ME 362
Navier-Stokes Equation
Page 1 of 2
Differential Equations of Mass and Momentum Conservations:
For incompressible flow:
Continuity:
u v

0
x y
NS equation:



DV

 g  P   2V
Dt
where
D 
 
 
 
(V )  (V )  u (V )  v (V )
Dt
t
x
y

2 
2 
 2 (V )  2 (V )  2 (V )
 x
 y
Solution Procedure:
Step 1: Assumptions

Used to simplify NS equation (reducing PDE to ODE)

Common assumptions are:
o Incompressible


o Steady state

o Fully developed

 and  are constant

V
0
t
V
0
x
(time independent)
(space independent)
Other assumptions may include:
o Neglect gravity

o No pressure gradient 

g0
P  0
Step 2: Boundary Conditions

Needed for solving integration constants in Step 5

Sketching the problem will help to define boundary conditions
ME 362
Navier-Stokes Equation
Page 2 of 2
Step 3: Continuity Equation

u v

0
x y
Step 4: Momentum Equation

x – momentum:

y – momentum:
D
P
(u )  g x 
  2 (u )
Dt
x
D
P
 (v)  g y 
  2 (v)
Dt
y

Step 5: Find the Velocity Profile


Integrate the resulting momentum equation(s) from Step 4
Apply boundary conditions from Step 2 to solve for integration constants
Step 6: Solve for Flow Rate or Wall Shear Stress

flow rate:
Q   u ( y )dy

wall shear stress:
 yx   (
u v
 )
y x
Example:
A conveyor belt, which moves at constant velocity Uc, transports oil from an oil bath to a
conduit above. The vacuum pump assists in the process by imposing a negative pressure
gradient in the flow direction (i.e., dp/dx < 0). Assume the flow is fully developed in the
x-direction. Find the velocity profile and flow rate.
Vacuum pump
Conveyor
belt
L
Uc
x
y
H
g
Oil bath (, )