Lecture (1&2)
Fluid Mechanics Applications(MPE303)
Institute of Aviation Engineering and Technology
Mechanical Power Department
Course Instructor
Dr. Ahmed El-Degwy
Email: mpdfoecu_ahmed@yahoo.com
Course Outline
1. General partial differential equations for fluid flow-mass conservation equations
(continuity).
2. Linear and angular momentum conservation equations for incompressible and
compressible flow.
3. Incompressible viscous flow and applications on lubrication.
4. Incompressible inviscid flow and its applications on ideal flow around submerged
bodies.
5. Theory of flow around wings.
6. Boundary layer viscous flow and its applications without ant with heat transfer
across the wall.
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Course Grades
The grades are distributed as:
Assignments
Quizzes & Sections attendance
Mid-term Exam
Final Exam
(20)
(20)
(20)
(90)
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Textbooks:
1- fundamentals of fluid mechanics, 7th eduction, okiishi.
2- Fluid Mechanics, 4th eduction, Frank Whit.
3- Introduction to Fluid Mechanics, 7th eduction, Fox and McDonal’s.
4- Boundary Layer Theroy, Hermann Schlichting (Deceased) Klaus
Gersten, Ninth Edition.
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Chapter (1)
Differential Relations
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Objectives
• Understand how the differential equation of conservation of mass and the
differential linear momentum equation are derived and applied.
• Obtain analytical solutions of the equations of motion for simple flow fields.
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Contents of the Chapter:
Introduction
Acceleration Field
Differential equation conservation of mass
Stream function
Differential equation of linear momentum
Inviscid flow
Vorticity and Irrotationality
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1- Introduction
Fluid mechanics: is that discipline within the broad field of applied mechanics
concerned with the behavior of liquids and gases at rest or in motion.
• Fluids essential to life
• Human body 65% water
• Earth’s surface is 2/3 water
• Atmosphere extends 17km above the earth’s surface
• History shaped by fluid mechanics
• Human migration and civilization
• Modern scientific and mathematical theories and methods
• Warfare
• Affects every part of our lives
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1- Introduction
Fluid Mechanics
Gas
Liquids
Statics
F  0
i
Air, He, Ar,
N2, etc.
Compressibility Density
Water, Oils,
Alcohols,
etc.
Viscosity
Dynamics
 F  0 , Flows
Stability
Pressure Buoyancy
i
Compressible/
Incompressible
Surface
Laminar/
Tension
Turbulent
Steady/Unsteady
Vapor
Viscous/Inviscid
Pressure
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History
Faces of Fluid Mechanics
Archimedes
Newton
( 287-212 B.C)
(1642-1727)
Leibniz
Bernoulli
Euler
(1646-1716)
(1667-1748)
(1707-1783)
Taylor
Navier
Stokes
Prandtl
(1785-1836)
Reynolds
(1819-1903)
(1842-1912)
(1875-1953)
(1886-1975)
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Introduction
What is the difference between Integral and Differential analyses:
Differentiation and Integration are the two major concepts of calculus.
Differentiation is used to study the small change of a quantity with respect to unit
change of another. On the other hand, integration is used to add small and
discrete data, which cannot be added singularly and representing in a single
value.
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INTRODUCTION
The control volume technique is useful when
we are interested in the overall features of a
flow, such as mass flow rate into and out of the
control volume or net forces applied to bodies.
Differential analysis, on the other hand,
involves application of differential equations of
fluid motion to any and every point in the flow
field over a region called the flow domain.
Boundary conditions for the variables must
be specified at all boundaries of the flow
domain, including inlets, outlets, and walls.
If the flow is unsteady, we must march our
solution along in time as the flow field changes.
(a) In control volume analysis, the interior of
the control volume is treated like a black box,
but (b) in differential analysis, all the details of
the flow are solved at every point within the
flow domain.
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Assignment (1):
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CONSERVATION OF MASS—THE
CONTINUITY EQUATION
The net rate of change of mass within the
control volume is equal to the rate at
which mass flows into the control volume
minus the rate at which mass flows out of
the control volume.
To derive a differential
conservation equation, we
imagine shrinking a control
volume to infinitesimal size.
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Derivation Using the Divergence Theorem
The quickest and most straightforward way to derive the differential form of
conservation of mass is to apply the divergence theorem (Gauss’s theorem).
This equation is the compressible form of the continuity equation since we have not
assumed incompressible flow. It is valid at any point in the flow domain.
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Derivation Using an
Infinitesimal Control Volume
At locations away from the center of the box,
we use a Taylor series expansion about the
center of the box.
A
small
box-shaped
control volume centered at
point P is used for
derivation
of
the
differential equation for
conservation of mass in
Cartesian coordinates; the
blue dots indicate the
center of each face.
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The mass flow rate through
a surface is equal to VnA.
The inflow or outflow of mass
through each face of the differential
control volume; the red dots
indicate the center of each face.
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The divergence
operation in Cartesian
and cylindrical
coordinates.
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There is also a physical interpretation which relates to volume flow. From Fig. 4.8, we can
compute the volume flow dQ through an element ds of control surface of unit depth
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Thus the change in ψ across the element is numerically equal to the volume flow
through the element. The volume flow between any two points in the flow field is
equal to the change in stream function between those points:
Further, the direction of the flow can be ascertained by noting whether increases or
decreases. As sketched in Fig. 4.9, the flow is to the right if ψ upper is greater than ψ lower,
where the subscripts stand for upper and lower, as before; otherwise the flow is to the left.
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The Bernoulli Equation
For inviscid, incompressible fluids (commonly called ideal fluids), steady flow, and
flow along streamline the Bernoulli equation can be write:
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Continuity and Navier–Stokes Equations
in Cartesian Coordinates
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Continuity and Navier–Stokes Equations
in Cylindrical Coordinates
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