Basics of Fluid Mechanics, 2014a

10.3. POTENTIAL FLOW FUNCTIONS INVENTORY 349
(boundary must be solid) and hence it dictates the boundary conditions on the potential
equation. From mathematical point of view this boundary condition as
U n = dφ
dn = ∇φ·̂n =0 (10.109)
In this case, ̂n represents the unit vector normal to the surface.
A solution to certain boundary condition with certain configuration geometry and
shape is a velocity flow field which can be described by the potential function, φ. If
such function exist it can be denoted as φ 1 . If another velocity flow field exists which
describes, or is, the solutions to a different boundary condition(s) it is denoted as φ 2 .
The Laplacian of first potential is zero, ∇ 2 φ 1 =0and the same is true for the second
one ∇ 2 φ 2 =0. Hence, it can be written that
=0 =0
{ }} { { }} {
∇ 2 φ 1 + ∇ 2 φ 2 =0 (10.110)
Since the Laplace mathematical operator is linear the two potential can be combined
as
∇ 2 (φ 1 + φ 2 )=0 (10.111)
The boundary conditions can be also treated in the same fashion. On a solid boundary
condition for both functions is zero hence
dφ 1
dn = dφ 2
=0 (10.112)
dn
and the normal derivative is linear operator and thus
d (φ 1 + φ 2 )
=0 (10.113)
dn
It can be observed that the combined new potential function create a new velocity field.
In fact it can be written that
U = ∇(φ 1 + φ 2 )=∇φ 1 + ∇φ 2 = U 1 + U 2 (10.114)
The velocities U 1 and U 2 are obtained from φ 1 and φ 2 respectively. Hence, the superposition
of the solutions is the characteristic of the potential flow.
Source and Sink Flow or Doublet Flow
In the potential flow, there is a special case where the source and sink are combined
since it represents a special and useful shape. A source is located at point B which is
r 0 from the origin on the positive x coordinate. The flow rate from the source is Q 0
and the potential function is
Q 1 = Q ( )
0
2 π ln rB
(10.115)
r 0