Basics of Fluid Mechanics, 2014a

332 CHAPTER 10. POTENTIAL FLOW
10.2 Potential Flow Function
The two different Bernoulli equations suggest that some mathematical manipulations
can provide several points of understating. These mathematical methods are known as
potential flow. The potential flow is defined as the gradient of the scalar function (thus
it is a vector) is the following
U ≡ ∇φ (10.33)
The potential function is three dimensional and time dependent in the most expanded
case. The vorticity was supposed to be zero for the first Bernoulli equation. According
to the definition of the vorticity it has to be
Ω = ∇×U = ∇×∇φ (10.34)
The above identity is shown to be zero for continuous function as
∇φ
{(
}} {
∇× i ∂φ
)
∂x + j∂φ ∂y + k∂φ ∂z
( ∂ 2 φ
= i
+ j
)
∂y∂z − ∂2 φ
∂z∂y
( ∂ 2 φ
∂z∂x − ∂2 φ
∂x∂z
) ( ∂ 2 )
φ
+ k
∂y∂x − ∂2 φ
∂x∂y
(10.35)
According to Clairaut’s theorem (or Schwarz’s theorem) 2 the mixed derivatives are
identical ∂ xy = ∂ yx . Hence every potential flow is irrotational flow. On the reverse
side, it can be shown that if the flow is irrotational then there is a potential function
that satisfies the equation (10.33) which describes the flow. Thus, every irrotational
flow is potential flow and conversely. In these two terms are interchangeably and no
difference should be assumed.
Substituting equation (10.33) into (10.24) results in
(
(∇φ) 2
∂∇φ
∂t
+ ∇
2
∫ ( dP
+ g l +
ρ
It can be noticed that the order derivation can be changed so
∂∇φ
∂t
= ∇ ∂φ
∂t
) ) =0 (10.36)
(10.37)
Hence, equation (10.36) can be written as
(
∫ ( ) ) ∂φ
∇
∂t + (∇φ)2
dP
+ g l +
=0 (10.38)
2
ρ
2 Hazewinkel, Michiel, ed. (2001), ”Partial derivative”, Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4