Basics of Fluid Mechanics, 2014a

10.2. POTENTIAL FLOW FUNCTION 335
the direction of −ĵ as shown in the Figure 10.2(a). In this case, the stream function
differential is
⎛ ⎞
bs
dψ = ∂ψ ∂ψ
) {}}{ dl
dx +
(î
∂x ∂y dy = U x + ĵ ⎜ ⎟
{}}{
U y · ⎝− ĵ ⎠ dx = −U y dx (10.45)
In this case, the conclusion is that
∂ψ
∂x = −U y (10.46)
y
2
α
U
ψ 2
1 ŝ dl
ψ 1
x
y
U
2
α
dl ŝ ψ 2
1 ψ 1
x
(a) Streamlines with element in
X direction.
(b) Streamlines with straight
in Y direction.
Fig. -10.2. Streamlines with different element in different direction to explain stream function.
On the other hand, if dl in the y direction as shown in Figure 10.16(b) then ŝ = îi
as shown in the Figure.
dψ = ∂ψ ∂ψ
)
dx +
(î
∂x ∂y dy = U x + ĵ U y
⎛ ⎞
bs
{}}{
⎜ ⎟
· ⎝ î ⎠
dl
{}}{
dy = U x dy (10.47)
In this case the conclusion is the
∂ψ
∂y = U x (10.48)
Thus, substituting equation (10.46) and (10.48) into (10.43) yields
U x dy − U y dx =0 (10.49)
It follows that the requirement on U x and U y have to satisfy the above equation which
leads to the conclusion that the full differential is equal to zero. Hence, the function
must be constant ψ =0.