Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 359
The other two lines are the horizontal coordinates. The flow does not cross any stream
line, hence the stream line represented by r = a can represent a cylindrical solid body.
For the case where ψ ≠0the stream function can be any value. Multiplying
equation (10.159) by r and dividing by U 0 a 2 and some rearranging yields
r
a
ψ
( r
) 2
= sin θ − sin θ (10.161)
aU 0 a
It is convenient, to go through the regular dimensionalzing process as
r ψ =(r) 2 sin θ − sin θ or r 2 − ψ r − 1=0 (10.162)
sin θ
The radius for other streamlines can found or calculated for a given angle and
given value of the stream function. The radius is given by
√ ( )2
ψ ψ
sin θ ± +4
sin θ
r =
(10.163)
2
It can be observed that the plus sign must be used for radius with positive values (there
are no physical radii which negative absolute value). The various value of the stream
function can be chosen and drawn. For example, choosing the value of the stream
function as multiply of ψ =2n (where n can be any real number) results in
√ ( ) 2
2 n 2 n
sin θ ± +4
sin θ
r =
= n csc(θ)+ √ n
2
2 csc 2 (θ)+1 (10.164)
The various values for of the stream function are represented by the ratios n. For
example for n =1the (actual) radius as a function the angle can be written as
(
r = a csc(θ)+ √ )
csc 2 (θ)+1
(10.165)
The value csc(θ) for θ = 0 and θ = π is equal to infinity (∞) and for values of
csc(θ = π/2)=1. Similar every line can be evaluated. The lines are drawn in Figure
10.13.
The velocity of this flow field can be found by using the equations that were
developed so far. The radial velocity is
U r = 1 r
∂ψ
∂θ = U 0 cos θ
(1 − a2
r 2 )
(10.166)
The tangential velocity is
U r = − ∂ψ
)
∂r = U 0 sin θ
(1+ a2
r 2
(10.167)