CE 150
Fluid Mechanics
G.A. Kallio
Dept. of Mechanical Engineering,
Mechatronic Engineering &
Manufacturing Technology
California State University, Chico
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Fluid Kinematics
Reading: Munson, et al.,
Chapter 4
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Introduction
• In this chapter we consider fluid
kinematics, which addresses the
behavior of fluids while they are
flowing without concern of the actual
forces necessary to produce the
motion
• Specifically, we will address
– fluid velocity
– fluid acceleration
– flow pattern description and
visualization
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Fluid Models
• Continuum model: fluids are a
collection of fluid particles that
interact with each other and
surroundings; each particle contains
a sufficient number of molecules
such that fluid properties (e.g.,
velocity) can be defined.
• Molecular model: the motions of
individual fluid molecules are
accounted for; not a practical model
unless fluid density is very small or
flow over very small objects are
considered.
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Flow Descriptions
• Lagrangian description: properties
of individual fluid particles are
defined as a function of time as they
move through the fluid; the overall
fluid motion is found by solving the
EOMs for all fluid particles.
• Eulerian description: properties are
defined at fixed points in space as
the fluid flows past these points; this
is the most common description and
yields the field representation of
fluid flow.
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The Velocity Field
• Consider an array of sensors that can
simultaneously measure the
magnitude and direction of fluid
velocity at many fixed points within
the flow as a function of time; in the
limit of measuring velocity at all
points within the flow, we would
have sufficient information to define
the velocity vector field:

V  u( x, y , z, t )ˆi  v ( x, y , z, t )ˆj 
w( x, y , z, t )kˆ
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The Velocity Field
• u, v, and w are the x, y, and z
components of the velocity vector
• The magnitude of the velocity, or
speed, is denoted by V as

V  V  u 2  v 2  w2
• Velocity field may be one- (u), two(u,v)or three- (u,v,w) dimensional
• Steady vs. unsteady flows:

V
0
t

V
0
t
(steady)
(unsteady)
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Visualization of
Fluid Flow
• Three basic types of lines used to
illustrate fluid flow patterns:
– Streamline: a line that is everywhere
tangent to the local velocity vector at a
given instant.
– Pathline: a line that represents the
actual path traversed by a single fluid
particle.
– Streakline: a line that represents the
locus of fluid particles at a given instant
that have earlier passed through a
prescribed point.
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Streamlines
• Streamlines are useful in fluid flow
analysis, but are difficult to observe
experimentally for unsteady flows
• For 2-D flows, the streamline
equation can be determined by
integrating the slope equation:
dy v
 ,
dx u
where u  u ( x, y , t )
v  v ( x, y , t )
– The resulting equation is normally
written in terms of the stream function:
 (x,y) = constant
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Pathlines & Streaklines
• The pathline is a Lagrangian concept
that can be visualized in the
laboratory by “marking” a fluid
particle and taking a time exposure
photograph of its trajectory
• The streakline can be visualized in
the laboratory by continuously
marking all fluid particles passing
through a fixed point and taking an
instantaneous photograph
• Streamlines, pathlines, and streaklines are identical for steady flows
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Acceleration Field
• Acceleration is the time rate of
change of velocity:

 dV du ˆ dv ˆ dw ˆ
a

i  j
k
dt
dt
dt
dt
• Using the Eulerian description, we
note that the total derivative of each
velocity component will consist of
four terms, e.g.,
du u
u
u
u

u v w
dt t
x
y
z
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Acceleration Field
• Collecting derivative terms from all
velocity components,





DV
V
V
V
V

u
v
w
Dt
t
x
y
z



V

 V  V
t


D( )
Dt
– The operator
is termed the
material, or substantial, derivative; it
represents the rate at which a variable
(V in this case) changes with time for a
given fluid particle moving through the
flow field
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Acceleration Field

• The term V
is called the local
t
acceleration; it represents the
unsteadiness of the fluid velocity and
is zero for steady flows.

V
x

V
y

V
z
• The terms u , v , w
are
called convective accelerations; they
represent the fact that the velocity of
the fluid particle may vary due to the
motion of the particle from one point
in space to another; it can occur for
both steady and unsteady flows.
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The Control Volume
• A control volume is a volume in space
through which fluid may flow; in some
cases, the volume may move or
deform
• The control volume has a boundary
which separates it from the surroundings and defines a control surface
• In the study of fluid dynamics, the
control volume approach is used to
analyze fluid flow and fluid machinery
• The control volume approach is
consistent with the Eulerian
description
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The Reynolds
Transport Theorem
• The basic laws governing the motion
of a fluid (e.g., conservation of mass,
momentum, and energy) are usually
written in terms of a fixed quantity
of mass, or system*
• Because a control volume does not
always have constant mass, the basic
laws must be rephrased
• The Reynolds Transport Theorem is
a tool that allows one to shift from a
system viewpoint (fixed mass) to a
control volume viewpoint
* In thermodynamics, a system is defined more
generally as a fixed mass or control volume
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The Reynolds
Transport Theorem
• Let B = any fluid parameter, such as
mass, velocity, temperature,
momentum, etc.
• Let b = B/m, a fluid parameter per
unit mass
• The mass m may be that contained in
a system or a control volume
Bsys   bdV
sys
Bcv   bdV
cv
Bsys  Bcv
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The Reynolds
Transport Theorem
• Example 4.7 (B = m, b = 1)
dBsys
dt

dmsys
dt
d   dV 
sys

 0

dt
dBcv dmcv d


dt
dt

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cv
d V
dt
 0
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The Reynolds
Transport Theorem
Bsys Bcv (t  t )  Bcv (t )


t
t
BI (t  t ) BII (t  t )

t
t
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The Reynolds
Transport Theorem
• Reynolds Transport Theorem (RTT)
for fixed control volume with one
inlet, one exit and uniform properties:
Bcv 

 Bout  Bin
Dt
t
Bcv

  2 A2V2b2  1 A1V1b1
t
DBsys
– LHS term is Lagrangian
– RHS terms are Eulerian
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The Reynolds
Transport Theorem
• A general control volume may have
multiple inlets and outlets, threedimensional flow, and nonuniform
properties; the general form of the
RTT is:
DBsys
Dt




  bdV   b V  nˆ dA
cs
t cv
– for a control volume moving at
constant velocity Vcv, replace V
by V-Vcv
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Physical Interpretation
• The RTT allows one to translate the
time rate of change of some
parameter B of the system in terms of
the time rate of change of B of the
control volume and the net flow rate
of B across the control surface
• A material derivative is used because
the translation consists of an
unsteady term ( )/t and convective
effects associated with the flow of
the system across the control surface
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Steady Flow
• For steady flow,



DBsys
Bcv
 0, 
  b V  nˆ dA
cs
t
Dt
• For B = m (mass), the LHS is zero
since the mass of a system is
constant
• For B = V (velocity), the LHS is
nonzero in general
• For B = T (temperature), the LHS is
also nonzero in general
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