ENGR 309 - Fluid Mechanics
CHAPTER 1 & 2
INTRODUCTION &
FLUID PROPERTIES
Lecture Outcomes
 Describe fluid mechanics.
 Contrast gases and liquids by describing similarities
and differences.
 Explain the continuum assumption.
 Define density, specific gravity, viscosity, surface
tension, vapor pressure.
 Describe how shear stress, viscosity, and the velocity
distribution are related.
Fluid
 A fluid is a substance whose molecules move
freely past each other.
 A fluid is a substance that will continuously
deform and flow under the action of a shear
stress.
 A fluid can be either liquid or gas
 A solid will deform under the action of a shear
stress but will not flow like a fluid.
Mechanics
 Mechanics is the field of science focused on
the motion and forces producing motion.
 When mechanics applies to material bodies in
the solid phase, the discipline is called solid
mechanics.
 When the material body is in the gas or liquid
phase, the discipline is called fluid mechanics.
Fluid Mechanics
 Fluid Mechanics is the science that
deals with the action of forces on
fluids, either at rest (statics) or in
motion (dynamics).
The Continuum Assumption



All matters are made up of atoms.
In gas phase atoms are widely spaced. Yet it is very
convenient to disregard the atomic nature of a
substance and view it as a continuous, homogeneous
matter with no holes, that is, a continuum.
A fluid often behaves as if it was comprised of
continuous matter that is infinitely divisible into smaller
and smaller parts. This idea is called the continuum
assumption
The Continuum Assumption
 The continuum idealization allows us to treat properties
as point functions and to assume the properties vary
continually in space with no jump discontinuities.
 This idealization is valid as long as the size of the
system we deal with is large relative to the space
between the molecules
Units
 In this course we will use SI Units.
Fluid Mechanics Applications
and Connections
 Hydraulics
 Hydraulics is the study of the flow of water
through pipes, rivers, and open-channels.
 Hydraulics includes pumps and turbines and
applications such as hydropower.
Fluid Mechanics Applications
and Connections
 Hydraulics
Fluid Mechanics Applications
and Connections
 Design of hydraulic structures
 Dams and breakwaters.
 Sewage conduits.
Fluid Mechanics Applications
and Connections
 Aerodynamics
 Aerodynamics is the study of air flow.
 Aerodynamics is important for the design of
vehicles and airplanes.
Fluid Mechanics Applications
and Connections
Aerodynamics
Fluid Mechanics Applications
and Connections
 HVAC
 HVAC stands for heating, ventilation, and
air conditioning.
Fluid Mechanics Applications
and Connections
 HVAC
Fluid Mechanics Applications
and Connections
 Bio-fluid mechanics
 Bio-fluid mechanics include the
study of air flow in lungs and
blood flow in veins and arteries,
development of artificial heart
valves, stents.
 Bio-fluid mechanics is important
for advancing health care.
Fluid Mechanics Applications
and Connections
 Bio-fluid mechanics
Fluid Mechanics Applications
and Connections
 Petroleum engineering
 Petroleum engineering is the application of
engineering to the exploration and
production of petroleum.
 Movement of oil in the ground involves flow
through a porous medium.
 Oil pipelines involve pumps
and conduit flow.
Fluid Mechanics Applications
and Connections
 Environmental engineering
 Environmental engineering involves
the application of science to protect
or improve the environment (air,
water, and or land resources) or to
remediate polluted sites.
 Environmental engineers design
water supply and wastewater
treatment systems for communities.
Fluid Mechanics Applications
and Connections
 Environmental engineering
Fluid Properties
 Mass Density ρ (kg/m3)
 It is defined as the ratio of mass to volume
at a point
 Specific volume: is given by:
Properties of the Fluid

 The gravitational force per unit volume of
fluid, or simply the weight per unit volume, is
defined as specific weight
𝒎𝒈
𝜸=
= 𝝆𝒈
𝑽
Properties of the Fluid
 Specific Gravity, S [S.G]
 The ratio of the specific weight of a given
fluid to the specific weight of water at the
standard reference temperature 4°C is
defined as specific gravity, S:
Ideal Gas
 Ideal Gas Law
• The equation of state for an ideal gas can be
expressed as:
• The value of (R) is the gas constant which is
characteristic of the gas itself.
• Values of (R) are given in Table A.2.
• Although no gas is ideal, most gases that we deal with
behave like ideal gases.
Properties Involving Thermal
Energy
 Specific Heat, c for liquids and solids
 The property that describes the capacity of a
substance to store thermal energy is called
specific heat.
 By definition, it is the amount of thermal energy
that must be transferred to a unit mass of
substance to raise its temperature by one
degree
Properties Involving Thermal
Energy
 Specific Heat, for gases
 Cv Specific heat at constant volume.
The specific volume v of the gas remains constant while
the temperature changes.
 Cp Specific heat at constant pressure
The pressure is held constant during the change in
state.
 The ratio
is given the symbol k [specific heat ratio].
Properties Involving Thermal
Energy
 Internal Energy 𝒖 (𝑱/𝒌𝒈)
 The internal energy is energy that a substance
possesses because of the state of the molecular
activity in the substance.
 Internal energy is usually expressed as a specific
quantity—that is, internal energy per unit mass.
 The internal energy 𝑢 is generally a function of
temperature and pressure.
 For an ideal gas, it is a function of temperature
alone.
Properties Involving Thermal
Energy
Viscosity
Viscosity is a measure of a fluid’s resistance to flow. It
determines the fluid strain rate that is generated by a
given applied shear stress. [Fluid resistance to shear
stresses]
It is easy to move through air, which has very low
viscosity, but movement is more difficult in water, which
has 50 times higher viscosity. Still more resistance is
found in SAE 30 oil, which is 300 times more viscous than
water.
Viscosity
Consider a fluid is placed between two parallel plates as shown. The
bottom plate is fixed, but the upper plate is free to move. Now a
constant force F is applied to the upper plate, it will move
continuously with a velocity, V, and the fluid in contact with the
bottom fixed plate has a zero velocity. The fluid between the two
plates moves with velocity that would be found to vary linearly as
illustrated. Thus, a velocity gradient, is
developed in the fluid between the
plates. In this particular case the velocity
gradient is a constant. The shear stress τ
acting on the upper fluid layer is
Viscosity
where A is the contact area
between the plate and the fluid.
The rate of deformation of a fluid element is equivalent to the
velocity gradient du/dy. Its found that the rate of deformation is
directly proportional to the shear stress τ,
OR
where μ is called the coefficient of viscosity or the dynamic (or
absolute) viscosity of the fluid, whose unit is kg/m · s, or N · s/m2 or
Pa . s
Viscosity
Temperature Dependency of
Viscosity
 The viscosity of liquids
decreases as the
temperature increases.
 The viscosity of gases
increases with increasing
temperature.
Newtonian and Non-Newtonian
Fluids
 Newtonian fluids: the
shear stress is directly
proportional to the rate of
strain.
 Non-Newtonian fluids:
the shear stress may not
be directly proportional to
the rate of strain.
Bulk Modulus of Elasticity or
Compressibility
The bulk modulus of elasticity, Ev, or (k)
is a property that relates changes in
pressure to changes in volume (e.g.,
expansion or contraction)
The bulk modulus of elasticity of water is
approximately 2.2 GPa which corresponds
to a 0.05% change in volume for a change
of 1 MPa in pressure.
Surface Tension
• Molecules of liquid below the surface act on each
other by forces that are equal in all directions.
• However, molecules near the surface have a greater
attraction for each other than they do for molecules
below the surface because of the presence of a
different substance above the surface.
• The attraction forces between molecules are called
cohesion forces
• Adhesive forces are
attractive forces between
molecules of different
materials. Ex- water
molecule and glass
Surface Tension
 Cohesion forces produces a layer of surface molecules
on the liquid that acts like a stretched membrane.
Surface Tension
 Surface tension is a material property whereby a liquid at
a material interface, usually liquid-gas, exerts a force per
unit length along the surface, this force is a surface
tension force due to cohesion and adhesion forces.
 The surface tension force acts in the plane of the surface,
and is given by:
where L is the length over which the surface tension acts.
 The surface tension force usually balance by external
forces like weight or pressure forces.
 Surface tension for a water–air surface is 0.073 N/m at
room temperature.
Surface Tension applications
1- capillary rise
 The effect of surface tension is
clear in the case of capillary
action (rise above or below a
static liquid level at atmospheric
pressure) in a small tube
 The relatively greater attraction of
the water molecules for the glass
rather than the water causes the
water surface to curve upward in
the region of the glass wall.
 The opposite occurs for the
mercury.
Surface Tension applications
1- capillary rise
The capillary rise h can be found by
force balance
𝑊 ↓ = 𝐹𝜎 ↑
𝑚𝑔 = 𝜎 × (2𝜋𝑅) × 𝑐𝑜𝑠∅
𝜌 × 𝜋𝑅2 ℎ × 𝑔 = 𝜎 × (2𝜋𝑅) × 𝑐𝑜𝑠∅
2𝜎 × 𝑐𝑜𝑠∅
ℎ=
𝜌×𝑔×𝑅
Surface Tension applications
2- spherical droplet
3- spherical bubble
2 2𝜋𝑟 𝜎 = 𝑝𝜋𝑟 2
𝑝=
4𝜎
𝑟
Surface Tension applications
4- cylinder
A cylinder supported by
surface-tension forces. The
liquid does not wet the cylinder
surface. The maximum weight
the surface tension can support
is:
where L is the length of the cylinder.
5- ring
A ring being pulled out of a
liquid. This is a technique to
measure surface tension. The
force due to surface tension
on the ring is
Vapor Pressure
 The pressure at which a liquid will vaporize, or
boil, at a given temperature, is called
vapor pressure.
 Water boils at temperature of 100 °C at
atmospheric pressure (101.3 kPa).
Vapor Pressure
 However, boiling can also
occur in water at temperatures
much below 100 °C if the
pressure in the water is
reduced to the vapor pressure
of water corresponding to that
lower temperature.
Examples
EXAMPLE 2.1 DENSITY OF AIR
Air at standard sea-level pressure (p 101 kN/ m2) has a temperature of 4°C.
What is the density of the air?
Solution
EXAMPLE 2.3 MODELING A BOARD
SLIDING ON A LIQUID LAYER
A board 1 m by 1 m that weighs 25 N slides
down an inclined ramp (slope 20°) with a
velocity of 2 cm/s. The board is separated from
the ramp by a thin film of oil with a viscosity of
0.05 N.s/m2. Neglecting edge effects, calculate
the space between the board and the ramp.
Solution
1- Free body diagram analysis.
For constant velocity
𝒂 = 𝟎 → 𝚺𝑭 = 𝟎 → 𝑭𝒕𝒂𝒏𝒈 = 𝑭𝒔𝒉𝒆𝒂𝒓
𝑾 𝒔𝒊𝒏𝟐𝟎° = 𝝉 𝑨 = 𝝁
𝒅𝑽
𝑨
𝒅𝒚
2- Assume linear velocity distribution
𝒅𝑽 ∆𝑽 𝑽𝟐 − 𝑽𝟏 𝟎. 𝟎𝟐 − 𝟎 𝟎. 𝟎𝟐
=
=
=
=
𝒅𝒚 ∆𝒚 𝒚𝟐 − 𝒚𝟏
𝒚𝟐 − 𝟎
𝒚𝟐
3- Sub values in above equation
𝟐𝟓𝑵 𝐬𝐢𝐧 𝟐𝟎° = 𝟎. 𝟎𝟓𝑵. 𝒔/𝒎𝟐 ×
𝟎. 𝟎𝟐𝒎/𝒔
× 𝟏𝒎𝟐
𝒚𝟐
4- Solve for 𝑦2
𝒚𝟐 = 𝟎. 𝟎𝟎𝟎𝟏𝟏𝟕 𝒎 = 𝟎. 𝟏𝟏𝟕 𝒎𝒎
EXAMPLE 2.4 CAPILLARY RISE IN A TUBE
To what height above the reservoir level will water (at
20°C) rise in a glass tube, as shown, if the inside
diameter of the tube is 1.6 mm?
Solution
1- Properties of water at 20 oC [Table A.5], σ = 0.073
N/m, γ = 9790 N/m3
2- Force balance: Weight of water (down) is balanced
by surface tension force (up).
𝑭𝝈 − 𝑾 = 𝟎
𝝅𝒅𝟐
𝝈𝝅𝒅𝒄𝒐𝒔𝜽 − 𝜸(∆𝒉) (
)=𝟎
𝟒
The contact angle for water against glass is so small, it can be assumed to be
0°; therefore 𝑐𝑜𝑠𝜃 ≅ 0
3- Solve the equation for ∆ℎ, and sub values
∆𝒉 =
𝟒𝝈
𝟒 × 𝟎. 𝟎𝟕𝟑 𝑵/𝒎
=
= 𝟎. 𝟎𝟏𝟖𝟔𝒎 = 𝟏𝟖. 𝟔𝒎𝒎
𝜸𝒅 𝟗𝟕𝟗𝟎 𝑵 × 𝟏. 𝟔 × 𝟏𝟎−𝟑 𝒎
𝒎𝟑
Problems
2.10
A 10 m3 oxygen tank is at 15°C and 800 kPa. The valve is opened, and some
oxygen is released until the pressure in the tank drops to 600 kPa. Calculate the
mass of oxygen that has been released from the tank if the temperature in the
tank does not change during the process.
Solution
2- Ideal gas law
1- RO2 = 260 J/kg·K [App Table A2]
3- Density and mass for case 1
5. Mass released from tank
4- Density and mass for case 2
2.37
Suppose that glycerin is flowing (T =
20°C) and that the pressure gradient
dp/dx is –1.6 kN/m3 What are the
velocity and shear stress at a distance of
12 mm from the wall if the space B
between the walls is 5.0 cm? What are
the shear stress and velocity at the wall?
The velocity distribution for viscous flow
between stationary plates is
Solution
2.45
A pressure of 2×106 N/m2 is applied to a mass of water that initially filled a 2000
cm3 volume. Estimate its volume after the pressure is applied.
Solution
2.51
A water bug is suspended on the surface of a pond by
surface tension (water does not wet the legs). The
bug has six legs, and each leg is in contact with the
water over a length of 5 mm. What is the maximum
mass (in grams) of the bug if it is to avoid sinking?
Solution
Practice these Problems
Q1: The pressure in an automobile tire depends on the
temperature of the air in the tire. When the air
temperature is 25°C, the pressure gage reads 210 kPa.
If the volume of the tire is 0.025 m3, determine the
pressure rise in the tire when the air temperature in the
tire rises to 50°C. Also, determine the amount of air
that must be bled off to restore pressure to its original
value at this temperature. Assume the atmospheric
pressure to be 100 kPa.
Q2: A cylinder falling inside a pipe that is filled with oil,
as depicted in the figure. The small space between the
cylinder and the pipe is lubricated with an oil film that
has viscosity μ. Derive a formula for the steady rate of
descent of a cylinder with weight W, diameter d, and
length l sliding inside a vertical smooth pipe that has
inside diameter D. Assume that the cylinder is
concentric with the pipe as it falls. Use the general
formula to find the rate of descent of a cylinder 100 mm
in diameter that slides inside a 100.5 mm pipe. The
cylinder is 200 mm long and weighs 15 N. The lubricant
is SAE 20W oil at 10°C.
Q3: Calculate the pressure increase that must be applied to water to reduce its
volume by 2%.
Q4: The surface tension of a liquid is to be measured using a liquid film
suspended on a U-shaped wire frame with an 8-cm-long movable side. If the
force needed to move the wire is 0.012 N, determine the surface tension of this
liquid in air.