Program : B.

Tech
Subject Name: Fluid Mechanics
Subject Code: ME-404
Semester: 4th
 FLUID MECHANICS (ME – 404)
UNIT – 1 (FLUID STATICS)
I. PHYSICAL PROPERTIES OF FLUID
Due to the development of industries there arose a need for the study of fluids other than water. Theories
like boundary layer theory were developed which could be applied to all types of real fluids, under various
conditions of flow. The combination of experiments, the mathematical analysis of hydrodynamics and the
new theories is known as ‘Fluid Mechanics’. Fluid Mechanics encompasses the study ofall types of fluids
under static, kinematic and dynamic conditions.

The study of properties of fluids is basic for the understanding of flow or static condition of fluids. The
important properties are density, viscosity, surface tension, bulk modulusand vapour pressure. Viscosity
causes resistance to flow. Surface tension leads to capillaryeffects. Bulk modulus is involved in the
propagation of disturbances like sound waves in fluids. Vapour pressure can cause flow disturbances due
to evaporation at locations of low pressure. It plays an important role in cavitation studies in fluid
machinery.

A fluid is defined as a material which will continue to deform with the application of shear force however
small the force may be.

COMPRESSIBLE & INCOMPRESSIBLE FLUID

If the density of a fluid varies significantly due to moderate changes in pressure or temperature, then the
fluid is called compressible fluid. Generally gases and vapoursunder normal conditions can be classified as
compressible fluids. In these phases the distance between atoms or molecules is large and cohesive forces
are small. So increase in pressure or temperature will change the density by a significant value.

If the change in density of a fluid is small due to changes in temperature and or pressure, then the fluid
is called incompressible fluid. All liquids are classified underthis category.

When the change in pressure and temperature is small, gases and vapours are treated as incompressible
fluids. For certain applications like propagation of pressure disturbances, liquids should be considered as
compressible.

CONTINUUM

As gas molecules are far apart from each other and as there is empty space between molecules doubt
arises as to whether a gas volume can be considered as a continuous matter like a solid for situations
similar to application of forces.

Under normal pressure and temperature levels, gases are considered as a continuum (i.e., as if no empty
spaces exist between atoms). The test for continuum is to measure properties like density by sampling at
different locations and also reducing the sampling volume to low levels. If the property is constant
irrespective of the location and size of sample volume, then the gas body can be considered as a
continuum for purposes of mechanics (application of force, consideration of acceleration, velocity etc.) and
for the gas volume to be considered as a single body or entity. This is a very important test for the
application of all laws of mechanics to a gas volume as a whole. When the pressure is extremely low, and
when there are only few molecules in a cubic meter of volume, then the laws of mechanics should be
applied to the molecules as entities and not to the gas body as a whole.

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 BASIC FLUID TERMINOLOGIES

Density (mass density):The mass per unit volume is defined as density. The unit used is kg/m3.The
measurement is simple in the case of solids and liquids. In the case of gases and vapours it is rather
involved. The symbol used is ρ. The characteristic equation for gases provides a means to estimate the
density from the measurement of pressure, temperature and volume.
Specific Volume:The volume occupied by unit mass is called the specific volume of thematerial. The
symbol used is v, the unit being m3/kg. Specific volume is the reciprocal of density.

In the case of solids and liquids, the change in density or specific volume with changes in pressure and
temperature is rather small, whereas in the case of gases and vapours, density will change significantly due
to changes in pressure and/or temperature.

Weight Density or Specific Weight:The force due to gravity on the mass in unitvolume is defined as Weight
Density or Specific Weight. The unit used is N/m3. The symbol used is y. At a location where g is the local
acceleration due to gravity, Specific weight, γ = g ρ

In the above equation direct substitution of dimensions will show apparent non-homogeneity as the
dimensions on the LHS and RHS will not be the same. On the LHS the dimension will be N/m3 but on the
RHS it is kg/m2 s2. The use of go will clear this anomaly. As seen in section 1.1, go = 1 kg m/N s2. The RHS of
the equation 1.3.1 when divided by go will lead to perfect dimensional homogeneity. The equation should
preferably be written as, Specific weight, γ = (g/go) ρ

Since newton (N) is defined as the force required to accelerate 1 kg of mass by 1/s 2, it can also be
expressed as kg m/s2. Density can also be expressed as Ns2/m4 (as kg = Ns2/m). Beam balances compare
the mass while spring balances compare the weights. The mass is the same (invariant) irrespective of
location but the weight will vary according to the local gravitational constant. Density will be invariant
while specific weight will vary with variations in gravitational acceleration.

Specific Gravity or Relative Density:The ratio of the density of the fluid to thedensity of water—usually
1000 kg/m3 at a standard condition—is defined as Specific Gravity or Relative Density 6 of fluids. This is a
ratio and hence no dimension or unit is involved.

VISCOSITY

A fluid is defined as a material which will continue to deform with the application of a shear force.
However, different fluids deform at different rates when the same shear stress (force/ area) is applied.

Viscosity is that property of a real fluid by virtue of which it offers resistance to shear force. For a given
fluid the force required varies directly as the rate of deformation. As the rate of deformation increases the
force required also increases.

The force required to cause the same rate of movement depends on the nature of the fluid. The resistance
offered for the same rate of deformation varies directly as the viscosity of the fluid. As viscosity increases
the force required to cause the same rate of deformation increases.Newton’s law of viscosity states that
the shear force to be applied for a deformation rate of (du/dy) over an area A is given by,

F = µ A (du/dy)

(F/A) = τ = µ (du/dy) = µ (u/y)

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 Where, F is the applied force in N, A is area in m2, du/dy is the velocity gradient (or rate of deformation),
1/s, perpendicular to flow direction, here assumed linear, and µ is the proportionality constant defined as
the dynamic or absolute viscosity of the fluid.

ub> ua, Fb> Faµa=µb ua= ub,µa<µb, Fb> Fa

(i) same fluid (ii) same velocity
The dimensions for dynamic viscosity µ can be obtained from the definition as Ns/m2 or kg/ms. The
first dimension set is more advantageously used in engineering problems. However, if the dimension of
N is substituted, then the second dimension set, more popularly used by scientists can be obtained.
The numerical value in both cases will be the same.
N = kg m/s2; µ = (kg m/s2) (s/m2) = kg/ms
The popular unit for viscosity is Poise named in honor of Poiseuilli.
Poise = 0.1 Ns/m2
Centipoise (cP) is also used more frequently as cP = 0.001 Ns/m2

For water the viscosity at 20°C is nearly 1 cP. The ratio of dynamic viscosity to the density is defined as
kinematic viscosity, ν, having a dimension of m2/s. Later it will be seen to relate to momentum transfer.
Because of this kinematic viscosity is also called momentum diffusivity. The popular unit used is stokes
(in honor of the scientist Stokes). Centistoke is also often used.
1 stoke = 1 cm2/s = 10–4 m2/s
Of all the fluid properties, viscosity plays a very important role in fluid flow problems. The velocity
distribution in flow, the flow resistance etc. are directly controlled by viscosity. In the study of fluid
statics (i.e., when fluid is at rest), viscosity and shear force are not generally involved.

Newtonian and Non Newtonian Fluids

An ideal fluid has zero viscosity. Shear force is not involved in its deformation. An ideal fluid has to be
also incompressible. Shear stress is zero irrespective of the value of du/dy. Bernoulli equation can be
used to analyses the flow.

Real fluids having viscosity are divided into two group namely Newtonian and non-Newtonian fluids. In
Newtonian fluids a linear relationship exists between the magnitude of the applied shear stress and the
resulting rate of deformation. It means that the proportionality parameter (in equation, τ = µ (du/dy)),
viscosity, µ is constant in the case of Newtonian fluids (other conditions and parameters remaining the
same). The viscosity at any given temperature and pressure is constant for a Newtonian fluid and is
independent of the rate of deformation.

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 Non Newtonian fluids can be further classified as simple non Newtonian, ideal plastic and shear
thinning, shear thickening and real plastic fluids. In non-Newtonian fluids the viscosity will vary with
variation in the rate of deformation. Linear relationship between shear stress and rate of deformation
(du/dy) does not exist. In plastics, up to a certain value of applied shear stress there is no flow. After
this limit it has a constant viscosity at any given temperature. In shear thickening materials, the
viscosity will increase with (du/dy) deformation rate. In shear thinning materials viscosity will decrease
with du/dy. Paint, tooth paste, printers ink are some examples for different behaviors. Many other
behaviors have been observed which are more specialized in nature. The main topic of study in this
text will involve only Newtonian fluids.
Viscosity and Momentum Transfer

In the flow of liquids and gases molecules are free to move from one layer to another. When the velocity in
the layers are different as in viscous flow, the molecules moving from the layer at lower speed to the layer
at higher speed have to be accelerated. Similarly the molecules moving from the layer at higher velocity to
a layer at a lower velocity carry with them a higher value of momentum and these are to be slowed down.
Thus the molecules diffusing across layers transport a net momentum introducing a shear stress between
the layers. The force will be zero if both layers move at the same speed or if the fluid is at rest.

When cohesive forces exist between atoms or molecules these forces have to be overcome, for relative
motion between layers. A shear force is to be exerted to cause fluids to flow.

Viscous forces can be considered as the sum of these two, namely, the force due to momentum transfer
and the force for overcoming cohesion. In the case of liquids, the viscous forces are due more to the
breaking of cohesive forces than due to momentum transfer (as molecular velocities are low). In the case
of gases viscous forces are more due to momentum transfer as distance between molecules is larger and
velocities are higher.

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 Effect of Temperature on Viscosity

When temperature increases the distance between molecules increases and the cohesive force decreases.
So, viscosity of liquids decrease when temperature increases.

In the case of gases, the contribution to viscosity is more due to momentum transfer. As temperature
increases, more molecules cross over with higher momentum differences. Hence, in the case of gases,
viscosity increases with temperature.

Significance of Kinematic Viscosity

Kinematic viscosity, ν=µ/ρ,

The unit in SI system is m2/s. (Ns/m2) (m3/ kg) = [(kg m/s2) (s/m2)] [m3/kg] =m2/s
Popularly used unit is stoke (cm2/s) = 10–4 m2/s named for the scientist Stokes.

Centi stoke is also popular = 10–6 m2/s.

Kinematic viscosity represents momentum diffusivity. It may be explained by modifying equation
τ = µ (du/dy) = (µ/ρ) × {d (ρu/dy)} = ν × {d (ρu/dy)}
Where, d (ρu/dy) represents momentum flux in the y direction.

So, (µ/ρ) = ν kinematic viscosity gives the rate of momentum flux or momentum diffusivity.
With increase in temperature kinematic viscosity decreases in the case of liquids and increases in the case
of gases. For liquids and gases absolute (dynamic) viscosity is not influenced significantly by pressure. But
kinematic viscosity of gases is influenced by pressure due to change in density. In gas flow it is better to use
absolute viscosity and density, rather than tabulated values of kinematic viscosity, which is usually for 1
atm.

SURFACE TENSION
Many of us would have seen the demonstration of a needle being supported on water surface without it
being wetted. This is due to the surface tension of water.
All liquids exhibit a free surface known as meniscus when in contact with vapour or gas. Liquid molecules
exhibit cohesive forces binding them with each other. The molecules below the surface are generally free
to move within the liquid and they move at random. When they reach the surface they reach a dead end in
the sense that no molecules are present in great numbers above the surface to attract or pull them out of
the surface. So they stop and return back into the liquid. A thin layer of few atomic thickness at the surface
formed by the cohesive bond between atoms slows down and sends back the molecules reaching the
surface. This cohesive bond exhibits a tensile strength for the surface layer and this is known as surface
tension. Force is found necessary to stretch the surface.
Surface tension may also be defined as the work in Nm/m2 or N/m required to create unit surface of the
liquid. The work is actually required for pulling up the molecules with lower energy from below, to form
the surface.
Another definition for surface tension is the force required to keep unit length of the surface film in
equilibrium (N/m). The formation of bubbles, droplets and free jets are due to the surface tension of the
liquid.

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 Surface Tension Effect on Solid-Liquid Interface
In liquids cohesive forces between molecules lead to surface tension. The formation of droplets is a direct
effect of this phenomenon. So also the formation of a free jet, when liquid flows out of an orifice or
opening like a tap. The pressure inside the droplets or jet is higher due to the surface tension.
Wall
Liquid surface
Liquid
surface Wall

Adhesive Adhesive
 forces forces
higher lower




Liquids also exhibit adhesive forces when they come in contact with other solid or liquid surfaces. At the
interface this leads to the liquid surface being moved up or down forming a curved surface. When the
adhesive forces are higher the contact surface is lifted up forming a concave surface. Oils, water etc.
exhibit such behavior. These are said to be surface wetting. When the adhesive forces are lower, the
contact surface is lowered at the interface and a convex surface results as in the case of mercury. Such
liquids are called non-wetting.
The angle of contact “β” defines the concavity or convexity of the liquid surface. It can be shown that if the
surface tension at the solid liquid interface (due to adhesive forces) is σs1 and if the surface tension in the
liquid (due to cohesive forces) is σ11 then

At the surface this contact angle will be maintained due to molecular equilibrium. The result of this
phenomenon is capillary action at the solid liquid interface. The curved surface creates a pressure
differential across the free surface and causes the liquid level to be raised or lowered until static
equilibrium is reached.

COMPRESSIBILITY AND BULK MODULUS

Bulk modulus, Ev is defined as the ratio of the change in pressure to the rate of change of volume due
to the change in pressure. It can also be expressed in terms of change of density.
Ev = – dp/(dv/v) = dp/(dρ/ρ)
Where dp is the change in pressure causing a change in volume dv when the original volume was v. The
unit is the same as that of pressure, obviously. Note that dv/v = – dρ/ρ.
The negative sign indicates that if dp is positive then dv is negative and vice versa, so that the bulk modulus
is always positive (N/m2). The symbol used in this text for bulk modulus is Ev (K is more popularly used).
This definition can be applied to liquids as such, without any modifications. In the case of gases, the value
of compressibility will depend on the process law for the change of volume and will be different for
different processes.
The bulk modulus for liquids depends on both pressure and temperature. The value increases with
pressure as dv will be lower at higher pressures for the same value of dp. With temperature the bulk
modulus of liquids generally increases, reaches a maximum and then decreases. For water the maximum is

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 at about 50°C. The value is in the range of 2000 MN/m2 or 2000 × 106 N/m2 or about 20,000 atm. Bulk
modulus influences the velocity of sound in the medium, which equals (go×Ev/ρ)0.5.

PRESSURE

Pressure is a measure of force distribution over any surface associated with the force. Pressure is a surface
phenomenon and it can be physically visualized or calculated only if the surface over which it acts is
specified. Pressure may be defined as the force acting along the normal direction on unit area of the
surface. However a more precise definition of pressure, P is as below:
P = lim(∆/∆) =
→
F is the resultant force acting normal to the surface area A. ‘a’ is the limiting area which will give results
independent of the area. This explicitly means that pressure is the ratio of the elemental force to the
elemental area normal to it.
The force dF in the normal direction on the elemental area dA due to the pressure P isdF = P dA
The unit of pressure in the SI system is N/m2 also called Pascal (Pa). As the magnitude is small kN/m2 (kPa)
and MN/m2 (MPa) are more popularly used. The atmospheric pressure is approximately 105 N/m2 and is
designated as ‘‘bar’’. This is also a popular unit of pressure. In the metric system the popular unit of
pressure is kgf/cm2. This is approximately equal to the atmospheric pressure or 1 bar.
PASCAL’S LAW
In fluids under static conditions pressure is found to be independent of the orientation of the area. This
concept is explained by Pascal’s law which states that the pressure at a point in a fluid at rest is equal in
magnitude in all directions. Tangential stress cannot exist ifa fluid is to be at rest. This is possible only if
the pressure at a point in a fluid at rest is the same in all directions so that the resultant force at that point
will be zero.
The proof for the statement is given below.

Consider a wedge shaped element in a volume of fluid as shown in Fig. 2.3.1. Let the thickness
perpendicular to the paper be dy. Let the pressure on the surface inclined at an angle θto vertical bePθ and
its length be dl. Let the pressure in the x, y and z directions be Px, Py, Pz.
First considering the x direction. For the element to be in equilibrium,
Pθ× dl × dy × cosθ= Px× dy × dz
But, dl × cosθ= dz So, Pθ= Px
When considering the vertical components, the force due to specific weight should be considered.
Pz× dx × dy = Pθ× dl × dy × sinθ+ 0.5 ×y× dx × dy × dz

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 The second term on RHS of the above equation is negligible, its magnitude is one order less compared
to the other terms.
Also, dl × sinθ= dx
So, Pz= Pθ
Hence, Px = Pz = Pθ
Note that the angle has been chosen arbitrarily and so this relationship should hold for all angles. By
using an element in the other direction, it can be shown that
Py = Pθ and so Px = Py = Pz
Hence, the pressure at any point in a fluid at rest is the same in all directions. The pressure at a point has
only one value regardless of the orientation of the area on which it is measured. This can be extended to
conditions where fluid as a whole (like a rotating container) is accelerated like in forced vortex or a tank of
water getting accelerated without relative motion between layers of fluid. Surfaces generally experience
compressive forces due to the action of fluid pressure.
PRESSURE VARIATION IN STATIC FLUID (HYDROSTATIC LAW)

It is necessary to determine the pressure at various locations in a stationary fluid to solve engineering
problems involving these situations. Pressure forces are called surface forces.
Gravitational force is called body force as it acts on the whole body of the fluid.

Consider an element in the shape of a small cylinder of constant area dAs along the s direction inclined
at angle θ to the horizontal, as shown in Fig. 2.4.1. The surface forces are P at section s and P + dp at
section s + ds. The surface forces on the curved area are balanced. The body force due to gravity acts
vertically and its value is y × ds × dAs. A force balance in the s direction (for the element to be in
equilibrium) gives
P × dAs– (P + dp) × dAs–y× dAs× ds × sinθ= 0
Simplifying, dp/ds = –y× sinθor, dp = –y× ds × sinθ
This is the fundamental equation in fluid statics.
The variation of specific weight y with location or pressure can also be taken into account, if these
relations are specified as = y (P, s)
For x axis, θ = 0 and sin θ = 0, dP/dx = 0
In a static fluid with no acceleration, the pressure gradient is zero along any horizontal line i.e., planes
normal to the gravity direction.
In y direction, θ = 90 and sin θ = 1, dP/dy = –y= –ρg/go
Rearranging and integrating between limits y1 and y, dP = –y dy
If y is constant as in the case of liquids, these being incompressible,
P – P1= –y× (y – y1) = –ρ g (y – y1)/go, (As P1, y1 and y are specified)
For any given situation, P will be constant if y is constant

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 This leads to the statement,The pressure will be the same at the same level in any connected static fluid
whose density is constant or a function of pressure only.
A consequence is that the free surface of a liquid will seek a common level in any container, where the
free surface is everywhere exposed to the same pressure.

PRESSURE MEASUREMENT
FLUID PRESSURE
In a stationary fluid the pressure is exerted equally in all directions and is referred to as the static
pressure. In a moving fluid, the static pressure is exerted on any plane parallel to the direction of motion.
The fluid pressure exerted on a plane right angles to the direction of flow is greater than the static
pressure because the surface has, in addition, to exert sufficient force to bring the fluid to rest. The
additional pressure is proportional to the kinetic energy of fluid; it cannot be measured independently of
the static pressure.

When the static pressure in a moving fluid is to be determined, the measuring surface must be parallel to
the direction of flow so that no kinetic energy is converted into pressure energy at the surface. If the fluid
is flowing in a circular pipe the measuring surface must be perpendicular to the radial direction at any
point. The pressure connection, which is known as a piezometer tube, should flush with the wall of the
pipe so that the flow is not disturbed: the pressure is then measured near the walls were the velocity is a
minimum and the reading would be subject only to a small error if the surface were not quite parallel to
the direction of flow.

The static pressure should always be measured at a distance of not less than 50 diameters from bends or
other obstructions, so that the flow lines are almost parallel to the walls of the tube. If there are likely to
be large cross-currents or eddies, a piezometer ring should be used. This consists of 4 pressure tapings
equally spaced at 90o intervals round the circumference of the tube; they are joined by a circular tube
which is connected to the pressure measuring device. By this means, false readings due to irregular flow
or avoided. If the pressure on one side of the tube is relatively high, the pressure on the opposite side is
generally correspondingly low; with the piezometer ring a mean value is obtained.

BAROMETER

A barometer is a device for measuring atmospheric pressure. A simple barometer consists of a tube more
than 30 inch (760 mm) long inserted in an open container of mercury with a closed and evacuated end at
the top and open tube end at the bottom and with mercury extending from the container up into the tube.

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 Strictly, the space above the liquid cannot be a true vaccum. It contains mercury vapor at its saturated
vapor pressure, but this is extremely small at room temperatures (e.g. 0.173 Pa at 20oC).
The atmospheric pressure is calculated from the relation Patm = ρgh where ρ is the density of fluid in the
barometer.

Piezometer
For measuring pressure inside a vessel or pipe in which liquid is there, a tube may be attached to the
walls of the container (or pipe) in which the liquid resides so liquid can rise in the tube. By determining
the height to which liquid rises and using the relation P1 = ρgh, gauge pressure of the liquid can be
determined. Such a device is known as piezometer. To avoid capillary effects, a piezometer's tube should
be about 1/2 inch or greater.It is important that the opening of the device to be tangential to any fluid
motion, otherwise an erroneous reading will result.

MANOMETER
A somewhat more complicated device for measuring fluid pressure consists of a bent tube containing one
or more liquid of different specific gravities. Such a device is known as manometer.
In using a manometer, generally a known pressure (which may be atmospheric) is applied to one end of
the manometer tube and the unknown pressure to be determined is applied to the other end.
In some cases, however, the difference between pressures at ends of the manometer tube is desired
rather than the actual pressure at the either end. A manometer to determine this differential pressure is
known as differential pressure manometer.

Manometers – Various Form

(i) Simple U - tube Manometer
(ii) Inverted U - tube Manometer
(iii) U - tube with one leg enlarged
(iv) Two fluid U - tube Manometer
(v) Inclined U - tube Manometer
Simple U - tube Manometer
Equating the pressure at the level XX'(pressure at the same level in a continuous body of fluid is equal),

For the left hand side:

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Page no: 10
 Px = P1 + g(a+h)
For the right hand side:
Px' = P2 + ga + mgh
Since Px = Px'
P1 + g(a+h)฀฀P2 + ga + mgh
P1 - P2 = mgh - gh
i.e. P1 - P2 = (m - gh.

The maximum value of P1 - P2 is limited by the height of the manometer. To measure larger pressure
differences we can choose a manometer with heigher density, and to measure smaller pressure
differences with accuracy we can choose a manometer fluid which is having a density closer to the fluid
density.

Inverted U - tube Manometer

Inverted U-tube manometer is used for measuring pressure differences in liquids. The space above the
liquid in the manometer is filled with air which can be admitted or expelled through the tap on the top, in
order to adjust the level of the liquid in the manometer.

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Page no: 11
 Equating the pressure at the level XX'(pressure at the same level in a continuous body of static fluid is
equal),
For the left hand side:

Px = P1 - g(h+a)
For the right hand side:
Px' = P2 - (ga + mgh)
Since Px = Px'
P1 - g(h+a) = P2 - (ga + mgh)
P1 - P2 = ( - m)gh
If the manometric fluid is choosen in such a way that m <<  then,
P1 - P2 = gh.

For inverted U - tube manometer the manometric fluid is usually air.

U - Tube Manometer with one leg enlarged

Industrially, the simple U - tube manometer has the disadvantage that the movement of the liquid in
both the limbs must be read. By making the diameter of one leg large as compared with the other, it is
possible to make the movement the large leg very small, so that it is only necessary to read the
movement of the liquid in the narrow leg.

In figure, OO' represents the level of liquid surface when the pressure difference P1 - P2 is zero. Then
when pressure is applied, the level in the right hand limb will rise a distance h vertically.

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 Volume of liquid transferred from left-hand leg to right-hand leg = h(4)d2
Where d is the diameter of smaller diameter leg. If D is the diameter of larger diameter leg, then, fall in
level of left-hand leg= Volume transferred/Area of left-hand leg = (h(4)d2) / ((/4)D2) = h(d/D)2
For the left-hand leg, pressure atX, i.e. Px = P1 + g(h+a) + g h(d/D)2
For the right-hand leg, pressure at X', i.e. Px' = P2 + ga + g(h + h(d/D)2)
For the equality of pressure at XX',
P1 + g(h+a) + g h(d/D)2 = P2 + ga + mg(h + h(d/D)2)
P1 - P2 = mg(h + h(d/D)2) - gh - g h(d/D)2
If D>>d then, the term h(d/D)2 will be negligible( i.e approximately about zero)
Then P1 - P2 = (m - )gh.
Where h is the manometer liquid rise in the right-hand leg.
If the fluid density is negligible compared with the manometric fluid density (e.g. the case for air as the
fluid and water as manometric fluid), then P1 - P2 = m g h.

Manometers - Advantages and Limitations

 The manometer in its various forms is an extremely useful type of pressure measuring instrument, but
suffers from a number of limitations.
 While it can be adapted to measure very small pressure differences, it cannot be used conveniently for
large pressure differences - although it is possible to connect a number of manometers in series and to
use mercury as the manometric fluid to improve the range. (limitation)
 A manometer does not have to be calibrated against any standard; the pressure difference can be
calculated from first principles. ( Advantage)
 Some liquids are unsuitable for use because they do not form well-defined menisci. Surface tension can
also cause errors due to capillary rise; this can be avoided if the diameters of the tubes are sufficiently
large - preferably not less than 15 mm diameter. (limitation)
 A major disadvantage of the manometer is its slow response, which makes it unsuitable for measuring
fluctuating pressures.(limitation)
 It is essential that the pipes connecting the manometer to the pipe or vessel containing the liquid
under pressure should be filled with this liquid and there should be no air bubbles in the
liquid.(important point to be kept in mind)

STABILITY OF FLOATING BODY

If an object is immersed in or floated on the surface of fluid under static conditions a force acts on it due to
the fluid pressure. This force is called buoyant force. The calculation of this force is based on Archimedes
principle.

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 Archimedes principle can be stated as
(i) A body immersed in a fluid is buoyed upby a force equal to the weight of the fluid displaced
(ii) A floating body displaces its own weight of the liquid in which it floats.

Other possible statements are: The resultant pressure force acting on the surface of a volume partially or
completely surrounded by one or more fluids under non flow conditions is defined as buoyant force and
acts vertically on the volume. The buoyant force is equal to the weight of the displaced fluid and acts
upwards through the centre of gravity of thedisplaced fluid. This point is called the centre of buoyancy
for the body.
This principle directly follows from the general hydrostatic equation, F = yAh and is applied in the design of
ships, boats, balloons and other such similar systems. The stability of such bodies against tilting over due
to small disturbance can be also checked using this principle.

BUYOUNCY:Up thrust on body = weight of fluid displaced by the body

If the body is immersed so that part of its volume V1 is immersed in a fluid of density 1 and the rest of its
volume V2 in another immiscible fluid of mass density 2,
Up thrust on upper part, R1 = 1gV1 acting through G1, the centroid of V1,
Up thrust on lower part, R2 = 2gV2 acting through G2, the centroid of V2,
Total up thrust = 1gV1 + 2gV2.
The positions of G1 and G2 are not necessarily on the same vertical line, and the Centre of buoyancy of the
whole body is, therefore, not bound to pass through the centroid of the whole body.

TOTAL HYDROSTATIC FORCE ON SURFACE

TOTAL HYDROSTATIC FORCE ON PLANE SURFACE

For horizontal plane surface submerged in liquid, or plane surface inside a gas chamber, or any plane
surface under the action of uniform hydrostatic pressure, the total hydrostatic force is given by

Page no: 14
 F=p A
Where p is the uniform pressure and A is the area.
In general, the total hydrostatic pressure on any plane surface is equal to the product of the area of the
surface and the unit pressure at its center of gravity.
F=Pcg A
Where pcg is the pressure at the center of gravity. For homogeneous free liquid at rest, the equation can be
expressed in terms of unit weight γ of the liquid.
F=yh¯A
Where h¯ is the depth of liquid above the centroid of the submerged area.
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 The figure shown below is an inclined plane surface submerged in a liquid. The total area of the plane
surface is given by A, cg is the center of gravity, and cp is the center of pressure.

The differential force dF acting on the element dA is

dF=pdA
dF=yhdA

From the figure h=ysinθ

dF=y(ysinθ)dA

Integrate both sides and note that y and θ are constants,

F=ysinθ ∫ydA

Recall from Calculus that ∫ydA=Ay¯

F=(ysinθ)Ay¯
F=y(y¯sinθ)A

From the figure, y¯sinθ=h¯ thus,

F=yh¯A
The product yh¯yh¯ is a unit pressure at the centroid at the plane area, thus, the formula can be
expressed in a more general term below.
F=Pcg A
Location of Total Hydrostatic Force (Eccentricity)
From the figure above, S is the intersection of the prolongation of the submerged area to the free liquid
surface. Taking moment about point S.

Fyp=∫ydF
Where
dF=y(ysinθ)dA
F=y(y¯sinθ)A

[y(y¯sinθ)A]yp=∫y[y(ysinθ)dA]
(ysinθ)Ay¯yp=(ysinθ)∫y2dA

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 Ay¯yp=∫y2dA

Again from Calculus, ∫y2dA∫y2dA is called moment of inertia denoted by I Since our reference point is S,
Ay¯yp=IS

Thus,
I
yp
¯
=
By transfer formula for moment of inertia I = I + ỹ2, the formula for yp, will become
I+ ỹ2
I =
ỹ

TOTAL HYDROSTATIC FORCE ON CURVED SURFACE

In the case of curved surface submerged in liquid at rest, it is more convenient to deal with the horizontal
and vertical components of the total force acting on the surface.

Horizontal Component
FH=Pcg A
Vertical Component
The vertical component of the total hydrostatic force on any surface is equal to the weight of either real or
imaginary liquid above it.

FV= y V
Total Hydrostatic force = √H2 + 2
K
Direction of F will be tan θ =

STABILITY OF FLOATING AND SUBMERGED BODIES

There are three possible situations for a body when immersed in a fluid.
(i) If the weight of the body is greater than the weight of the liquid of equal volume then the body
will sink into the liquid (To keep it floating additional upward force is required).
(ii) If the weight of the body equals the weight of equal volume of liquid, then the body will
submerge and may stay at any location below the surface.
(iii) If the weight of the body is less than the weight of equal volume of liquid, then the body will be
partly submerged and will float in the liquid.
Comparison of densities cannot be used directly to determine whether the body will float or sink unless
the body is solid over the full volume like a lump of iron. However the apparent density calculated by the
ratio of weight to total volume can be used to check whether a body will float or sink. If apparent density is
higher than that of the liquid, the body will sink. If these are equal, the body will stay afloat at any location.
If it is less, the body will float with part above the surface.

A submarine or ship though made of denser material floats because, the weight/volume of the ship will be
less than the density of water. In the case of submarine its weight should equal the weight of water
displaced for it to lay submerged.

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 Stability of a body: A ship or a boat should not overturn due to small disturbances butshould be stable and
return, to its original position. Equilibrium of a body exists when there is no resultant force or moment on
the body. A body can stay in three states of equilibrium.
(i) Stable equilibrium: Small disturbances will create a correcting couple and the body will go back to its
original position prior to the disturbance.
(ii) Neutral equilibrium Small disturbances do not create any additional force and so the body remains in
the disturbed position. No further change in position occurs in this case.
(iii) Unstable equilibrium: A small disturbance creates a couple which acts toincrease the disturbance and
the body may tilt over completely.

Under equilibrium conditions, two forces of equal magnitude acting along the same line of action, but in
the opposite directions exist on a floating/submerged body. These are the gravitational force on the body
(weight) acting downward along the centroid of the body and buoyant force acting upward along the
centroid of the displaced liquid. Whether floating or submerged, under equilibrium conditions these two
forces are equal and opposite and act along the same line.
When the position of the body is disturbed or rocked by external forces (like wind on a ship), the position
of the centre of gravity of the body (with respect to the body) remains at the same position. But the shape
of the displaced volume of liquid changes and so its centre of gravity shifts to a new location. Now these
two forces constitute a couple which may correct the original tilt or add to the original tilt. If the couple
opposes the movement, then the body will regain or go back to the original position. If the couple acts to
increase the tilt then the body becomes unstable.

Figure: Stability of Floating and Submerged Bodies

CONDITIONS FOR THE STABILITY OF FLOATING BODIES

(i) When the centre of buoyancy is above the centre of gravity of the floatingbody, the body is
always stable under all conditions of disturbance. A righting couple isalways created to bring the body
back to the stable condition.
(ii) When the centre of buoyancy coincides with the centre of gravity, the two forces act at the same
point. A disturbance does not create any couple and so the body justremains in the disturbed position.
There is no tendency to tilt further or to correct the tilt.
(iii) When the centre of buoyancy is below the centre of gravity as in the case of ships, additional
analysis is required to establish stable conditions of floating.

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Page no: 18
 This involves the concept of metacenter and metacentric height. When the body is disturbed the centre of
gravity still remains on the centroidal line of the body. The shape of the displaced volume changes and the
centre of buoyancy moves from its previous position.
The location M at which the line of action of buoyant force meets the centroidal axis of the body, when
disturbed, is defined as metacenter. The distance of this point from the centroid of the body is called
metacentric height.
If the metacenter is above the centroid of the body, the floating body will be stable. If it is at the centroid,
the floating body will be in neutral equilibrium. If it is below the centroid, the floating body will be
unstable.

Figure: Metacenter Height and stable condition

When a small disturbance occurs, say clockwise, then the centre of gravity moves to the right of the
original centre line. The shape of the liquid displaced also changes and the centre of buoyancy also
generally moves to the right. If the distance moved by the centre of buoyancy is larger than the distance
moved by the centre of gravity, the resulting couple will act anticlockwise, correcting the disturbance. If
the distance moved by the centre of gravity is larger, the couple will be clockwise and it will tend to
increase the disturbance or tilting.
The distance between the metacenter and the centre of gravity is known is metacentricheight. The
magnitude of the righting couple is directly proportional to the metacentric height.Larger the metacentric
height, better will be the stability.
The centre of gravity G is above the centre of buoyancy B. After a small clockwise tilt, the centre of
buoyancy has moved to B′. The line of action of this force is upward and it meets the body centre line at
the metacenter M which is above G. In this case metacentric height is positive and the body is stable. It
may also be noted that the couple is anticlockwise. If M falls below G, then the couple will be clockwise
and the body will be unstable.
METACENTER HEIGHT

In equilibrium
Due to symmetry of the situation, displaced volume remains unaltered and hence the buoyancy force

Thus,
For couple calculation, can be equivalently taken as a sum of
(Upward) due to added volume of fluid RQQ'
(Downward) due to decreased volume RPP'
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 Let DC be the couple due to these forces. Taking an element of area dA on the surface RQQ' at a distance x
from the center line. Corresponding volume element is x DθdA and the buoyant force is ρgxDθ dA. This
produces the couple 2ρgx DθdA (due to symmetrically located element on ‘P’).

Therefore, Integrating

Now we have where r is moment arm of about B.

Therefore,

Now from fig (d)

Thus

This is the metacentric height.

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