Mechanics of Fluids

368 The flow of an inviscid fluid
Potential flow
For irrotational flow the circulation is zero and so
� P
↑
Q qsds =
↑
A
� P
↑
R qsds
↑
A
In other words, the value of the integral depends only on the position of P
relative to A.
Putting
−φ =
� P
A
qsds (9.7)
we have δφ =−qsδs or qs =−∂φ/∂s. (The reason for the minus sign will
appear in a moment.) The function φ is termed the velocity potential. Ifthe
line element, of length δs, is perpendicular to a streamline, then qs = 0
and so δφ = 0. Thus the velocity potential is constant along lines perpendicular
to streamlines. These lines of constant velocity potential are known
as equipotential lines. The velocity potential provides an alternative means
of expressing the velocity components parallel to the coordinate axes in
irrotational flow:
u =− ∂φ
; v =−∂φ
(9.8)
∂x ∂y
The minus signs in these equations arise from the convention that the velocity
potential decreases in the direction of flow just as electrical potential
decreases in the direction in which current flows. (Some writers adhere to the
opposite convention by which δφ = qsδs.) The analogy with electrical potential
should not, however, be pushed further than this. Velocity potential is
simply a quantity defined mathematically by eqn 9.7. It is not a physical
quantity that can be directly measured. Its zero position, like that of the
stream function, may be arbitrarily chosen.
Whereas the stream function applies to both rotational and irrotational flow,
the velocity potential only has meaning in irrotational flow. This is because
it is only in irrotational flow that the value of � P
A qsds is independent of the
path traversed from A and P. For this reason irrotational flow is often termed
potential flow.
If the expressions for u and v from eqn 9.8 are substituted into the
continuity relation 9.1 we obtain
∂2φ ∂x2 + ∂2φ = 0 (9.9)
∂y2 This is the two-dimensional form of Laplace’s equation which finds application
in many branches of science. All flows that conform to the principle of
continuity therefore satisfy Laplace’s equation if they are irrotational.
Similar substitutions in the expression for vorticity (9.5) yield
ζ = ∂
�
−
∂x
∂φ
�
−
∂y
∂
�
−
∂y
∂φ
�
=−
∂x
∂2φ ∂x∂y + ∂2φ = 0
∂y∂x
The vorticity is shown to be zero when a velocity potential φ exists.