Fluid Mechanics I

(ME 321)

Introduction to Fluid Mechanics

1
 Definition

 Mechanics is the oldest physical science that deals

with both stationary and moving bodies under the
influence of forces.
 The branch of mechanics that deals with bodies at rest
is called statics, while the branch that deals with
 bodies in motion is called dynamics.
 The subcategory fluid mechanics is defined as the
science that deals with the behavior of fluids at rest
(fluid statics) or in motion (fluid dynamics), and the
interaction of fluids with solids or other fluids at the
boundaries.
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 Definition
 The study of fluids at rest is called fluid statics.
 The study of f1uids in motion, where pressure forces are not
considered, is called fluid kinematics and if the pressure
forces are also considered for the fluids in motion. that
branch of science is called fluid dynamics.
 Fluid mechanics itself is also divided into several categories.
 The study of the motion of fluids that are practically
incompressible (such as liquids, especially water, and gases
at low speeds) is usually referred to as hydrodynamics.
 A subcategory of hydrodynamics is hydraulics, which deals
with liquid flows in pipes and open channels.

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 Definition
 Gas dynamics deals with the flow of fluids that undergo
significant density changes, such as the flow of gases
through nozzles at high speeds.
 The category aerodynamics deals with the flow of gases
(especially air) over bodies such as aircraft, rockets, and
automobiles at high or low speeds.
 Some other specialized categories such as meteorology,
oceanography, and hydrology deal with naturally
occurring flows.

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 What is a fluid?
A fluid is a substance in the gaseous or liquid form that
deforms continuously under the application of shear force.
Distinction between solid and fluid?
Solid: can resist an applied shear by deforming. Stress is
proportional to strain
Fluid: deforms continuously under applied shear. Stress
is proportional to strain rate

Solid Fluid
F F V
    
A A h
Figure.
Deformation of a rubber eraser placed between two
parallel plates under the influence of a shear force.
What is a fluid?

Stress is defined as the

force per unit area.
Normal component:
normal stress
In a fluid at rest, the
normal stress is called
pressure
Tangential component:
shear stress
What is a fluid?

A liquid takes the shape of

the container it is in and
forms a free surface in the
presence of gravity
A gas expands until it
encounters the walls of the
container and fills the entire
available space. Gases
cannot form a free surface
Gas and vapor are often
used as synonymous
words
What is a fluid?

solid liquid gas

No-slip condition
No-slip condition: A fluid in
direct contact with a solid
``sticks'‘ to the surface due to
viscous effects
Responsible for generation of
wall shear stress w, surface
drag D= ∫w dA, and the
development of the boundary
layer
The fluid property responsible
for the no-slip condition is
viscosity
Important boundary condition
in formulating initial boundary
value problem (IBVP) for
analytical and computational
fluid dynamics analysis
Classification of Flows

We classify flows as a tool in making simplifying

assumptions to the governing partial-differential
equations, which are known as the Navier-
Stokes equations
Conservation of Mass

Conservation of Momentum
Viscous vs. Inviscid Regions of Flow
Regions where frictional
effects are significant are
called viscous regions.
They are usually close to
solid surfaces.
Regions where frictional
forces are small
compared to inertial or
pressure forces are called
inviscid
Internal vs. External Flow

Internal flows are

dominated by the
influence of viscosity
throughout the
flowfield
For external flows,
viscous effects are
limited to the
boundary layer and
wake.
Compressible vs. Incompressible Flow
A flow is classified as
incompressible if the density
remains nearly constant.
Liquid flows are typically
incompressible.
Gas flows are often
compressible, especially for
high speeds.
Mach number, Ma = V/c is a
good indicator of whether or
not compressibility effects are
important.
Ma < 0.3 : Incompressible
Ma < 1 : Subsonic
Ma = 1 : Sonic
Ma > 1 : Supersonic
Ma >> 1 : Hypersonic
1
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Compressible versus Incompressible Flow…

 A pressure of 210 atm, for example, causes the density of
liquid water at 1 atm to change by just 1 percent.
 Gases, on the other hand, are highly compressible.
 A pressure change of just 0.01 atm, for example, causes a
change of 1 percent in the density of atmospheric air.
 Gas flows can often be approximated as incompressible if
the density changes are under about 5 percent.
 The compressibility effects of air can be neglected at
speeds under about 100 m/s.
Laminar vs. Turbulent Flow
Laminar: highly ordered
fluid motion with smooth
streamlines.
Turbulent: highly
disordered fluid motion
characterized by velocity
fluctuations and eddies.
Transitional: a flow that
contains both laminar and
turbulent regions
Reynolds number, Re=
rUL/m is the key
parameter in determining
whether or not a flow is
laminar or turbulent.
1
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Laminar versus Turbulent Flow

 The flow of low-viscosity fluids
such as air at high velocities is
typically turbulent.
 A flow that alternates between
being laminar and turbulent is
called transitional.
Steady vs. Unsteady Flow
Steady implies no change at
a point with time. Transient
terms in N-S equations are
zero
Unsteady is the opposite of
steady.
Transient usually describes a
starting, or developing flow.
Periodic refers to a flow which
oscillates about a mean.
Unsteady flows may appear
steady if “time-averaged”
The term uniform implies no
change with location over a
specified region.
One-, Two-, and Three-Dimensional Flows

N-S equations are 3D vector equations.

Velocity vector, U(x,y,z,t)= [Ux(x,y,z,t),Uy(x,y,z,t),Uz(x,y,z,t)]
Lower dimensional flows reduce complexity of analytical and
computational solution
Change in coordinate system (cylindrical, spherical, etc.) may
facilitate reduction in order.
Example: for fully-developed pipe flow, velocity V(r) is a function of
radius r and pressure p(z) is a function of distance z along the pipe.
System and Control Volume
A system is defined as a
quantity of matter or a
region in space chosen
for study.
A closed system consists
of a fixed amount of
mass.
An open system, or
control volume, is a
properly selected region
in space.
We'll discuss control
volumes in more detail in
Chapter 6.
 Application areas of Fluid Mechanics
 Mechanics of fluids is extremely important in many areas
of engineering and science. Examples are:
 Biomechanics
 Blood flow through arteries and veins
 Airflow in the lungs
 Flow of cerebral fluid
 Households
 Piping systems for cold water, natural gas, and sewage
 Piping and ducting network of heating and air-
conditioning systems
 Meteorology and Ocean Engineering
 Movements of air currents and water currents
 Chemical Engineering
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 Design of chemical processing equipment
 Application areas of Fluid Mechanics
 Mechanical Engineering
 Design of pumps, turbines, air-conditioning equipment,
pollution-control equipment, etc.
 Design and analysis of aircraft, boats, submarines,
rockets, jet engines, wind turbines, biomedical devices,
the cooling of electronic components, and the
transportation of water, crude oil, and natural gas.
 Civil Engineering
 Transport of river sediments
 Pollution of air and water
 Design of piping systems
 Flood control systems

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 Properties of Fluids
 Any characteristic of a system is called a property.
 Some familiar properties are pressure P, temperature T,
volume V, and mass m.
 Other less familiar properties include viscosity, thermal
conductivity, modulus of elasticity, thermal expansion
coefficient, electric resistivity, and even velocity and
elevation.
 Properties are considered to be either intensive or extensive.
Intensive properties are those that are independent of the mass
of a system, such as temperature, pressure, and density.
 Extensive properties are those whose values depend on the
size—or extent—of the system. Total mass, total volume V, and
total momentum are some examples of extensive properties.

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 Properties of Fluids

 An easy way to determine

whether a property is
intensive or extensive is to
divide the system into two
equal parts with an imaginary
partition.
 Each part will have the same
value of intensive properties
as the original system, but
half the value of the extensive
properties.

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 Properties of Fluids
Density or Mass Density.
 Density or mass density of a fluid is defined as the ratio of
the mass of a f1uid to its volume. Thus mass per unit
volume of a fluid is called density. It is denoted the symbol
ρ (rho). The unit of mass density in SI unit is kg per cubic
meter, 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.
Massof fluid
r
Volume of fluid
 The value of density of water is 1 gm/cm3 or 1000 kg/m3.
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Continuum
Atoms are widely spaced in the
gas phase.
However, we can disregard the
atomic nature of a substance.
View it as a continuous,
homogeneous matter with no
holes, that is, a continuum.
This allows us to treat properties
as smoothly varying quantities.
Continuum is valid as long as
size of the system is large in
comparison to distance between
molecules.
 Properties of Fluids
Density or Mass Density.
 The density of a substance, in general, depends on
temperature and pressure.
 The density of most gases is proportional to pressure and
inversely proportional to temperature.
 Liquids and solids, on the other hand, are essentially
incompressible substances, and the variation of their
density with pressure is usually negligible.

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 Properties of Fluids
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 and it is denoted by the symbol γ.
 Mathematically,
Weight of fluid (Massof fluid) x Accelerati on due to gravity
γ 
Volume of fluid Volume of fluid
Massof fluid x g

Volume of fluid
rx g
γ  rg
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 Properties of Fluids
Specific Volume.
 Specific volume of a fluid is defined as the volume
of a fluid occupied by a unit mass or volume per
unit mass of a fluid is called specific volume.
 Mathematically, it is expressedas

Volume of fluid 1 1
Specific volume   
Massof fluid Massof fluid r
Volume
 Thus specific volume is the reciprocal of mass
density. It is expressed as m3/kg.
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 It is commonly applied to gases.
 Properties of Fluids
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.
 For liquids, the standard fluid is taken water and for gases, the
standard fluid is taken air. Specific gravity is also called relative
density. It is dimensionless quantity and is denoted by the symbol S.
Weight density (density)of liquid
S(for liquids) 
Weight density (density)of water
Weight density (density)of gas
S(for gases) 
Weight density (density)of air
Thus weight density of a liquid  S x Weight density of water
 S x 1000 x 9.81 N/m3
Thus density of a liquid  S x Density of water
25  S x 1000 kg/m 3
 Properties of Fluids
Specific Gravity.
 If the specific gravity of a
fluid is known, then the
density of the fluid will be
equal to specific gravity of
fluid multiplied by the
density of water.
 For example the specific
gravity of mercury is 13.6,
hence density of mercury
= 13.6 x 1000 = 13600
kg/m3.

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Vapor Pressure and Cavitation
Vapor Pressure Pv is defined
as the pressure exerted by its
vapor in phase equilibrium
with its liquid at a given
temperature
If P drops below Pv, liquid is
locally vaporized, creating
cavities of vapor.
Vapor cavities collapse when
local P rises above Pv.
Collapse of cavities is a
violent process which can
damage machinery.
Cavitation is noisy, and can
cause structural vibrations.
 Properties of Fluids
Example 1.
Calculate the specific weight, density and specific gravity of one
liter of a liquid which weighs 7 N.

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 Example 2. Calculate the density, specific weight and weight of
one liter of petrol of specific gravity = 0.7

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 Properties of Fluids
Viscosity
 Viscosity is defined as the property of a fluid which offers
resistance to the movement of one layer of fluid over another
adjacent layer of the fluid.
 When two layers of a fluid, a distance 'dy' apart move one over
the other at different velocities say u and u+ du as shown in Fig.
1.1 , the viscosity together with relative velocity causes a shear
stress acting between the fluid layers:

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 Properties of Fluids
Viscosity
 The top layer causes a shear stress on the adjacent lower
layer while the lower layer causes a shear stress on the
adjacent top layer.
 This shear stress is proportional to the rate of change of
velocity with respect to y. It is denoted by symbol τ called
Tau.
 Mathematically,

 or

  m du (1.2)
dy
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 Properties of Fluids
 where μ (called mu) is the constant of proportionality
and is known as the coefficient of dynamic viscosity or
only viscosity.
du
 dy represents the rate of shear strain or rate of shear
deformation or velocity gradient.
 From equation (1.2) we have

m 
(1.3)
du
dy
 Thus viscosity is also defined as the shear stress
required to produce unit rate of shear strain.
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 Properties of Fluids
Unit of Viscosity.
 The unit of viscosity is obtained by putting the
dimension of the quantities in equation ( 1.3)

Newton second N.s

SI unit of viscosity  
m2 m2
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A common viscosity unit is poise, which is equivalent to 0.1 Pa.s
 Properties of Fluids
Kinematic Viscosity.
 It is defined as the ratio between the dynamic viscosity and
density of fluid.lt is denoted by the Greek symbol (ν) called
'nu' . Thus, mathematically,
Viscosity m
  
Density r
We can derive the SI units for kinematic viscosity by
substituting the previously developed units for μ and ρ:

Two common units of kinematic viscosity are m2/s and stoke

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(1 stoke = 1 cm2/s = 0.0001 m2/s).
 Properties of Fluids
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 proportionality is called the co-efficient viscosity.
Mathematically, it is expressed as given by equation (1 . 2).

Fluids for which the rate of deformation is linearly proportional to the shear
stress are called Newtonian fluids after Sir Isaac Newton, who expressed it
first in 1687. Most common fluids such as water, air, gasoline, and oils are
Newtonian fluids.

42 Blood and liquid plastics are examples of non-Newtonian fluids.

The rate of deformation (velocity gradient) of a Newtonian fluid
is proportional to shear stress, and the constant of
proportionality is the viscosity.
 Properties of Fluids
 Fluids which obey the above relation are known as
Newtonian fluids and the fluids which do not obey the
above relation are called Non-newtonian fluids.
Variation of Viscosity with Temperature
 Temperature affects the viscosity.
 The viscosity of liquids decreases with the increase of
temperature while the viscosity of gases increases with
increase of temperature. This is due to reason that the
viscous forces in a fluid are due to cohesive forces and
molecular momentum transfer.
 In liquids the cohesive forces predominates the molecular
momentum transfer due to closely packed molecules and
with the increase in temperature, the cohesive forces
decreases with the result of decreasing viscosity.
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The viscosity of liquids decreases and the viscosity of gases
increases with temperature.
 Properties of Fluids
 But in the case of gases the cohesive force are small and
molecular momentum transfer predominates. With the
increase in temperature, molecular momentum transfer
increases and hence viscosity increases. The relation between
viscosity and temperature for liquids and gases are:

The viscosity of gases is expressed as a function of

temperature by the Sutherland correlation (from The U.S.
Standard Atmosphere) as

where T is absolute temperature and a and b are

experimentally determined constants.
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For air at atmospheric conditions:

For liquids, the viscosity is approximated as

 Types of Fluids
1. Ideal Fluid. 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.
2. Real fluid. A fluid, which possesses viscosity, is known as
real fluid. All the fluids: in actual practice, are real fluids.
3. Newtonian Fluid. A real fluid, in which the shear stressis
directly, proportional to the rate of shear strain (or
velocity gradient), is known as a Newtonian fluid.
4. Non-Newtonian fluid. A real fluid, in which shear stress
is not proportional to the rate of shear strain (or velocity
gradient), known as a Non-Newtonian fluid.

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The study of the deformation and flow characteristics of
substances is called rheology, which is the field from which we
learn about the viscosity of fluids.

For non-Newtonian fluids, the relationship between shear stress

and rate of deformation is not linear, as shown in the figure
below. The slope of the curve on the taw versus du/dy chart is
referred to as the apparent viscosity of the fluid.
Fluids for which the apparent viscosity increases with the rate of
deformation (such as solutions with suspended starch or sand) are
referred to as dilatant or shear thickening fluids.

And those that exhibit the opposite behavior (the fluid becoming
less viscous as it is sheared harder, such as some paints, polymer
solutions, and fluids with suspended particles) are referred to as
pseudoplastic or shear thinning fluids. (Eg. Blood plasma, latex,
inks )
Some materials such as toothpaste can resist a finite shear stress
and thus behave as a solid, but deform continuously when the
shear stress exceeds the yield stress and behave as a fluid. Such
materials are referred to as Bingham plastics after Eugene C.
Bingham (1878–1945), who did pioneering work on fluid viscosity
for the U.S. National Bureau of Standards in the early twentieth
century. (Eg: Chocolate, mayonnaise, toothpaste)
 Example 3
If the velocity distribution over a plate is given by
2
u  y  y2
3
in which u is velocity in metre per second at a distance y
metre above the plate, determine the shear stress at y = 0
and y= 0.15 m. Take dynamic viscosity of fluid as 8.63
poises.

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53
 Example 4
Calculate the dynamic viscosity of an oil, which is used for
lubrication between a square plate of size 0.8 m x 0.8 m and an
inclined plane with angle of inclination 30o as shown in Fig. 1.4.
The weight of the square plate is 300 N and it slides down the
inclined plane with a uniform velocity of 0.3 m/s. The thickness
of oil film is 1.5 mm.

Fig.1.4

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55
 Example 5
The space between two square flat parallel plates is filled with
oil. Each side of the plate is 60 cm. The thickness of the oil
film is 12.5 mm. The upper plate, which moves at 2.5 metre per
sec requires a force of 98.1 N to maintain the speed.
Determine : ·
i.the dynamic viscosity of the oil, and
ii.the kinematic viscosity of the oil if the specific gravity of the
oil is 0.95.

Solution. Given:
Each side of a square plate = 60 cm = 0.6 m
Area A= 0.6 x 0.6 = 0.36 m2
Thickness of oil film dy = 12.5 mm = 12.5 x 10-3 m
Velocity of upper plate u = 2.5 m/s
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57
 Thermodynamic Properties
 Fluids consist of liquids or gases. But gases are compressible
fluids and hence thermodynamic properties play an important
role.
 With the change of pressure and temperature, the gases undergo
large variation in density.
 The relationship between pressure (absolute), specific volume
and temperature (absolute) of a gas is given by the equation of
state as

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 Thermodynamic Properties
J
 The value of gas constant R is R  287
kg.K

 Isothermal Process. If the changes in density occurs at

constant temperature, then the process is called isothermal
and relationship between pressure (p) and density (ρ) is
given by p
 constant
ρ
 Adiabatic Process. If the change in density occurs with no
heat exchange to and from the gas, the process is called
adiabatic. And if no heat is generated within the gas due to
friction, the relationship between pressure and density is
given by
p
 constant
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ρ k
 Thermodynamic Properties
 where k = Ratio of specific heat of a gas at constant
pressure and constant volume.
 k = 1.4 for air

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 Compressibility and Bulk Modulus
 Compressibility is the reciprocal of the bulk modulus of
elasticity, K which is defined as the ratio of compressive
stress to volumetric strain.
 Consider a cylinder fitted with a piston as shown in the Fig.
 Let V= Volume of a gas enclosed in the cylinder
p =Pressure of gas when volume is V
 Let the pressure is increased to p+ dp, the volume of gas
decreases from V to V – dV.
 Then increase in pressure = dp
 Decrease in volume = dV
 Volumetric strain = - dV/V

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 Compressibility and Bulk Modulus
 - ve sign means the volume
decreases with increase of pressure.

Increase of pressure
Bulk modules K
Volumetric strain
dp dp
  V
- dV dV
V
 Compressibility is given by = 1/K

Relation between Bilk Modulus,

pressure and density
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 Surface Tension and Capillarity
 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 is created due to the unbalanced cohesive
forces acting on the liquid molecules at the fluid surface.
 Molecules in the interior of the fluid mass are surrounded
by molecules that are attracted to each other equally.
 However, molecules along the surface are subjected to a net
force toward the interior.
 The apparent physical consequence of this unbalanced
force along the surface is to create the hypothetical skin or
membrane.
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 Surface Tension and Capillarity
 A tensile force may be
considered to be acting in the
plane of the surface along any
line in the surface.
 The intensity of the molecular
attraction per unit length along
any line in the surface is
called the surface tension.
 It is denoted by Greek letter σ
(called sigma).
 The SI unit is N/m.

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 Surface Tension and Capillarity
Surface Tension on liquid Droplet and
Bubble
 Consider a small spherical droplet of a
liquid of radius ‘R'. On the entire
surface of the droplet, the tensile force
due to surface tension will be acting.
 Let σ = surface tension of the liquid
 P= Pressure intensity inside the
droplet (in excess of the outside
pressure intensity)
 R= Radius of droplet.
 Let the droplet is cut into two halves.
The forces acting on one half (say left
half) will be
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 Surface Tension and Capillarity
 (i) tensile force due to
surface tension acting
around the circumference
of the cut portion as
shown and this is equal to
= σ x Circumference
= σ x 2πR
 (ii) pressure force on the
area (π/4)d2 and
 = P x πR2 as shown

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 Surface Tension and Capillarity
 These two forces will be equal and opposite under
equilibrium conditions, i.e.,

 A hollow bubble like a soap bubble in air has two surfaces

in contact with air, one inside and other outside. Thus two
surfaces are subjected surface tension.

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 Surface Tension……. Example 1
 Find the surface tension in a soap bubble of 40 mm
diameter when the inside pressure is 2.5 N/m2 above
atmospheric pressure.

68
 Surface Tension……. Example 2
 The pressure outside the droplet of water of diameter
0.04 mm is 10.32 N/cm2 (atmospheric pressure).
Calculate the pressure within the droplet if surface
tension is given as 0.0725 N/m of water.

69
 Surface Tension and Capillarity
Capillarity
 Capillarity is defined as a phenomenon of rise or fall of a
liquid surface 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.
 The attraction (adhesion) between the wall of the tube and
liquid molecules is strong enough to overcome the mutual
attraction (cohesion) of the molecules and pull them up the
wall. Hence, the liquid is said to wet the solid surface.
 It is expressed in terms of cm or mm of liquid. Its value
depends upon the specific weight of the liquid, diameter of
the tube and surface tension of the liquid.
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 Surface Tension and Capillarity

Expression for Capillary Rise

 Consider a glass tube of small
diameter ‘d’ opened at both ends
and is inserted in a liquid, say water.
 The liquid will rise in the tube
above the level of the liquid.
 Let h = the height of the liquid in
the tube . Under a state of
equilibrium, the weight of the liquid
of height h is balanced by the force
at the surface of the liquid in the
tube. But the force at the surface of
the liquid in the tube is due to
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surface tension.
 Expression for Capillary Rise…
 Let σ = Surface tension of liquid
θ = Angle of contact between the liquid and glass tube
 The weight of the liquid of height h in the tube
= (Area of the tube x h) x ρ x g

72
 Expression for Capillary Rise…

 The value of θ between water and clean glass tube is

approximately equal to zero and hence cos θ is equal to
unity. Then rise of water is given by

 Contact angle depends on both the liquid and the solid.

 If θ is less than 90o, the liquid is said to "wet" the solid.
However, if θ is greater than 90o, the liquid is repelled by
the solid, and tries not to "wet" it.
 For example, water wets glass, but not wax. Mercury on the
other hand does not wet glass.
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 Capillarity

Expression for Capillary Fall

 lf the glass tube is dipped in mercury, the revel of
mercury in the tube will be lower than the general level
of the outside liquid as shown above.

74
 Capillarity
Expression for Capillary Fall
 Let h = Height of depression in
tube.
 Then in equilibrium, two forces
arc acting on the mercury inside
the tube.
 First one is due to surface tension
acting in the downward direction
and is equal to σ x πd x cos θ.
 Second force is due to hydrostatic
force acting upward and is equal
to intensity of pressure at a depth
'h' x Area
75
 Capillarity
Expression for Capillary Fall

Value of θ for mercury and glass tube is 128o

76
 Capillarity…Example 1
 Calculate the capillary rise in a glass tube of 2.5 mm
diameter when immersed vertically in (a) water and (b)
mercury. Take surface tensions σ = 0.0725 N/m for
water and σ = 0.52 N/m for mercury in contact with
air. The specific gravity for mercury is given as 13.6
and angle of contact = 130o.

77
 Capillarity…Example 1

78
 Capillarity…Example 2
 Find out the minimum size of glass tube that can be used to
measure water level if the capillary rise in the tube is to be
restricted to 2 mm. Consider surface tension of water in
contact with air as 0.073575 N/m.

79
 Dimensions and Units
 Fluid mechanics deals with the measurement of many
variables of many different types of units. Hence we
need to be very careful to be consistent.
Dimensions and Base Units
 The dimension of a measure is independent of any
particular system of units. For example, velocity may
be in metres per second or miles per hour, but
dimensionally, it is always length per time, or L/T =
LT−1 .
 The dimensions of the relevant base units of the
Système International (SI) system are:

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 Dimensions and Units

Derived Units

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Unit Table

Quantity SI Unit English Unit

Length (L) Meter (m) Foot (ft)
Mass (m) Kilogram (kg) Slug (slug) =
lb*sec2/ft
Time (T) Second (s) Second (sec)
Temperature ( ) Celcius (oC) Farenheit (oF)
Force Newton Pound (lb)
(N)=kg*m/s2

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 Dimensions and Units…

 1 Newton – Force required to accelerate a 1 kg of mass

to 1 m/s2
 1 slug – is the mass that accelerates at 1 ft/s2 when acted
upon by a force of 1 lb
 To remember units of a Newton use F=ma (Newton’s 2nd
Law)
 [F] = [m][a]= kg*m/s2 = N
 To remember units of a slug also use F=ma => m = F / a
 [m] = [F] / [a] = lb / (ft / sec2) = lb*sec2 / ft

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