Example:
A jet of water is deflected by a vane mounted on a cart. The water jet has an area, A, everywhere and is
turned by an angle with respect to the horizontal. The pressure everywhere within the jet is atmospheric.
The incoming jet velocity with resp
Example:
A box with a hole of area, A, moves to the right with velocity, ubox, through an incompressible fluid as
shown in the figure. If the fluid has a velocity of ufluid which is at an angle, , to the vertical, determine
how long it will take to fill t
Example:
A spherical balloon is filled through an area, A1, with air flowing at velocity, V1, and constant density, 1.
The radius of the balloon, R(t), can change with time, t. The average density within the balloon at any
given time is b(t). Determine th
We could have also chosen a fixed control volume through which the free surface moves. Using this type
of control volume, conservation of mass is given by:
d
dV + u rel dA = 0
(3.39)
dt
where
d
dt
CV
CS
dV = 0
(the mass of fluid in the fixed control vol
Example:
Water enters a cylindrical tank with diameter, D, through two pipes at volumetric flow rates of Q1 and Q2
and leaves through a pipe with area, A3, with an average velocity, V 3 . The level in the tank, h, does not
remain constant. Determine the t
Example:
Water enters a cylindrical tank through two pipes at volumetric flow rates of Q1 and Q2. If the level in the
tank remains constant, calculate the average velocity of the flow leaving the tank through a pipe with an
area, A3.
Q1
Q2
h=constant
V3 =
Example
An incompressible flow in a pipe has a velocity profile given by:
r2
u (r ) = u c 1 2
R
where uc is the centerline velocity and R is the pipe radius. Determine the average velocity in the pipe.
r
u
R
SOLUTION:
The volumetric flow rate using th
Example:
Consider the flow of an incompressible fluid between two parallel plates separated by a distance 2H. If the
velocity profile is given by:
y2
u = u c 1 2
H
where uc is the centerline velocity, determine the average velocity of the flow, u . As
Example:
Calculate the mass flux through the control surface shown below. Assume a unit depth into the page.
D
V
control surface
SOLUTION:
dA = Rd
The mass flux through the surface is given by:
m=
= 2
u rel dA =
)
V cos + sin ( Rd )
i
i
j
= u rel
= 2
3.
Conservation of Mass (COM)
In words and in mathematical terms, COM for a system is:
D
The mass of a system remains constant.
dV = 0
(3.24)
Dt
Vsystem
mass of the system
where D/Dt is the Lagrangian derivative (implying that were using the rate of c
Example:
The market price, P (in dollars), of used cars of a certain model is found to be:
P = $1000 + ( $0.02 / mile ) x ( $2 / day ) t
where x is the distance in miles west of Detroit, MI and t is the time in days. If a car of this model is driven
from
Example:
A fluid velocity field is given by:
u = 2te x
Will a fluid particle accelerate in this flow? Why?
SOLUTION:
The acceleration is given by:
Du u
u
u
u
a=
=
+ ux
+ uy
+ uz
Dt
t
x
y
z
= 2e x
2t
=0
=0
=0
Hence, for the given flow:
a = 2e x Yes, fluid
Example:
Determine the rate at which fluid mass collects inside the room shown below in terms of , V1, A1, V2, A2,
Vc, R, and . Assume the fluid moving through the system is incompressible.
2R
V1
room
r2
V = Vc 1 2
R
A1
A2
V2
SOLUTION:
Apply conserva
Now look at the Ydirection:
d
uY dV + uY ( u rel dA ) = FB ,Y + FS ,Y
dt
where
d
dt
CV
CS
u
Y dV
(3.59)
= 0 (the momentum within the control volume doesnt change with time)
CV
=u rel
=A
uY ( u rel dA ) = Vjet sin Vjet cos + sin A cos + sin
i
j
i
j
CS
=u
Problem 13.43:
Problem 13.44:
Problem 13.49:
Problem 13.49: (cont)
Problem 13.52:
Problem 12.1(a):
Problem 12.2(b):
Problem 12.3:

Problem 5.18:
Problem 12.4(a):
Problem 12.5:
Problem 12.6:
Problem H.15:
Problem 10.4:
Problem 10.5:
Problem 10.6:
Problem 10.7:
Problem 10.7  Alternate Method:
Problem 10.14:
Problem 10.14: (cont)

Alternate Graphical Technique:
Problem 10.15:
Problem 10.15: (cont)
Alternate Graphical Technique:
Problem 10.16:
Problem 10.16
Problem C.17:
Problem C.23:
Problem C.23  Alternate Method:
Problem C.28:
Problem C.34:
Problem C.34  Alternate Method:
Problem C.25:
Problem C.25  Alternate Method:
Problem C.26:
Problem C.26: (cont)
Problem C.29(b):
Problem 10.9:
Example:
A fluid velocity field is given by:
u = (cy 2 ) + (cx 2 )
i
j
where c is a constant. Determine
a. the components of the acceleration and
b. the points in the flow field where the acceleration is zero.
SOLUTION:
The acceleration of a fluid elemen
Notes:
1.
There is no known closedform solution to this ODE so we resort to solving it numerically (using a
RungeKutta method for example). A plot of the solution looks like (plot from Panton, R.L.,
Incompressible Flow, 2nd ed., Wiley):
u
U
y
U
x
The an
2.
Reynolds Transport Theorem (RTT)
Recall that we can look at the behavior of small pieces of fluid in two ways: the Eulerian perspective or the
Lagrangian perspective. Often were interested in the behavior of an entire system of fluid (many pieces of
fl
DT T
T
T
T
=
+ ux
+ uy
+ uz
Dt
t
x
y
z
(3.4)
T
=
+ (u )T
t
The notation, D/Dt, indicating a Lagrangian (also sometimes referred to as the material or substantial)
derivative, has been used in Eqn. (3.4) to indicate that were following a particular piece o