Problem 5.6

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Problem 5.6 [Difficulty: 2]
Given: The x-component of velocity in a steady, incompressible flow field 
Find: The simplest y-component of velocity for this flow field
Solution: We will check this flow field against the continuity equation
      0




t
w
z
v
y
u
x

Governing
Equations: (Continuity equation)
Assumptions: (1) Incompressible flow (ρ is constant)
(2) Two dimensional flow (velocity is not a function of z)
0


y
v
x
uBased on the two assumptions listed above, the continuity equation reduces to:
2x
A
x
u 

2x
A
x
u
y
v 

The partial of u with respect to x is: Therefore from continuity, we have
Integrating this expression will yield the y-component of velocity: v y
A
x2



d f x( ) A y
x2
f x( )
The simplest version of this velocity component would result when f(x) = 0: v
A y
x2
