Basics of Fluid Mechanics, 2014a

372 CHAPTER 10. POTENTIAL FLOW
This flow contains two extremes cases discussed earlier horizontal and vertical flow.
Flow in a Sector
The uniform flow presentation seem to be just repeat of what was done in the
presentation without the complex numbers. In sector flow is an example where the
complex number presentation starts to shine. The sector flow is referred to as a flow in
sector. Sector is a flow in opening with specific angle.The potential is defined as
F (z) =U 0 z n (10.222)
where n ≥ 1 the relationship between the n and opening angle will be established in
this development. The polar represented is used in this derivations as z = re iθ and
substituting into equation (10.222) provides
The potential function is
and the stream function is
F (z) =U 0 r n cos(nθ)+iU 0 r n sin(nθ) (10.223)
φ = U 0 r n cos(nθ) (10.224)
ψ = U 0 r n sin(nθ) (10.225)
The stream function is zero in two extreme cases: one when the θ =0and two when
θ = π/n. The stream line where ψ =0are radial lines at the angles and θ =0and
θ = π/n. The zone between these two line the streamline are defined by the equation
of ψ = U 0 r n sin(nθ). The complex velocity can be defined as the velocity along these
lines and is
W (z) =nU 0 z n−1 = nU 0 r n−1 e i (n−1)θ =
= nU 0 r n−1 cos(nθ)+inU 0 r n−1 sin(nθ) e iθ (10.226)
Thus the velocity components are
and
U r = nU 0 r n−1 cos(nθ) (10.227)
U θ = −nU 0 r n−1 sin(nθ) (10.228)
It can be observed that the radial velocity is positive in the range of 0