Basics of Fluid Mechanics, 2014a

334 CHAPTER 10. POTENTIAL FLOW
Example 10.2:
What are stream lines that should be obtained in Example 10.1.
Solution
Utilizing equation (10.42) results in
dy
dx = U y −4 y3
=
U x 2 x
(10.II.a)
The solution of the non–linear ordinary differential obtained by separation of variables
as
− dy
2 y 3 = dx
(10.II.b)
2 x
The solution of equation streamLineSimple:separation is obtained by integration as
1
=lnx + C (10.II.c)
4 y2 End Solution
From the discussion above it follows that streamlines are
continuous if the velocity field is continuous. Hence, several
streamlines can be drawn in the field as shown in Figure 10.1. If
two streamline (blue) are close an arbitrary line (brown line) can
be drawn to connect these lines. A unit vector (cyan) can be
drawn perpendicularly to the brown line. The velocity vector is
almost parallel (tangent) to the streamline (since the streamlines
are very close) to both streamlines. Depending on the orientation
of the connecting line (brown line) the direction of the unit vector
is determined. Denoting a stream function as ψ which in the two
dimensional case is only function of x, y, that is
y
U
1
α
ŝ
dl
2
ψ 1
ψ 2
Fig. -10.1. Streamlines
to explain stream function.
x
ψ = f (x, y) =⇒ dψ = ∂ψ ∂ψ
dx + dy (10.43)
∂x ∂y
In this stage, no meaning is assigned to the stream function. The differential of stream
function is defied as
dψ = U · ŝdl (10.44)
The term ,dl refers to a small straight element line connecting two streamlines close
to each other. It could be viewed as a function as some representing the accumulative
of the velocity. The physical meaning is needed to be connected with the previous
discussion of the two dimensional function. If direction of the l is chosen in a such
away that it is in the direction of x as shown in Figure 10.2(a). In that case the ŝ in