FM chapter-1.pdf

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FLUID MECHANICS
Jaspal Singh

CHAPTER-1
FLUID AND FLUID PROPERTIES
• Fluid mechanics is the branch of science which defines the behaviour of
fluids (liquids or gases) at rest as well as in motion.
NOTE: Fluid mechanics is analyzed in following forms:
1. Fluid statics: It deals with fluid in rest condition.
2. Fluid kinematics: It deals with fluid in motion without considering the force
responsible for motion.
3. Fluid dynamics: It deals with fluid in motion by considering the forces
responsible for motion.
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• A substance in liquid or gaseous phase is referred to as fluid.
• Fluid has the capability to deform continuously under the action of
shear stress.
• In contrary to solids, where stress is proportional to strain, in fluids,
stress is proportional to rate of strain.
• Fluid can also be stated to be in continuum for its analysis.
• In macro system, when the intermolecular distances are very small as
compared to dimensions of medium or system, we can assume that
there is one molecule adjacent to another molecule without any space/
void in between.
• Hence, the entire fluid mass can be considered as continuous
distribution of mass, which is termed as “continuum”.
• Fluids can be classified as liquid, gas, ideal fluid, real/ practical fluid.
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Q. A fluid is one which can be defined as a substance that
a) Has same shear stress at all points
b) Can deform indefinitely under the action of the smallest shear force.
c) Has the small shear stress in all directions
d) Is practically incompressible
[GATE: 1996]
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Q. Continuum approach in fluid mechanics is valid when
a) The compressibility is very high
b) Viscosity is low
c) The mean free path of the molecule is much smaller compared to the
characteristic dimension.
d) M>>1, where M is the Mach number
[GATE: 1992]
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PROPERTIES OF FLUID
1. DENSITY OR MASS DENSITY:
• Density or mass density of a fluid is defined as the ratio of the mass of a fluid to
its volume.
• It is donated by the symbol ρ (rho). The unit of mass density in SI unit is kg per
cubic metre, i.e., Kg/m3 .
• The density of liquids may be considered as constant while that of gases
changes with the variation of pressure and temperature.
Mathematically, mass density is written as:
𝛒 =
𝐦𝐚𝐬𝐬 𝐨𝐟 𝐟𝐥𝐮𝐢𝐝
𝐯𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝐟𝐥𝐮𝐢𝐝
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• With increase in temperature, molecular activity and spacing between the
molecules increases, hence lesser number of molecules would pack/ present in
same volume, which results in reduced mass density.
𝛒𝛂
𝟏
𝐭𝐞𝐦𝐩𝐞𝐫𝐚𝐭𝐮𝐫𝐞
• With increase in pressure, large number of molecules can be forced into the given
volume, which results in higher mass density.
𝛒𝛂 𝐩𝐫𝐞𝐬𝐬𝐮𝐫𝐞
NOTE: The value of density of water is 1gm/cm3 or 1000Kg/m3 at STP, while that
of air is 1.292 kg/m3.
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2. SPECIFIC WEIGHT OR WEIGHT DENSITY:
• Specific weight or weight density of a fluid is the ratio between the weight of
a fluid to its volume.
• Thus weight per unit volume of a fluid is called weight density .
• Thus mathematically,
Y =
weight of fluid
volume of fluid
=
mass of fluid × g
volume of fluid
𝐘 = 𝛒 × 𝐠
• It signifies the force exerted by gravity over the unit volume of fluid.
• It depends upon acceleration due to gravity, temperature and pressure of fluid.
NOTE: The value of specific weight or weight density (w) for water is 9.81×1000
Newton/m3 in SI units and that of air is 12.670 N/m3.
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3. SPECIFIC VOLUME:
Specific volume of a fluid is defined as the volume of a fluid occupied by a unit
weight of a fluid.
𝐒𝐩𝐞𝐜𝐢𝐟𝐢𝐜 𝐯𝐨𝐥𝐮𝐦𝐞 =
𝐕𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝐟𝐥𝐮𝐢𝐝
𝐖𝐞𝐢𝐠𝐡𝐭 𝐨𝐟 𝐟𝐥𝐮𝐢𝐝
=
𝟏
𝐖𝐞𝐢𝐠𝐡𝐭𝐨𝐟 𝐟𝐥𝐮𝐢𝐝
𝐕𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝐟𝐥𝐮𝐢𝐝
=
𝟏
𝐘
Thus, specific volume is the reciprocal of Weight density. It is expressed as m3/N. It
is commonly applied to gases.
4. SPECIFIC GRAVITY:
• Specific gravity is defined as the ratio of the weight density (or density) of a fluid
to the weight density( or density) of a standard fluid of same volume.
• Specific gravity is also called relative density. It is dimensionless quantity and is
donated by the symbol G.
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Mathematically,
𝐆 𝐟𝐨𝐫 𝐥𝐢𝐪𝐮𝐢𝐝𝐬 =
𝐰𝐞𝐢𝐠𝐡𝐭 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐨𝐟 𝐥𝐢𝐪𝐮𝐢𝐝
𝐰𝐞𝐢𝐠𝐡𝐭 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐨𝐟 𝐰𝐚𝐭𝐞𝐫
𝐆 𝐟𝐨𝐫 𝐠𝐚𝐬𝐞𝐬 =
𝐰𝐞𝐢𝐠𝐡𝐭 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐨𝐟 𝐠𝐚𝐬
𝐰𝐞𝐢𝐠𝐡𝐭 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐝𝐞𝐧𝐬𝐢𝐭𝐲 𝐨𝐟 𝐚𝐢𝐫
• For liquids, standard fluid is generally considered as pure water at 4oC and for
gases, standard fluid is generally considered as air or hydrogen at specified
temperature and pressure.
NOTE: Since weight density and mass density varies with temperature, hence
temperature is also reported when specific gravity is reported.
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Q. If 850 kg liquid occupies volume of one cubic meter, than 0.85 represents its
a) Specific weight
b) Specific mass
c) Specific gravity.
d) Specific density
[SSC: 2016]
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5. VISCOSITY:
• Viscosity is defined as the property of fluid which offers resistance to the
movement of one layer of fluid over another adjacent layer of fluid.
• It is due to cohesion between the molecules in case of liquids, and due to
momentum exchange between fluid layers in case of gases.
• The viscosity together with relative velocity causes a shear stress, which acts
between the fluid layers.
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NEWTON’S LAW OF VISCOSITY
• It states that the shear stress(τ) on a fluid element layer is directly proportional to
the rate of shear strain.
• The constant of proportionally is called the coefficient of viscosity.
Mathematically, it is expressed as given by:
𝛕 = 𝛍
𝐝𝐮
𝐝𝐲
• Fluids which obey the above relation are known as Newtonian fluids like water,
air, mercury, petrol and gasoline, and the fluids which do not obey the above
relation are called Non- Newtonian fluids.
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• This shear stress (𝜏) is proportional to the rate of change of velocity with respect
to y.
τα
du
dy
α
dθ
dt
𝛕 = 𝛍
𝐝𝐮
𝐝𝐲
where, 𝛍 is the constant of proportionality and is known as the co-efficient of
dynamic viscosity or only viscosity.
𝐝𝐮
𝐝𝐲
represents the rate of shear strain or rate of shear deformation or velocity
gradient.
Thus, viscosity is also defined as the shear stress required to produce unit rate of
shear strain.
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• If flow is taking place between two bodies, the variation of velocity between plate
is not linear in general, but if thickness of fluid is considered to be small, then this
variation may be assumed to be linear. This is termed as “Linearization of
Newton’s law of viscosity”.
• At y= 0,
du
dy
is maximum; shear stress is maximum
• At y= H,
du
dy
= 0; shear stress is minimum (zero)
• When a real fluid flows over a solid body, the fluid particles at the surface of the
body flows with the same velocity as that of surface of solid, so as to satisfy “no-
slip condition”. Hence, relative velocity of fluid particles at the surface of solid
body is zero.
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Q. The property which a liquid opposes relative motion between its different layers
is called
a) Surface tension
b) Coefficient of viscosity
c) Viscosity.
d) Osmosis
[SSC: 2016]
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NOTE: Units of viscosity:
NOTE: 10 poise = 1kg/m-sec or N-sec/m2
1 poise = 0.1kg/m-sec or N-sec/m2
At NTP, 𝛍𝐰𝐚𝐭𝐞𝐫 = 𝟏. 𝟎𝟎𝟐 𝐜𝐞𝐧𝐭𝐢𝐩𝐨𝐢𝐬𝐞; 𝛍𝐚𝐢𝐫 = 𝟏. 𝟖𝟑 × 𝟏𝟎−𝟐 𝐜𝐞𝐧𝐭𝐢𝐩𝐨𝐢𝐬𝐞
Hence, water is approximately 55 times more viscous than air.
System of unit Unit of viscosity
1. MKS System kgf − sec
m2
2. CGS System dyne − sec
cm2
= poise
3. SI Unit Newton − sec
m2
=
N − sec
m2
= Pa − s
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6. KINEMATIC VISCOSITY:
• It is defined as the ratio between the dynamic viscosity and density of fluid.
• In kinematic viscosity, only flow of fluid is under consideration, but the force
causing the flow is not considered.
• It is denoted by Greek symbol(𝜐) called ‘nu’.
𝐊𝐢𝐧𝐞𝐦𝐚𝐭𝐢𝐜 𝐯𝐢𝐬𝐜𝐨𝐬𝐢𝐭𝐲 = 𝛖 =
𝐃𝐲𝐧𝐚𝐦𝐢𝐜 𝐯𝐢𝐬𝐜𝐨𝐬𝐢𝐭𝐲
𝐝𝐞𝐧𝐬𝐢𝐭𝐲
=
𝛍
𝛒
• In MKS and SI, the unit of kinematic viscosity is m2 /sec while in CGS unit, it is
written as cm2/sec. In CGS units, kinematic viscosity is also known as “stokes”
1 stokes = 10-4 m2/s
NOTE: At NTP, 𝛎𝐰𝐚𝐭𝐞𝐫 = 𝟏𝟎−𝟐𝐬𝐭𝐨𝐤𝐞𝐬; 𝛎𝐚𝐢𝐫 = 𝟏. 𝟓 × 𝟏𝟎−𝟏𝐬𝐭𝐨𝐤𝐞𝐬
Hence, kinematic viscosity of air is approximately 15 times more than that of
water.
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VARIATION OF VISCOSITY OF FLUID w.r.t.
TEMPERATURE AND PRESSURE
I. EFFECT OF TEMPERATURE
Dynamic viscosity (μ)
Liquids Gases
With increase in temperature, molecules get
energized, hence can resist large intermolecular
cohesion, hence can move more freely, thus
viscosity of liquid decreases.
μliquid = a10
b
T−c
Where, a, b, c are constants
𝛍𝐥𝐢𝐪𝐮𝐢𝐝𝛂
𝟏
With increase in temperature, molecules get
energized, hence molecular collision per unit
volume per unit time also increases, thereby
offering greater resistance to flow.
μgas =
a T
1 +
b
T
Where, a, b are constants
𝛍𝐠𝐚𝐬 𝛂 𝐭𝐞𝐦𝐩𝐞𝐫𝐚𝐭𝐮𝐫𝐞
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Kinematic viscosity (𝛎)
Liquids Gases
With increase in temperature, dynamic viscosity
and density of liquid decreases, but decrease in
dynamic viscosity is more than decrease in
density, hence, kinematic viscosity of liquid also
decreases with temperature.
𝛎𝐥𝐢𝐪𝐮𝐢𝐝𝛂
𝟏
With increase in temperature, dynamic viscosity
of gas increases and at constant pressure, its
density decreases, hence kinematic viscosity of
gases increase at faster rate with increase in
temperature (even at faster rate than dynamic
viscosity)
𝛎𝐠𝐚𝐬 𝛂 𝐭𝐞𝐦𝐩𝐞𝐫𝐚𝐭𝐮𝐫𝐞
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Dynamic viscosity (𝛍)
Liquids Gases
If liquid is considered to be incompressible, then
with change of pressure, there is no change in
intermolecular cohesion between particles,
hence dynamic viscosity remains unchanged.
𝛍𝐥𝐢𝐪𝐮𝐢𝐝 = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭
For gases, dynamic viscosity is generally
independent of pressure, particularly at low to
moderate pressure (3-4 atm) but if pressure is
increased exceptionally, dynamic viscosity of
gases increases.
Kinematic viscosity (𝛎)
With increase in pressure, there is no change in
dynamic viscosity and density of liquid (if it is
considered to be incompressible), hence,
kinematic viscosity remains constant.
𝝂𝐥𝐢𝐪𝐮𝐢𝐝 = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭
With increase in pressure, dynamic viscosity of
gases remains constant (for low- medium
pressure), but its density increases significantly,
hence kinematic viscosity decreases.
𝛎𝐠𝐚𝐬𝛂
𝟏
𝐩𝐫𝐞𝐬𝐬𝐮𝐫𝐞
II. EFFECT OF PRESSURE
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Q. Poise has the unit of
a) Dyne- cm/s2
b) Dyne- cm/s
c) Dyne- s/cm
d) Dyne- s/cm2.
[IES: 2011]
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Q. On increasing the temperature of a liquid, the viscosity of the liquid
a) Decrease.
b) Increase
c) First decrease, then increase
d) Remains same
[IES: 2017]
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Q. In which of the following units is the kinematic viscosity of fluid expressed?
a) m2/s.
b) N.s/m
c) N/m2.s
d) N.s/m2
[SSC: 2018]
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Q. Statement (I): The movement of two blocks of wood joined with hot glue
requires greater and greater effort as the glue is drying up
Statement (II): Viscosity of liquid varies inversely with temperature
Choose the correct option:
a) Both Statement (I) and Statement (II) are correct and Statement (II) is correct
explanation of Statement (I).
b) Both Statement (I) and Statement (II) are correct but Statement (II) is not the
correct explanation of Statement (I)
c) Statement (I) is correct but Statement (II) is incorrect
d) Statement (I) is incorrect but Statement (II) is correct
[IES:2010]
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Q. Shear stress in the Newtonian fluid is proportional to
a) Pressure
b) Strain
c) Strain rate.
d) Inverse of viscosity
[GATE: 1996]
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Q. Newton’s law of viscosity relates to
a) Intensity of pressure and rate of angular deformation
b) Viscosity and rate of angular deformation.
c) Among shear stress, viscosity and temperature
d) None of these
[SSC: 2004]
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TYPES OF FLUIDS
Types of
fluids
Ideal fluid Real fluid
Newtonian
fluid
Non-
Newtonian
fluid
Ideal plastic
fluid
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A Fluid which is incompressible and is having no viscosity, is
known as an ideal fluid. Ideal fluid is only an imaginary fluid
as all the fluids, which exist, have some viscosity.
Ideal fluid
• A fluid, which possesses viscosity, is known as real fluid. All
the fluids, in actual practice, are real fluids.
Real fluid
• A real fluid, in which the shear stress is directly proportional
to the rate of shear strain( or velocity gradient) is known as a
Newtonian fluid.
Newtonian fluid
• A real fluid in which the shear stress is not proportional to the
rate of shear strain ( or velocity gradient), is known as a Non-
Newtonian fluid.
Non- Newtonian fluid
• A fluid, in which shear stress is more than the yield value and
is proportional to the rate of shear strain( or velocity gradient)
is known a ideal plastic fluid.
Ideal plastic fluid
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NON- NEWTONIAN FLUIDS
• These are the type of fluids which do not follow law of viscosity, i.e. in this case,
shear stress between the layers of the fluid is not directly proportional to rate of
change of shear strain/ rate of angular deformation/ velocity gradient.
• In general, for any liquid, shear stress varies as follows:
𝛕 = 𝐀
𝐝𝐮
𝐝𝐲
𝐧
+ 𝐁
Where, A = consistency index; B = yield stress ; n = flow behavior index
τ = shear stress ;
du
dy
= velocity gradient
NOTE: For Newtonian fluid, A = μ, n = 1, B = 0
dτ
d
du
dy
i.e. rate of change of shear stress with change in velocity gradient (which
signifies slope of τ-
du
dy
curve) is termed as “apparent viscosity”.
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𝐝𝛕
𝐝
𝐝𝐮
𝐝𝐲
= 𝛍𝐚𝐩𝐩𝐚𝐫𝐞𝐧𝐭 = 𝐀𝐧
𝐝𝐮
𝐝𝐲
𝐧−𝟏
For Newtonian fluid, μapparent = μ
Non- Newtonian fluids are further classified as:
I. TIME INDEPENDENT NON- NEWTONIAN FLUIDS
• It is type of non- Newtonian fluid, apparent viscosity of which does not depends
on time.
• For them, yields stress (B) is zero, i.e.
𝝉 = 𝐀
𝐝𝐮
𝐝𝐲
𝐧
These are further classified as:
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(i) Dilatant fluids: These are also termed as “shear thickening fluids”. These are
the type of fluids for which apparent viscosity increases with velocity gradient,
i.e. n>1.
• For eg- solution of suspended starch, sand, sugar in water.
(ii) Pseudo plastic fluids: These are also termed as “shear thinning fluids”.
These are the type of fluids, apparent viscosity of which decreases with velocity
gradient, i.e. n<1.
• For eg – blood, paper pulp, syrup, milk, gelatin, paint, polymer solution.
(iii) Bingham plastic fluids: These are also termed as “Ideal plastic fluids”. It is a
type of fluid which requires a certain minimum shear stress (termed as yield
stress, 𝜏𝑜) before they start flowing.
For these fluids, B = 𝜏𝑜; A = 𝜇; n = 1
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𝛕 = 𝛍
𝐝𝐮
𝐝𝐲
+ 𝛕𝐨
Here also, μapparent = μ, i.e. they are independent of velocity gradient.
For eg: Sewage Sludge , Drilling Mud , Toothpaste.
II. TIME DEPENDENT NON- NEWTONIAN FLUIDS
• These are the type of fluids, apparent viscosity of which depends on time.
• For these fluids, yield stress (B) is not zero.
𝛕 = 𝐀
𝐝𝐮
𝐝𝐲
𝐧
+ 𝐁
These are further classified as:
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(i) Thixotropic fluids: These are the type of fluids, apparent viscosity of which
decreases with time.
• For them, flow behavior index (n) is less than 1 (n<1).
• For eg- some paints, enamel, ketchup, printer’s ink.
(ii) Rheopectic fluids: These are the type of fluids, apparent viscosity of which
increases with time.
• For them, n>1.
• For eg – lubricants, gypsum paste.
III. IDEAL FLUIDS
It is an imaginary fluids which do not possess surface tension, viscosity (inviscid)
and is incompressible.
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Q. The characteristic of an ideal fluid is:
a) One which satisfies continuity equation
b) One which flows with least friction
c) One which obeys Newton’s law of viscosity
d) Frictionless and incompressible.
[SSC: 2012]
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Q. Statement (I): The shear strain graph for a Newtonian fluid is linear.
Statement (II): The coefficient of viscosity of the fluid is not a constant.
Choose the correct option:
a) Both Statement (I) and Statement (II) are correct and Statement (II) is correct
explanation of Statement (I)
b) Both Statement (I) and Statement (II) are correct but Statement (II) is not the
correct explanation of Statement (I)
c) Statement (I) is correct but Statement (II) is incorrect.
d) Statement (I) is incorrect but Statement (II) is correct
[IES: 2016]
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Q. Which of the following fluids can be classified as non- Newtonian?
1. Kerosene oil
2. Diesel oil
3. Human blood
4. Toothpaste
5. Water
Select the correct answer using the codes given below:
a) 1 and 2
b) 3 and 4.
c) 2 and 5
d) 1 and 5
[IES: 2003]
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Q. Match list (I) with list (II) and select the correct answer:
Codes: A B C D
a) 2 4 1 3
b) 3 1 4 2
c) 2 1 4 3.
d) 3 4 1 2
[IES: 2002]
List- I List- II
A. Newtonian fluid 1. Frictionless and incompressible
B. Ideal fluid 2. Viscosity invariant with shear stress
C. Thixotropic fluid 3. Viscosity decreases at higher shear stress
D. Rheopectic fluid 4. Viscosity increases at higher shear stress
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Q. Match list- I with list- II and select the correct answer:
Codes: A B C D
a) 1 2 3 4.
b) 1 2 4 3
c) 2 1 3 4
d) 2 1 4 3
[IES: 2001]
List -I List- II
A. Concentrated sugar solution 1. Dilatant fluid
B. Sewage sludge 2. Bingham plastic fluid
C. Blood 3. Pseudoplastic fluid
D. Air 4. Newtonian fluid
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COMPRESSIBILITY AND BULK MODULUS
• The property of fluid to undergo volume change on application of pressure is
termed as “compressibility”, which is quantitatively related with “bulk modulus
of elasticity (K)”.
• Compressibility is the reciprocal of the bulk modulus of elasticity, K which is
defined as the ratio of compressive stress (change in pressure) to volumetric strain.
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Bulk modulus (K) =
Increase of pressure
Volumetric strain
K =
dp
−
dV
V
= −
dP
dV
V
Units: N/m2, kg(f)/m2, gm(f)/cm2, dyne/cm2
NOTE: At NTP, Kwater = 2.06×109 N/m2 and Kair = 1.03×105 N/m2
Hence, it can be stated that air is “20,000 times” more compressible than water.
𝐂𝐨𝐦𝐩𝐫𝐞𝐬𝐬𝐢𝐛𝐢𝐥𝐢𝐭𝐲 = 𝛃 =
𝟏
𝐊
Units: m2/N
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• Bulk modulus (K) for fluids, increases with increase in pressure, as when a fluid
mass is compressed, its molecules becomes close together and its resistance to
further compression increases.
NOTE: Bulk modulus of water almost doubles when pressure is raised from 1 atm
to 3500 atm.
• In liquids, with increase in temperature, intermolecular bond between molecules
decreases, hence resistance against volume change decreases, thereby more
volume change is observed, hence bulk modulus decreases.
• In gases, with increase in temperature, random motion of particles increases,
that further increases resistance to volume change, thereby less volume change is
observed, hence bulk modulus increases.
In general, for gases, −
dV
V
=
dρ
ρ
𝐊 =
𝛒 𝐝𝐏
𝐝𝛒
; 𝐇𝐞𝐧𝐜𝐞, 𝛃 =
𝐝𝛒
𝛒 𝐝𝐏
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• If
𝐝𝛒
𝐝𝐏
= 𝟎; 𝛃 = 𝟎, means fluid is incompressible
• If
𝐝𝛒
𝐝𝐏
≠ 𝟎; 𝛃 ≠ 𝟎, means fluid is compressible
1. ISOTHERMAL COMPRESSIBILITY OF GASES
For constant temperature,
𝜕P
𝜕ρ
= RT
𝐊𝐢𝐬𝐨𝐭𝐡𝐞𝐫𝐦𝐚𝐥 = 𝐏 = 𝛃 =
𝟏
𝐊𝐢𝐬𝐨𝐭𝐡𝐞𝐫𝐦𝐚𝐥
=
𝟏
𝐏
2. ADIABATIC COMPRESSIBILITY OF GASES
γ = Adiabatic constant =
CP
CV
=
Specific heat at constant pressure
Specific heat at constant volume
𝐊𝒂𝒅𝒊𝒂𝒃𝒂𝒕𝒊𝒄 = 𝛄𝐏 = 𝛃 =
𝟏
𝐊𝒂𝒅𝒊𝒂𝒃𝒂𝒕𝒊𝒄
=
𝟏
𝛄𝐏
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Q. The bulk modulus of a fluid is given by 25GPa. What is the compressibility
(Pa-1) of that fluid?
a) 4×10-9
b) 4×10-11.
c) 25×10-9
d) 25×10-11
[SSC: 2017]
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Q. In the isothermal condition, the isothermal bulk modulus of an ideal gas is equal
to
a) Gas constant
b) Pressure.
c) Temperature
d) Viscosity
[SSC: 2017]
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SURFACE TENSION AND CAPPILARITY
• Surface tension is defined as the tensile force acting on the surface of a liquid in
contact with a gas or on the surface between two immiscible liquids such that the
contact surface behaves like a membrane under tension.
(𝐒𝐮𝐫𝐟𝐚𝐜𝐞 𝐭𝐞𝐧𝐬𝐢𝐨𝐧)𝛔 =
𝐅𝐨𝐫𝐜𝐞
𝐋𝐞𝐧𝐠𝐭𝐡
• In MKS units, it is expressed as Kg/m
while in SI units as N/m.
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• Since, surface tension is primarily due to cohesion, all the factors affecting
cohesion affects surface tension also.
• For eg- with increase in temperature of liquid, cohesive forces between molecules
reduces, hence surface tension also reduces.
• Due to surface tension, an object can be supported over the surface of liquid (eg-
insects, pin, coin) and formation of droplet over the leaves after rainfall.
• This surface tension also depends on the type on interface.
• Surface tension of water in contact with air at 20o C is approx. 0.0736N/m.
• Surface tension can also be defined as work done per unit increase in the surface
area of liquid, which is stored in the form of surface energy.
𝛔 =
𝐝𝐖
𝐝𝐀
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1. SURFACE TENSION ON A LIQUID DROPLET
Tensile force due to surface tension = pressure force
σ × πd = p ×
π
4
d2
𝐩 =
𝟒𝛔
𝐝
2. SURFACE TENSION ON A HOLLOW BUBBLE
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2(σ × πd) = p ×
π
4
d2
𝐩 =
𝟖𝛔
𝐝
3. SURFACE TENSION ON A LIQUID JET
σ × 2L = p × L × d
𝐩 =
𝟐𝛔
𝐝
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Q. The difference of pressure between the inside and outside of a liquid drop is
a) P = T× r
b) P = T/r
c) P = T/ 2r
d) P = 2T/r.
[SSC: 2016]
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Q. Surface tension
a) Acts in the plane of interface normal to any line in the surface.
b) Is also known as capillarity
c) Is a function of the curvature of the interface
d) Decreases with fall in temperature
[SSC: 2016]
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CAPPILARITY
• Capillarity is defined as a phenomenon of rise or fall of a liquid in a small tube
relative to the adjacent general level of liquid, when the tube is held vertically in the
liquid.
• The rise of liquid surface is known as capillary rise while the fall of the liquid
surface is known as capillary depression.
• It is due to both surface tension (cohesion) and adhesion.
• Its value depends upon the specific weight of the liquid, diameter of the tube and
surface tension of the liquid.
• Smaller is the bubble/ droplet, higher is the pressure difference.
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1. Capillary Rise:
Weight of liquid of height ‘h’ in tube = Vertical component of surface tension force
Area of tube × h × ρ × g = σ × circumference × cosθ
π
4
d2 × h × ρ × g = σ × πd × cosθ
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𝐡 =
𝟒𝛔𝐜𝐨𝐬𝛉
𝛒𝐠𝐝
• The value of θ between water and clean glass tube is approximately equal to zero
and hence cosθ is equal to unity. Then rise of water is given by:
𝐡 =
𝟒𝛔
𝛒𝐠𝐝
• It occurs when adhesion> cohesion
• Here, contact angle, 𝛉 < 𝟗𝟎𝒐
• Here, liquid wets the solid surface.
• Here, top surface is concave upwards.
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2. Capillary Fall:
If the glass tube is dipped in mercury, the level of mercury in the tube will be lower than
the general level of the outside liquid.
Pressure force of liquid of height ‘h’ in tube = vertical component of surface tension force
Area of tube × h × ρ × g = σ × circumference × cosθ
π
4
d2
× h × ρ × g = σ × πd × cosθ
𝐡 =
𝟒𝛔𝐜𝐨𝐬𝛉
𝛒𝐠𝐝
NOTE: value of 𝜃 for mercury and
glass tube is 128°
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• It occurs when cohesion> adhesion
• Here, contact angle, 𝛉 > 𝟗𝟎𝒐
• Here, liquid do not wet the solid surface.
• Here, top surface is convex upwards.
NOTE: Capillary rise/fall inversely depends upon the size of tube, hence capillary
effect of water is usually negligible in tubes whose size is greater than 10mm.
Due to this, pressure measuring devices are made with large tubes of size greater
than 10mm, to minimize capillary effect.
Interface Contact angle (𝛉)
1. Water glass air interface 0o
2. Mercury air glass interface 130o
3. Kerosene glass air interface 26o
4. Water paraffin air interface 107o
5. Mercury sodalime glass interface 140o
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SPECIAL CASES OF CAPILLARY
1. Capillary rise between parallel plates:
𝐡 =
𝟐𝛔 𝐜𝐨𝐬𝛉
𝛄𝐭
2. Capillary rise in annular space:
𝐡 =
𝟒𝛔 𝐜𝐨𝐬𝛉
𝛄(𝐃𝐨 − 𝐃𝐢)
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Q. Which one of the following expresses the height of rise or fall of a liquid in a
capillary tube?
a)
4wd
σ cosα
b)
σ cosα
4wd
c)
4σ cosα
wd
.
d)
wd
4σ cosα
w = specific weight of the liquid
𝛼= angle of contact of the liquid surface
𝜎= surface tension
[IES: 2007]
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Q. When the adhesion between molecules of a fluid is greater than adhesion
between fluid and the glass, then the free level of fluid in glass tube dipped in the
glass vessel will be
a) Same as the surface of the fluid
b) Lower than the surface of the fluid.
c) Higher than the surface of the fluid
d) Dependent on atmospheric pressure
[SSC: 2016]
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Q. Capillarity is due to:
i. Surface tension
ii. Cohesion
iii. Viscosity
iv. Vapor pressure
v. Weight density of liquid
a) ii, iii
b) iii
c) i.
d) ii, iii, iv
[SSC: 2014]
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VAPOUR PRESSURE AND CAVITATION
• A change from the liquid state to the gaseous state is known as vaporization.
• The vaporization (which depends upon the prevailing pressure and temperature
condition) occurs because of continuous escaping of the molecules through the
free liquid surface.
• When vaporization takes place, the molecules escapes from the free surface of the
liquid. These vapour molecules get accumulated in the space between the free
liquid surface and top of the vessel.
• These accumulated vapours exerts a partial pressure on the liquid surface. This
pressure is known as “vapour pressure” of the liquid or this is the pressure at
which the liquid is converted into vapours.
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• With increase in temperature, as molecular activity increases, vapor pressure also
increases.
• If external absolute pressure imposed over the liquid becomes less than vapor
pressure of liquid, boiling of liquid starts.
• Liquid with high vapor pressure evaporates readily and are termed as “volatile
liquids”, eg- benzene.
NOTE: Mercury (Hg) has low vapor pressure, hence it does not vaporizes even at
very low pressure, thereby it is used for measurement of pressure in equipments.
• Vapor pressure of water at different temperatures:
T (oC) Vapor pressure (kPa)
10 1.2
20 2.3
100 101.3 ≃ 1 atm
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• “Cavitation” is the phenomenon of formation of vapour bubbles of a
flowing liquid in a region where the pressure of the liquid falls below the
vapour pressure and sudden collapsing of these vapour bubbles in a region of
higher pressure.
• When the vapour bubbles collapse, a very high pressure is created. The
metallic surfaces, above which the liquid is flowing, is subjected to these
high pressures, which cause pitting action on the surface. Thus cavities are
formed on the metallic surface and hence the name is cavitation.
• With increase in temperature, vapor pressure increases, tendency of
vaporization increases, hence, cavitation also increases.
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Q. Which of the following statements is correct?
a) Dynamic viscosity is the property of a fluid which is not in motion
b) Surface energy is a fluid property giving rise to the phenomenon of capillary in
water.
c) Cavitation results from the action of very high pressure
d) Real fluids have lower viscosity than ideal fluids
[IES: 2011]
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Q. Consider the following statements which are related to the phenomenon of
cavitation in fluid flow:
1. Cavitation occurs when local velocity is decreased so that local pressure
increases to a high degree
2. Cavitation occurs if elevation is high thereby decreasing ambient pressure
3. Cavitation occurs if local velocity is increased so that the local pressure
decreases
4. Cavitation is dependent on vapor pressure of the fluid
Which of the above statements are correct?
a) 1, 2 and 3
b) 1, 2 and 4
c) 1, 3 and 4
d) 2, 3 and 4.
[IES: 2010]
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Q. A liquid would wet the solid, if adhesion forces as compared to cohesion forces
are
a) Less
b) More.
c) Equal
d) Less at low temperature and more at high temperature
[SSC: 2016]
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Q. Consider the following statements:
Cavitation generally results from a combination of several influences
1. By reduction of pressure intensity below a limiting value
2. By increase in either elevation or the velocity of flow
3. By reduction of pressure load in the system
4. By decrease in the velocity of flow
Which of the above statements are correct?
a) 1, 2 and 3
b) 1 and 2 only.
c) 2 and 3 only
d) 3 and 4
[IES: 2009]
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Q. Which one of the following statements is correct?
a) Dynamic viscosity of water is nearly 50 times that of air.
b) Kinematic viscosity of water is 30 times that of air
c) Water in soil is able to rise a considerable distance above the groundwater table
due to viscosity
d) Vapor pressure of a liquid is inversely proportional to the temperature
[IES: 2003]
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