Basics of Fluid Mechanics, 2014a

10.2. POTENTIAL FLOW FUNCTION 333
The integration with respect the space and not time results in the
Euler Equation or Inviscid Flow
∫ ( )
∂φ
∂t + (∇φ)2
dP
+ g l + = f(t)
2
ρ
(10.39)
Example 10.1:
The potential function is given by φ = x 2 − y 4 +5. Calculate the velocity component
in Cartesian Coordinates.
Solution
The velocity can be obtained by applying gradient on the potential U = ∇φ as
V x = ∂φ
∂x =2x
V y = ∂φ
∂y
= −4 y 3
V z = ∂φ
∂z
=0
(10.I.a)
End Solution
10.2.1 Streamline and Stream function
The streamline was mentioned in the earlier section and now the focus is on this issue.
A streamline is a line that represent the collection of all the point where the velocity is
tangent to the velocity vector. Equation (10.25) represents the unit vector. The total
differential is made of three components as
̂l = î U x
U + ĵ U y
U + ̂k U z
U = î dx
dl + ĵ dy dz
+ ̂k
dl dl
(10.40)
It can be noticed that dx /dl is x component of the unit vector in the direction of x.
The discussion proceed from equation (10.40) that
U x
dx = U y
dy = U z
dz
(10.41)
Equation (10.41) suggests a system of three ordinary differential equations as a way
to find the stream function. For example, in the x–y plane the ordinary differential
equation is
dy
dx = U y
U x
(10.42)