tube fitted with a frictionless piston of area of cross section A.
Work done to move the piston through a small distance X
W = F.X
But,
P – Pressure on the piston
W = PAX
Mass of liquid pushed
M = 
But
V = AX
This work done is stored in as pressure energy of a column of liquid
Pressure energy = PAX
* Pressure energy per unit mass = 
*Pressure energy per unit volume = 
Pressure energy per unit volume = P
Pressure energy of a volume V of a liquid is given by = PV
Total energy of moving liquid
Total energy per unit volume
Total energy per unit mass
But
BERNOULLI’S THEOREM
States that “for steady flow of an ideal liquid, the total energy per unit volume remains constant throughout the flow”.
P – Pressure within the fluid
– Density of the fluid h – Height of the fluidV – Velocity of the fluid
– Acceleration due to gravityBernoulli’s theorem is simply a statement of law of conservation of energy applied to a liquid.
The importance of this theorem lies in the fact it can be used to determine the variation of pressure in liquids as a function of velocity of the liquid and the elevation of the pipe through which the liquid is flowing.
PROOF OF BERNOULLI’S THEOREM
Consider an incompressible, non-viscous fluid flow and steadily through a pipe of variable cross – section. Consider two sections A and Q.
– The cross – sectional areas at P and Q
– Fluid velocity at P and Q
– Fluid pressure at P and Q
– Mean heights above ground level at P and Q respectivelyThe force acting on the liquid layer at P is
Under this force in time the layer moves through a distance
the layer moves through a distance = 
Work done on the fluid due to the force
= 
When the liquid moves from P to 
In a time at the same time the fluid at Q moves to 
, doing work against the pressure at B.
at the same time the fluid at Q moves to 
, doing work against the pressure 
at B.Work done by the fluid against pressure P2
= 
Network done on the fluid = work done on the fluid – work done by the place
This equation means, whatever mass of fluid enters the pipe at P in a certain time will leave the pipe at Q at the same time.
Total work done on the fluid = 
= (
Total energy = ΔK.E + ΔP.E
Increase in P.E. 
Work | done | on | the | |
 |
From work energy theorem ΔW = ΔK.E + ΔP.E
FLOW IN A HORIZONTAL PIPE
When the liquid flows through a horizontal pipe
h
P = Static pressure
DIFFERENT FORMS OF BERNOULLI’S THEOREM
1. 
2.
3.
Where by
LIMITATION OF BRNOLLI’S THEOREM
- In derivation of Bernoulli‘s equation, it is assumed that the liquid is non-viscous i.e. the liquid has zero viscosity (no friction).
However, a real liquid does have some viscosity so that a part of mechanical energy is lost to overcome liquid friction. This fact is not taken into account in this elevation.
- In derivation of Bernoulli‘s equation, it is assume that the rate of flow of liquid is constant
But this is not correct in actual practice. Thus in the case of liquid flowing through a pipe, the velocity of flaw is maximum at the center and goes on decreasing towards the walls of the pipe. Therefore, we should take the average velocity of the liquid.
- In derivation of Bernoulli‘s equation, it is assumed that there is no loss of energy when liquid is motion. In practice, this is not true e.g. A part of K.E of flowing liquid is converted into heat and is last forever.
- If the liquid is flowing along a curved path, the energy due to centrifugal force must be considered.
APPLICATIONS OF BERNOULL’S THEOREM
1. Flow meter – Venturimeter
Is a device that is used to measure the flow speed (flow rate) of a liquid through a pipe. It works on Bernoulli’s theorem.
It consists of two tubes A and C connected by a narrow coaxial tube B with a constriction called the throat.
Using the two tubes D and E the difference in pressure of the liquid flowing through A and B can be round out.
As the liquids flows from A to B the velocity increases, due to decreases in cross-sectional area.
Let the velocities at A and B isV1 and A1 and A2and V2 and cross-sectional areas and A and B be A1 and A2 respectively.
By equation of continuity
A1 V1 = A2V2
Applying the equation of continuity
Q = 
Where Q is the volume of liquid flowing in one second
Applying Bernoulli’s theorem at A and B
But, A1 > A2, V2> V1 P1> P2hence the level of the liquid in D is higher than that in E.
H – Difference of levels of the liquid in the tubes D and E.
From, 
If 
Q =
TORRICELLI’S THEOREM
States that “if the difference in levels between the hole and the upper liquid surface in a drum is h, then
the velocity with which the liquid emerges from the hole in 
States that the velocity of efflux is equal to the velocity which a body attains. in falling freely from the surface of the liquid to the orifice
States that the velocity of efflux is equal to the velocity which a body attains. in falling freely from the surface of the liquid to the orificeVelocity of efflux
Is the velocity of liquid at the orifice.
Or
Is the velocity of emerging fluid from the orifice
This theorem applies to a liquid flowing from a drum with a horizontal opening near the base.
This is the same velocity which freely falling object will acquire in falling from rest through a vertical distance h.
PROOF OF TORRICELLI’S THEOREM
Suppose an ideal liquid flows through a hole H at the bottom of a wide drum as shown below
Let 
be the density of the liquid.
be the density of the liquid. According to Bernoulli’s theorem, at any point of the liquid
At point 1
The point 1 is at the surface of the liquid in the drum
At point 2
The point 2 is at the place where liquid leaves the hole.
*The velocity of the liquid emerging from the hole depends only upon the depth h of the hole below the surface of the liquid.
*
Horizontal Range
The liquid flows out of the hole in the form of a parabolic jet and strikes the ground at a distance R from the base of the drum.
The distance R is the horizontal range of the liquid coming out of the hole.
At the hole P, the velocity V2 of the emerging liquid is along the horizontal direction.
T – Time taken by the parabolic jet to strike the ground after emerging from the hole P h1 – Height of hole above the bottom of the drum
Therefore, the vertical distance covered by the set in time T is h‘
PITOT TUBE.
Pitot tube is a device used for measuring the velocities of flowing liquids and hence the rates of flow of the liquids.
Its working is based on Bernoulli’s theorem.
Pitot tube is an open ended L-shaped tube immersed in the liquid with its aperture and its nose B facing the flow of liquid, so that the plane of the aperture is normal to the direction of another tube A with a small opening at its bottom.
The plane of aperture of A is parallel to the direction of flow, so that it measures the static pressure at A, which is the pressure of the undisturbed liquid.
at A, which is the pressure of the undisturbed liquid.The flow of liquid is stopped in the plane of aperture B, there by converting the Kinetic energy of the liquid into P.E.
So the liquid rises in the tube T as shown. The height of the liquid in this tube gives the total pressure or the stagnation pressure.
Applying Bernoulli’s Equation
Let the difference between the levels of the liquid in the two tubes be h
The velocity of flow of the liquid
The rate of flow of liquid
A – Area of cross-sectional of the pipe at the place where the pitot tube is placed.
Static pressure.
Static pressure is the actual pressure of the fluid at the point due to its rest position of fluid.
Dynamic pressure.
Dynamic pressure is the pressure exerted by fluid due to its own motion.
Let h be difference in the liquid levels in the two limbs.
