Assignment # 6 (45 pts)
Name: __________________________
1. (8 pts) Starting with Newton’s second law, F  ma , derive the Navier-Stokes equation of
motion for a non-rotating reference frame, 
^
Du
 p  g k . Show all steps, and state all
Dt
assumptions.
2. (5 pts) Write out all the terms of each of the component equations for u, v, and w. Do not use
the material derivative notation (expand it into the local and advective components. Also expand
advection into 3 terms).
3. (5 pts) When accelerations are observed to be zero, describe what that means for the resulting
forces and for fluid motion.

 0 ) with constant density (   o ) that neglects
t
friction and the rotation of the Earth, that the Navier-Stokes equation of motion takes the form:
4. (7 pts) Show for a steady state system (
 u   u  
where g  9.8
1
o
p  gkˆ
(1)
m
is a constant.
sec 2
5. (8 pts) If the advective term in (1) can be expanded into equation (2) as follows:
 u    u  u   
1
 u  u 
2
(2)
where    u , show that equation (1) can be re-written in the following way:
1

p
 u u 
 gz   u  
o
2

(3)
1
p
u u 
 gz  B , where B is a constant, is true
2
o
along a streamline of the flow field for an incompressible, steady-state fluid. This is Bernoulli’s
Equation.
6. (7 pts) Take u {equation(3)} to show that
7. (5 pts) If we neglect the effects of gravity in Bernoulli’s equation, we obtain that the quantity
1
p
u u 
must be constant along a streamline. Recall from Lab 5 the example of a constricting
2
o
pipe with an incompressible fluid flowing through it. Apply this relationship to Bernoulli’s
equation to discuss the dangers of two ships being too close together and colliding during
underway replenishment (i.e., explain what would cause them to collide). Assume the two
velocity components u1 and u2 are along the same streamline.
Ship 1
Ship 2