CHAPTER 3. Elementary Fluid Dynamics
- Understanding the physics of fluid in motion
- Derivation of the Bernoulli equation from Newton’s second law
 Basic Assumptions of fluid stream, unless a specific comment
1st assumption: Inviscid fluid (Zero viscosity = Zero shearing stress)
 No force by wall of container and boundary
 Applied force = Only Gravity + Pressure force
 Newton’s Second Law of Motion of a Fluid Particle
F  (Net pressure force) + (Gravity)

= ma  (Fluid mass)  (Acceleration)

2nd assumption:
Steady flow (?)
 No Change of flowing feature with time at a given location
 Every successive particle passing though the same point
: Same path (called streamline) &
Same velocity (tangential to the streamline)
 Additional Basic Terms in Analysis of Fluid Motion
 Streamline (Path of a fluid particle)
- Position of a particle

= f ( ro , v )
where ro : Initial position,

v : Velocity of particle
- No streamlines intersecting each other
 Two Components in Streamline Coordinates (See the figure)
1. Tangential coordinate:
s  s(t )
: Moving distance along streamline,
: Related to Particle’s speed (v  ds / dt )
n  n(t )
2. Normal coordinate:
: Local radius of curvature of streamline R  R(s)
: Related to Shape of the streamline
 Two Accelerations of a fluid particle along s and n coordinates
1. Streamwise acceleration ( Change of the speed)
as 
dv v ds v

 v
dt s dt s
2. Normal acceleration
v2
an 
R
using the Chain rule
(
Change of the direction)
(: Centrifugal acceleration)
Q. What generate these as and an?
(Pressure force and Gravity)
Part 1. Newton’s second law along a streamline ( ŝ direction)
Consider a small fluid particle of size s  n  y as shown
: Forces acting on
a fluid particle
sny
Newton’s second law in ŝ direction
v
v
 Vv
along ŝ direction
s
s
= Gravity force + Net Pressure force
 Fs  ma s  mv
where V : Volume of a fluid particle = s  n  y
(i) Gravity force
along ŝ direction
Ws  W sin   (V ) sin 
(ii) Pressure force
along ŝ direction
Let p: Pressure at the center of V
p s
: Average pressures at Left face (Decrease)
s 2
p s
p
: Average pressures at Right face (Increase)
s 2
p
Then, Net pressure force along ŝ direction, Fps  (Pressure)(Area)
Fps  ( p 
p s
p s
p
)ny  ( p 
)ny   sny
s 2
s 2
s
: Depends not on p itself, but on

p
(Rate of change in p)
s
 Total force in ŝ direction (Streamline)
 Fs  Ws  Fps

Vas  Vv
v
p
 ( sin   )V
s
s
Finally, Newton’s second law along a streamline ( ŝ direction)
 as  v
p
V
s
v
p
  sin  
s
s
Change of Particle’s speed
 Affected by Weight and Pressure Change
 Making this equation more familiar
v
p
v   sin  
s
s
because
or
or


dp
ds
dz
(See the figure above)
ds
p dp
p
p
p

using dp  ds  dn  ds (why?)
s ds
s
n
s
2
v
dv 1 dv
v v
=
s
ds 2 ds
sin  
1
dp  d (v 2 )  dz  0
2

dz
1 d (v 2 )
=

ds
2
ds
dp
1
 v 2  gz  constant
 2
(Divided by ds)
(By integration)
By assuming a constant  (Incompressible fluid):
1
 p  v 2  z  Constant
2
3rd assumption
along streamline ( ŝ direction)
: Bernoulli equation along a streamline
 Valid for (1) a steady flow of (2) incompressible fluid
(3) without shearing stress
c.f. If  is not constant (Compressible, e.g. Gases),
dp
    ( p) : Must be known to integrate  .

 What this Bernoulli Equation means? (Physical Interpretation)
For a Steady flow of Inviscid and Incompressible fluid,
1
p  v 2  z  Constant
along streamline
2
: Mathematical statements of Work-energy principle
 Unit of Eq. (1):
(1)
[N/m2] = [Nm/m3] = [Energy per unit volume]
p
= Works on unit fluid volume done by pressure
z
= Works on unit fluid volume done by weight
1 2
v  Kinetic energy per unit fluid volume
2
 Same Bernoulli Equations in different units
1. Eq (1)  

[Nm/m3]  [Nm/m3] = [m] = [Length unit]
v2
+
+ z = Constant

2g
p
(Head unit)
p
: Depth of a fluid column produce p
(Pressure head)

v2
: Height of a fluid particle to reach v from rest by free falling
2g
(Velocity head)
z: Height corresponding to Gravitational potential (Elevation head)
Part 2. Newton’s second law normal to a streamline ( n̂ direction)
Consider the same situation as Sec. 3.3 shown in figure
For a small fluid particle of size s  n  y as shown
: Forces acting on
a fluid particle
sny
Newton’s second law in n̂ direction
v2
v2
 Fn  man  m  V
R
R
= Gravity force + Net Pressure force
along n̂ direction
(i) Gravity force
along n̂ direction
Wn  W cos   (V ) cos 
(ii) Pressure force
along n̂ direction
By the same manner in the previous case,
Fpn  ( p 
p n
p n
)sy  ( p 
)sy
n 2
n 2

p
nsy
n

p
V
n
 Total force in n̂ direction (Normal to Streamline)
 Fn  Wn  Fpn
p
v2
 Van  V  ( cos   )V
n
R
v2
p

normal to streamline ( n̂ direction)
  cos  
R
n
Change of Particle’s direction of motion
 Affected by Weight and Pressure Change along n̂
Ex. If a fluid flow:
Steep direction change (R ) or fast flow (v )
or heavy (  ) fluid

Generate large force unbalance
 Special case: Standing close to a Tornado
i.e. Gas flow (Negligible 
) in horizontal motion (
dz p
v2



dn n
R
p
v2
   0
n
R

dz
= 0)
dn
(Attractive)
: Moving closer (R )

More dangerous (
p
)
n
 Making this equation more familiar
By the same manner as the previous case,
v2
p
   cos  
R
n
because cos  

v2
dz dp
  

R
dn dn
dz
(See the figure)
dn
p
p
p
dp
p
=
, since dp  ds  dn  dn
n
dn
s
n
n
or

v2

dn  gz = Constant
 R
dp
(normal to streamline)
By assuming a constant  (Incompressible fluid):
v2
 p    dn  z = Constant
R
3rd assumption
(normal to streamline)
: Bernoulli equation normal to streamline ( n̂ direction)
 Valid for (1) a steady flow of (2) incompressible fluid
(3) without shearing stress