Fluid Mechanics
FLUID MECHANICS
———————————————————————————————————
Fluid mechanics deals with the behaviour of fluids at rest and in motion. A fluid is a substance that
deforms continuously under the application of shear (tangential) stress no matter how small the shear
stress may be:
Thus, fluid comprise the liquid and gas (or vapour) phases of the physical forms in which matter exists.
We may alternatively define a fluid as a substance that cannot sustain a shear stress when at rest.
1. Density of a Liquid
Density () of any substance is defined as the mass per unit volume or
mass m
= =
volume V
2. Relative Density (RD)
In case of a liquid, sometimes an another term relative density (RD) is defined. It is the ratio of density
of the substance to the density of water at 4°C. Hence,
Density of substance
RD =
Density of water at 4C
RD is a pure ratio. So, it has no units. It is also sometimes referred as specific gravity.
Density of water at 4°C in CGS is 1g/cm 3. Therefore, numerically the RD and density of substance
(in CGS) are equal. In SI units the density of water at 4°C is 1000 kg/m 3.
Example 1. Relative density of an oil is 0.8. Find the absolute density of oil in CGS and SI units.
Solution : Density of oil (in CGS) = (RD)g/cm3 = 0.8 g/cm3 = 800 kg/m3
———————————————————————————————————
3. Pressure in a Fluid
dA
When a fluid (either liquid or gas) is at rest, it exerts a force
perpendicular to any surface in contact with it, such as a dF dF
container wall or a body immersed in the fluid.
While the fluid as a whole is at rest, the molecules that makes
up the fluid are in motion, the force exerted by the fluid is due
to molecules colliding with their surrounding.
If we think of an imaginary surface within the fluid, the fluid on the two sides of the surface exerts equal
and opposite forces on the surface, otherwise the surface would accelerate and the fluid would not
remain at rest.
Consider a small surface of area dA centered on a point on the fluid, the normal force exerted by the
fluid on each side is dF. The pressure P is defined at that point as the normal force per unit area, i.e.,
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,Fluid Mechanics
dF
P=
dA
If the pressure is the same at all points of a finite plane surface with area A, then
F
P=
A
where F is the normal force on one side of the surface. The SI unit of pressure is pascal
where 1 pascal = 1Pa = 1.0 N/m2
One unit used principally in meterology is the Bar which is equal to 105 Pa
1 Bar = 105 Pa
Note : Fluid pressure acts perpendicular to any surface in the fluid no matter how that surface is
oriented. Hence, pressure has no intrinsic direction of its own, its a scalar. By contrast, force is a vector
with a definite direction.
Atmospheric Pressure (P0)
It is pressure of the earth's atmosphere. This changes with weather and elevation. Normal atmospheric
pressure at sea level (an average value) is 1.013 × 10 5 Pa
Absolute pressure and Gauge Pressure
The excess pressure above atmospheric pressure is usually called gauge pressure and the total
pressure is called absolute pressure. Thus,
Gauge pressure = absolute pressure – atmospheric pressure
Absolute pressure is always greater than or equal to zero. While gauge pressure can be negative also.
Variation in pressure with depth
If the weight of the fluid can be neglected, the pressure in a fluid is the same throughout its volume. But
often the fluid's weight is not negligible and under such condition pressure increases with increasing
depth below the surface.
Let us now derive a general relation between the pressure P at any point in a fluid at rest and the
elevation y of that point. We will assume that the density and the acceleration due to gravity g are the
same throughout the fluid. If the fluid is in equilibrium, every volume element is in equilibrium.
Consider a thin element of fluid with height dy. The bottom and top surfaces each have area A, and
they are at elevations y and y + dy above some reference level where y = 0. The weight of the fluid
element is
(P+dP)A
dW = (volume) (density) (g) = (A dy) () (g)
or dW = gAdy
What are the other forces in y-direction on this fluid element ? Call the pressure at the bottom surface
P, the total y component of upward force is PA. The pressure at the top surface is P + dP and the total
y-component of downward force on the top surface is (P + dP)A. The fluid element is in equilibrium, so
the total y component of force including the weight and the forces at the bottom and top surfaces must
be zero.
Fy = 0
dP
PA – (P + dP)A – gA dy = 0 or – g
dy
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, Fluid Mechanics
This equation shows that when y increases, P decreases, i.e., as we move upward in the fluid pressure
decreases.
If P1 and P2 be the pressures at elevations y1 and y2 and if and g are constant, then integration
Equation (i), we get
or P2 – P1 = – g (y2 – y1) ...........(ii)
It's often convenient to express Equation (ii) in terms of the depth below the surface of a fluid. Take
point 1 at depth h below the surface of fluid and let P represents pressure at this point. Take point 2 at
the surface of the fluid, where the pressure is P0 (subscript for zero depth). The depth of point 1 below
the surface is,
h = y2 – y1
and equation (ii) becomes
P0 – P = – g (y2 – y1) = – gh
P = P0 + gh ..........(iii)
Thus pressure increases linearly with depth, if and g are uniform. A graph between P and h is shown
below.
Further, the pressure is the same at any two points at the same level in the fluid. The shape of the
container does not matter.
Example 2. The manometer shown below is used to measure the difference in water level between the two
tanks. Calculate this difference for the conditions indicated.
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FLUID MECHANICS
———————————————————————————————————
Fluid mechanics deals with the behaviour of fluids at rest and in motion. A fluid is a substance that
deforms continuously under the application of shear (tangential) stress no matter how small the shear
stress may be:
Thus, fluid comprise the liquid and gas (or vapour) phases of the physical forms in which matter exists.
We may alternatively define a fluid as a substance that cannot sustain a shear stress when at rest.
1. Density of a Liquid
Density () of any substance is defined as the mass per unit volume or
mass m
= =
volume V
2. Relative Density (RD)
In case of a liquid, sometimes an another term relative density (RD) is defined. It is the ratio of density
of the substance to the density of water at 4°C. Hence,
Density of substance
RD =
Density of water at 4C
RD is a pure ratio. So, it has no units. It is also sometimes referred as specific gravity.
Density of water at 4°C in CGS is 1g/cm 3. Therefore, numerically the RD and density of substance
(in CGS) are equal. In SI units the density of water at 4°C is 1000 kg/m 3.
Example 1. Relative density of an oil is 0.8. Find the absolute density of oil in CGS and SI units.
Solution : Density of oil (in CGS) = (RD)g/cm3 = 0.8 g/cm3 = 800 kg/m3
———————————————————————————————————
3. Pressure in a Fluid
dA
When a fluid (either liquid or gas) is at rest, it exerts a force
perpendicular to any surface in contact with it, such as a dF dF
container wall or a body immersed in the fluid.
While the fluid as a whole is at rest, the molecules that makes
up the fluid are in motion, the force exerted by the fluid is due
to molecules colliding with their surrounding.
If we think of an imaginary surface within the fluid, the fluid on the two sides of the surface exerts equal
and opposite forces on the surface, otherwise the surface would accelerate and the fluid would not
remain at rest.
Consider a small surface of area dA centered on a point on the fluid, the normal force exerted by the
fluid on each side is dF. The pressure P is defined at that point as the normal force per unit area, i.e.,
Corp. / Reg. Office : CG Tower, A-46 & 52, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) – 324005
Website : www.resonance.ac.in | E-mail :
ADVFL - 1
Toll Free : 1800 258 5555 | CIN : U80302RJ2007PLC024029
,Fluid Mechanics
dF
P=
dA
If the pressure is the same at all points of a finite plane surface with area A, then
F
P=
A
where F is the normal force on one side of the surface. The SI unit of pressure is pascal
where 1 pascal = 1Pa = 1.0 N/m2
One unit used principally in meterology is the Bar which is equal to 105 Pa
1 Bar = 105 Pa
Note : Fluid pressure acts perpendicular to any surface in the fluid no matter how that surface is
oriented. Hence, pressure has no intrinsic direction of its own, its a scalar. By contrast, force is a vector
with a definite direction.
Atmospheric Pressure (P0)
It is pressure of the earth's atmosphere. This changes with weather and elevation. Normal atmospheric
pressure at sea level (an average value) is 1.013 × 10 5 Pa
Absolute pressure and Gauge Pressure
The excess pressure above atmospheric pressure is usually called gauge pressure and the total
pressure is called absolute pressure. Thus,
Gauge pressure = absolute pressure – atmospheric pressure
Absolute pressure is always greater than or equal to zero. While gauge pressure can be negative also.
Variation in pressure with depth
If the weight of the fluid can be neglected, the pressure in a fluid is the same throughout its volume. But
often the fluid's weight is not negligible and under such condition pressure increases with increasing
depth below the surface.
Let us now derive a general relation between the pressure P at any point in a fluid at rest and the
elevation y of that point. We will assume that the density and the acceleration due to gravity g are the
same throughout the fluid. If the fluid is in equilibrium, every volume element is in equilibrium.
Consider a thin element of fluid with height dy. The bottom and top surfaces each have area A, and
they are at elevations y and y + dy above some reference level where y = 0. The weight of the fluid
element is
(P+dP)A
dW = (volume) (density) (g) = (A dy) () (g)
or dW = gAdy
What are the other forces in y-direction on this fluid element ? Call the pressure at the bottom surface
P, the total y component of upward force is PA. The pressure at the top surface is P + dP and the total
y-component of downward force on the top surface is (P + dP)A. The fluid element is in equilibrium, so
the total y component of force including the weight and the forces at the bottom and top surfaces must
be zero.
Fy = 0
dP
PA – (P + dP)A – gA dy = 0 or – g
dy
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, Fluid Mechanics
This equation shows that when y increases, P decreases, i.e., as we move upward in the fluid pressure
decreases.
If P1 and P2 be the pressures at elevations y1 and y2 and if and g are constant, then integration
Equation (i), we get
or P2 – P1 = – g (y2 – y1) ...........(ii)
It's often convenient to express Equation (ii) in terms of the depth below the surface of a fluid. Take
point 1 at depth h below the surface of fluid and let P represents pressure at this point. Take point 2 at
the surface of the fluid, where the pressure is P0 (subscript for zero depth). The depth of point 1 below
the surface is,
h = y2 – y1
and equation (ii) becomes
P0 – P = – g (y2 – y1) = – gh
P = P0 + gh ..........(iii)
Thus pressure increases linearly with depth, if and g are uniform. A graph between P and h is shown
below.
Further, the pressure is the same at any two points at the same level in the fluid. The shape of the
container does not matter.
Example 2. The manometer shown below is used to measure the difference in water level between the two
tanks. Calculate this difference for the conditions indicated.
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Website : www.resonance.ac.in | E-mail :
ADVFL - 3
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