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Dynamics of Inviscid Fluids 59

3. Irrotational Flows of Incompressible Inviscid

Fluids. The Plane Case

The Lagrange theorem, stated in the first section of this chapter,
establishes the conservation of the irrotational character of certain fluid
flows. An important application of this theorem is the case when the
fluid starts its flow from an initial rest state (where, obviously, w = 0).
If a fluid flow is irrotational, then from the condition rot v = 0 we
will deduce the existence of a scalar function ® (£1, 22,%3,t), defined

to within an additive function of time, such that v = grad®. Obvi-
ously, the determination of this function, called the velocity potential, is
synonymous with that of the velocity field. But from the equation of
continuity we also get 0 = divuv = div(grad®) = A®, while the slip
condition on a fixed wall (%), immersed in the fluid, becomes

d®
grad®@-
=
0=v-nl; n|, dnp’
=
that is the determination of ® comes to the solving of a boundary value
problem of Neumann type joined to the Laplace operator.
Obviously, if the domain flow is “unbounded” we need some behaviour

conditions at far distances (infinity) which, in the hypothesis of a fluid
stream “attacking” with the velocity vg. an obstacle whose boundary is
(5°), implies that
lim grad ® = Vo.
x? +22 423-400
So that in this particular case the flow determining comes either to a
Neumann problem for the Laplace operator (the same problem arises in
the tridimensional case too), that means A® = 0 in the fluid domain D
with ae | gp = 9, or to a Dirichlet problem for the same Laplace operator

(which is specific only in the 2-dimensional case) when Ay = 0 in D with
Wlap = constant.
In the conditions of an unbounded domain (the case of a flow past a
bounded body being included too), the above two problems should be
completed by information about the velocity (that is about grad® and
grad w respectively) at far distances (infinity).

Now we will show that in a potential flow past one or more body(ies),
the maximum value for the velocity is taken on the body(ies) boundary.
If M is an arbitrary point in the fluid which is also considered the origin
of a system of axes, the Oz axis being oriented as the velocity at M,
then we have v?(M) = ©2(M), while for any other point P, we have
v?(P) = (2 + &2 + 62)(P).
If the function ®, is harmonic and consequently it does not have an

extremum inside the domain, then there will always be some points P