1. Dr. Yaser H. Alahmadi
Mechanical Engineering Department
ME 3563
Fluid Mechanics
2. Recommended Books
Textbook
Fundamentals of Fluid Mechanics.
by Munson, B. R., D. F. Young, and T. H. Okiishi
Reference Books
Fluid Mechanics
by Frank M. White.
Introduction to Fluid Mechanics
by Robert W. Fox & Alan T. McDonald
Mechanics of Fluids
by Merle C. Potter & David C. Wiggert
Engineering to Fluid Mechanics
by Clayton T. Crowe, Donald F. Elger, Barbara C. Williams and John A. Robe
3. Course Outline
1. Introduction and fundamentals
2. Fluid statics (Hydrostatic)
3. Fluid Dynamics
4. Fluid kinematics
• Midterm
5. Fluid kinematics
6. Control volume analysis
7. Differential analysis
8. Viscose flow in pipes
• Final Exam Comprehensive
6. What is Fluid
Fluid:
a substance that deforms continuously under the application of a shear (tangential) stress of any
size.
What is stress?
It is the force per unit area.
What is shear stress?
It is stress parallel to the plane of the area.
Why of any size?
Mathematically we can say even if infinitesimal small shear stress acting on that surface, the
fluid will continuously deforms.
fluid
𝐹
𝑆𝑡𝑟𝑒𝑠𝑠 =
𝐹𝑜𝑟𝑐𝑒
𝐴𝑟𝑒𝑎
7. Fluids
A solid deforms when a shear stress is applied, but it does not deform continuously.
In the case of solid material, the deformation is proportional to the applied shear stress
𝜏 = 𝐹/𝐴, where A is the area of the contact surface to the plate.
In case of fluid, there is no slip at the boundary because the fluid in the direct contact
with the solid boundary has the same velocity as the boundary itself.
Solid behavior Fluid behavior F
𝑡0
𝑡1
𝑡2
F
𝑡0 𝑡𝑓
8. Why study fluid mechanics?
Understanding the basic principle and concept of fluid mechanics are essential to
analyze any system in which the fluid is the main working medium.
Examples:
1. Aircraft industries
2. Automobile industries
3. Bridge and skyscrapers design and building.
4. The design of propulsion system for space flight.
9. Why study fluid mechanics?
Ignoring the basic principles of the fluid mechanics can lead to a catastrophic failure,
the collapse of Tacoma Narrows Bridge in 1940 is just one example.
10. Basic equations to solve F.M problems
To solve a fluid mechanics problems, the fluid must satisfy basic law of physics and
mathematics.
1. Conservation of mass.
2. Newton’s 2nd law “ conservation of momentum”
3. 1st law of thermodynamics “ conservation of energy”
4. 2nd law of thermodynamics
11. Dimensions, Dimensional Homogeneity, and Units
Fluid characteristics can be described qualitatively and quantitively:
• The qualitative aspect identifies the nature, or type, of the characteristics (such as length, time,
stress, and velocity).
• The quantitative aspect provides a numerical measure of the characteristics.
• The qualitative description is conveniently given in terms of certain primary quantities, such as
length, L, time, T, mass, M, and temperature, Φ.
• These primary quantities can be used to provide a qualitative description of any other secondary
quantity: for example, area = 𝐿2, velocity = 𝐿𝑇−1 , density = 𝑀𝐿−3 and so on,
• The symbol is used to indicate the dimensions of the secondary quantity in terms of the primary
quantities. Thus, to describe qualitatively a velocity, V, we would write as:
V = 𝐿𝑇−1
• Thus, we can say that “the dimensions of a velocity equal length divided by time.”
12. Dimensions, Dimensional Homogeneity, and Units
All theoretically derived equations are dimensionally homogeneous—that is, the dimensions
of the left side of the equation must be the same as those on the right side, and all additive
separate terms must have the same dimensions. We accept as a fundamental premise that
all equations describing physical phenomena must be dimensionally homogeneous. If this
were not true, we would be attempting to equate or add unlike physical quantities, which
would not make sense. For example, the equation for the velocity, V, of a uniformly
accelerated body is:
V = 𝑉0 + 𝑎𝑡
𝐿𝑇−1 = 𝐿𝑇−1 + 𝐿𝑇−2𝑇
14. Fluid Properties
Density 𝝆 =
𝒎
𝑽
𝐤𝐠
𝐦𝟑
Specific volume 𝝂 =
𝟏
𝝆
𝐦𝟑
𝐤𝐠
Specific weight 𝜸 = 𝝆𝒈 =
𝒘𝒆𝒊𝒈𝒉𝒕
𝒗𝒐𝒍𝒖𝒎𝒆
𝑵
𝐦𝟑
Specific gravity 𝑺𝑮 =
𝝆
𝝆𝑯𝟐𝑶
15. Density
From equilibrium thermodynamics
𝜌 = 𝑓(𝑇, 𝑝)
The density is a function of temperature and pressure under two conditions:
1. They have to be intensive (intensive mean the property does not depend on the system size or the amount of material in the system)
2. They have to be independent (they cannot be related to each other in any way).
The pressure and the temperature can related in special cases i.e. mixture of liquid and vapor (dealing with mixture and multiphase fluid is
beyond the scope of this course). Therefore, p & T are intensive.
At normal condition (1atm, 300K)
𝜌𝐻2𝑂 = 1,000 𝑘𝑔/𝑚3
𝜌𝑎𝑖𝑟 = 1.220 𝑘𝑔/𝑚3
For ideal gas:
𝜌 =
𝑝
𝑅𝑇
16. Viscosity
The most important fluid property in fluid mechanics is the VISCOSITY.
Why???
The viscosity is the fluid property that has a significant effect on the deformation rate occur due
to the applied force.
The viscosity of a fluid is a measure of its resistance to deformation at a given shear strain rate.
17. No-Slip condition
No-slip condition defined as: The fluid on contact with a surface will have the same velocity of
that surface, and that is due to the viscos effects.
18. No-Slip condition
If a force is applied on the top wall for a very short period of time (𝑑𝑡), then the fluid attached to
the top surface will deforms and displace with amount of (𝑑𝑥), Thus:
𝑑𝑥 = 𝑈𝑑𝑡
tan(𝑑𝛽) =
𝑑𝑥
ℎ
≈ 𝑑𝛽 since 𝑑𝛽 ≪ 1
𝒉
𝑑𝑥
𝑑𝛽
𝑭, 𝑼
𝑦
𝑥
19. No-Slip condition
• The angular velocity (the rate of change of angle) is given by:
𝑑𝛽
𝑑𝑡
=
𝑑𝑥
ℎ𝑑𝑡
=
𝑈
ℎ
= 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 𝑟𝑎𝑡𝑒
• In words: when I shear a fluid, it will flow.
• The shear strain rate is nothing but the angle variation in time:
𝑑𝛽
𝑑𝑡
=
𝑈
ℎ
• The unit of shear strain rate:
𝑈
ℎ
=
𝑚/𝑠
𝑚
=
1
𝑠
20. No-Slip condition
Note that:
𝑢
ℎ
~
𝑑𝑢
𝑑𝑦
is actually represent the slope of the velocity profile.
If we consider a tiny small element of a fluid, then the SSR is the difference of the velocity
between the top and bottom surface divided by the vertical distance between these surfaces.
SSR =
du
dy
𝒅𝒚
𝑑𝑥
𝑑𝛽
𝑭, 𝑼
𝑦
𝑥
21. Connection between SSR and viscosity
The shear stress is given by:
𝜏 =
𝐹
𝐴
𝑁
𝑚2
The shear strain rate given by:
SSR =
𝑑𝑢
𝑑𝑦
If the SS is linear proportional to SSR, then the fluid is Newtonian:
𝜏 ∝
𝑑𝑢
𝑑𝑦
What is the constant of proportionality between SS and SSR?
The viscosity (𝜇) is the constant of proportionality between SS and SSR, thus:
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
Unit of viscosity:
𝜇 =
𝜏
𝑑𝑢
𝑑𝑦
=
N m2
1 s
=
N∙s
m2 =
kg
m∙s
A
𝑦 𝑢
𝐹
