Fluid Mechanics
Introduction
Fluids that include liquids and gases play a vital role in our daily lives as we breathe and
drink them. Cars run on fluid (Petrol, diesel), Oceans and rivers are full of fluid (water),
battery, AC’s, refrigerators too contain fluids (CFC’s etc)
DEFINITION
A fluid is a substance that can flow and conform to the boundaries of any container
containing it. This is because fluid can not sustain a force that is tangential to its surface.
More over fluid flows because it can not with stand a shearing stress.
DENSITY
To find the density of a fluid at any point, we isolate a small volume element V
around that point and measure the mass m of the fluid contained within that element.
The density is then
For uniform density we may write as
(Uniform density)
Where m and V are the mass and Volume of the sample
S.I unit of density is Kgcm-3
Density is a Scalar quantity.
PRESSURE IN A FLUID
(pressure of uniform force on flat area).
S.I unit of pressure is N/m2 called Pascal (Pa).
Pressure is a scalar quantity.
The atmosphere (atm) exerts certain pressure at a point depending on the column or
height of atmosphere lying above that point.
The average pressure of the atmosphere at sea level is known as atmospheric pressure
(atm) and
Now onwards we shall assume that the liquids we deal with are incompressible and non
viscous (though they are not strictly).
VARIATION OF PRESSURE WITH HEIGHT (DEPTH):
Fig (1)
Let us consider a cylinder filled with water. Consider surfaces S1 and S2 on which
pressure due to water is P1 and P2 respectively at heights y1 and y2 respectively from the
top of surface i.e. y=0 then P1 and P2 can be related as
P2 = P1+g (y1-y2) --------------------- (1)
Where = Density of water (or other liquid).
g = Acceleration due to gravity (9.8 m/sec2).
The equation can be used to find pressure both in a liquid (as a function of depth) and in
the atmosphere (as a function of altitude or height).
Thus, pressure at depth h can be taken as P where
P = P0 + gh ------------------------ (2)
Where P0 = Pressure on the surface where h = 0.
Note: The pressure at any point in a fluid in static equilibrium depends on the depth of
that point but not on any horizontal dimension of the fluid or its container.
Thus equation (2) holds no matter what the shape of the container.
ABSOLUTE PRESSURE:
In equation (2), P is said to be the total pressure or absolute pressure.
GAUGE PRESSURE:
The difference between an absolute pressure and an atmospheric pressure is called the
gauge pressure. Thus in equation (2) gauge pressure is gh as P0 is atmospheric pressure
because on surface of the vessel the pressure (P0) is solely due to atmosphere. Thus
Absolute Pressure Atmospheric pressure = Gauge Pressure ------------ (3).
PASCAL LAW:
If the pressure in a liquid is changed at a particular point, the change is transmitted to the
entire liquid without being diminished in magnitude.
Ex: Hydraulic lift is used to raise heavy loads such as car. It contains of two vertical
cylinders A and B of different cross sectional areas A1 and A2. Pistons are fitted in both
the cylinders as shown in fig (2).
Fig (2)
The load is kept on a platform fixed with the piston of larger area. A liquid is filled in the
equipment. A value V is filled in the horizontal tube which allows the liquid to go from A
to B when pressed from the A side. The piston is pushed by a force F1. The pressure in
the liquid increases every where by an amount F1/A1. Thus force on the larger piston in
the upward direction is
which raises the load upward.
Thus if A2>>A1, even a small force F1 is able to generate a large force F2 which can raise
the load.
Note:
There is no gain in terms of work. The work done by F1 is same as that by F2 if there is
no dissipation due to friction etc. Thus
F1.d1=F2.d2
But F2>>F1 thus d1>>d2
i.e piston with greater area traverses a smaller upward distance as compared to the piston
having smaller area that traverses larger downward distance.
Thus Pascal’s law is in consistence with the first law of Thermodynamics – “law of
conservation of energy”.
In short
“With a hydraulic lever, a given force applied over a given distance can be transformed to
a greater force applied over a smaller distance”
MEASURING PRESSURE:
a) The Mercury Barometer:
Figure (3) shows a very basic mercury barometer, a device used to measure the pressure
of the atmosphere. The long glass tube is filled with mercury and inverted with its open
end in a dish of mercury, as the figure shows. The space above the mercury column
contains only mercury vapour, whose pressure is so small at ordinary temperature that it
can be neglected.
Fig (3)
Thus P0=gh.
Where =density of the mercury.
The atmospheric pressure is often given as the length of mercury column in a barometer.
Thus, a pressure of 76cm of mercury means,
b) Manometer:
Manometer is a simple device to measure the pressure in a closed vessel containing a gas.
It consists of a U-shape tube having some liquid. One end of the tube is open to the
atmosphere and the other end is connected to the vessel as shown in figure (4).
Fig (4)
The pressure of the gas is equal to the pressure at A
= Pressure at B
= Pressure at C + hg
= P0 + hg
Where P0 is the atmospheric pressure h = BC is the difference in levels of the liquid in the
two arms and is the density of the liquid.
ARCHIMEDES PRINCIPLE:
When the body is partially or fully dipped into a fluid, the fluid exerts forces on the body.
At any small portion of the surface of the body, the force by the fluid is perpendicular to
the surface and is equal to the pressure at that point multiplied by the area as shown in
figure (5).
The resultant of all these contact forces is called the forces of buoyancy or buoyant force.
Archimedes principle states that when a body is partially or fully dipped into a fluid at
rest, the fluid exerts an upward force of buoyancy equal to the weight of the displaced
fluid.
Numerical:
1) A 700g solid cube having an edge of length 10cm floats in water. How much
volume of the cube is outside the water? Density of water = 1000 kg/m3.
Solution:
The weight of the cube is balanced by the buoyant force. The buoyant force is equal
to the weight of the water displaced. If a volume V of the cube is inside the water, the
weight of the displaced water = Vg, Where is the density of water,
Thus Vg = (0.7kg) g.
The total volume of the cube = (10cm) 3 = 1000cm3. The volume outside the water is
1000-700 = 300 cm3.
FLOATING:
When a body floats in a fluid, the magnitude Fb of the buoyant force on the body is equal
to the magnitude Fg of the gravitational force on the body.
Thus, Fb = Fg (floating).
Also, Fb = mfg = Fg.
Where, mf = mass of the fluid that is displaced by the body.
APPERENT WEIGHT IN A FLUID:
If an object is placed inside a fluid then,
(Apparent weight) = (Actual weight) – (Buoyant force).
FLOW OF IDEAL FLUIDS:
An ideal fluid is incompressible and lacks viscosity, and its flow is steady and
irrotational. A stream line is the path followed by an individual fluid particle.
EQUATION OF CONTINUITY:
It means that total mass of fluids going into the tube through any cross-section should be
equal to the total mass coming out of the same tube from any other cross section in the
same time.
Fig (6)
Thus A1V1t = A2V2t (as shown in figure (6))
Or A1V1 = A2V2.
The product of the area of cross section and the speed remains the same at all points of a
tube of flow. This is called the “equation of continuity” and expresses the law of
conservation of mass in fluid dynamics.
BERNOULLI’S EQUATION:
Bernoulli’s 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.
We here consider the case of irrotational and steady flow of an incompressible and non
viscous liquid.
According to Bernoulli’s Equation,
= Constant.
Application of Bernoulli’s Equation:
a) Hydrostatics:
If the speed of the fluid is zero every where, we get the situation of hydrostatics. Putting
V1 = V2 = 0 in the Bernoulli’s equation
P1+gh1 = P2+gh2.
P1 P2 = g (h1h2). As expected from hydrostatics.
b) Speed of Efflux:
Consider liquid of density in a tank of large cross sectional area A1. There is a hole of
cross-sectional area A2 at the bottom with liquid flowing out as shown in figure (7).
Fig (7)
Let V1 and V2 be the speed of the liquid at A1 and A2. Applying Bernoulli’s equation at
A1 and A2 we have
From equation of continuity A1V1 = A2V2
Solving above two equations we have
If A2<
The speed of liquid coming out though a hole at a depth ‘h’ below the free surface is
same as that of a particle fallen freely through the height ‘h’ under gravity. This is known
as Torricelli’s theorem. The speed of the liquid coming out is called the speed of efflux.
c) Change of plane of motion of a spinning Ball.
Quite often when swing bowlers of cricket deliver the ball; the ball changes its plane of
motion in air.
Such a deflection from the plane of projection may be explained on the basis of
Bernoulli’s equation.
Suppose a ball spinning about the vertical direction is going ahead with some velocity in
the horizontal direction in otherwise still air. Let us work in a frame in which the center
of the ball is at rest. In this frame the air moves fast the ball at a speed V in the opposite
direction. The situation is shown in Figure (8).
Fig (8)
The plane of the figure represents horizontal plane. The air that goes from the A side of
the ball in the figure is dragged by the spin of the ball and its speed increases. The air that
goes from the B side of the ball in the figure suffers an opposite drag and its speed
decreases. The pressure of air is reduced on the A side and is increased on the B side as
required by the Bernoulli’s theorem. As a result a net force F acts on the ball from the B
side to the A side due to this pressure difference. This force causes the deviation of the
plane of motion.