Page 109 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

94 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Of course the relationship between these solutions and the other solu-

tions of the Euler system is not very obvious but it could be established
rigorously under some carefully chosen hypotheses.
Now we remark that the above system forms also a Hamilton system.
Really, by defining H = -~ do C0; In|r;—r,;| , the system is equivalent
tj
with

dey OH ,, dyj OH. _ aw
[; p= N.
=
.
= —
J dt Oy;’ J dt az;’? 1,
Introduce the new variables

a, = JV ilzi,y; = J\Tilson (0) yt = 1,N ;

we get a real Hamilton system

B= Sh
=
a

dt dy,’ dt — Ax,
a
and, as in classical mechanics we have

> Olt ae OH dy,

>» Oy; ‘dt

1.e., H is a constant in time along a path line.
A consequence of this property is that if all the vortices have the same
sign for their strength, then they cannot collide during the motion. In

other terms, if |r;-r;| # 0,7 7, at t = 0, then this result remains
valid for all time since if |r;—r;] + 0 , H will become infinite.
We remark that the Euler equations themselves form a Hamiltonian
system (see, for instance, [2]) such that the Hamiltonian nature of the
vortex model (approximation) should not surprise. What might be of

great interest is to establish whether or not this system is completely in-
tegrable in the sense of Hamiltonian systems. There are some reasons to
suppose the existence of a certain Lie group that generates the equations
(in some sense) [19].
Let us generalize the previous case and imagine the N vortices moving
in a domain D with boundary QD. Following the same way as before we
must modify the flow of the j-th vortex (its velocity vj) so thatv-nlyp =
0. This could be done by adding a potential flow of velocity u,; such that

vj;;n = —u,-n. In other words, we choose a stream function wp; associated
with the 7-th vortex, which satisfies