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```Faculty of Engineering and Applied Science
Fluid Mechanics II – Engineering 5913
Winter 2002
Assignment # 1 (Navier Stokes Equations)
1) A velocity field is described by the vector

 
 

yz
2 xz
xy
 i   2
 j   2
k
V   2
2
2 
2
2 
2
2 
x y z  x y z  x y z 

Does the field satisfy the equation of continuity for an incompressible flow? If so,
compute the pressure gradient assuming frictionless flow with negligible body forces.
2) Solve the problem of gravity driven flow down an inclined slope assuming a free
surface. The governing equation we derived in class may be assumed:
d 2u
 2  g sin(  )  0
dy
The boundary conditions are
y  0, u  0
du
y  b,
0
dy
Once you have determined the solution to the velocity distribution, obtain an expression
for the mass flow rate and find the average velocity.
3) Solve for fully developed pressure driven flow in a two dimensional channel
containing two immiscible fluids. The channel has a width of 2h, and each layer has a
width of h. The following equation from the class notes may be assumed to govern the
flow of each fluid:
d 2ui 1 dp

dy 2  dx
where i=1,2. Place your co-ordinate axes at the interface of the two fluids. At the fluid
interface the following boundary conditions are specified:
y  0, u1  u2
du
du
y  0, 1 1   2 2
dy
dy
At the channel walls the following no-slip conditions are required:
y  h, u 2  0
y  h, u1  0
Show that the average velocity in each layer is:
dp h 2  7 1   2 


u1  
dx 121  1   2 
dp h 2  1  7  2 


u2  
dx 12 2  1   2 
4) Find the solution for fully developed flow in the annular space between concentric
pipes. The inner pipe has a diameter of 2a, while the outer pipe has a diameter of 2b. The
governing equation in polar co-ordinates is:
d 2u 1 du 1 dp


dr 2 r dr  dx
The governing equation is subject to the no-slip condition at r=a, and r=b. Hint: the
equation may be integrated if the left hand side is written as:
d 2u 1 du 1 d  du 


r 
dr 2 r dr r dr  dr 
Find the volumetric flow Q, and the radius where the velocity is maximum. What is the
maximum velocity?
5) Problem 6.81 from the text.
6) Problem 6.84 from the text.
Assignment is to be passed in by 4:00 PM February 14, 2002.
```
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