D O C . 2 9 3 O N C E N T E N A R Y O F K E L V I N 2 9 7
This result evidently applies irrespective of the conditions for producing the curva-
ture in the liquid’s surface.
Proof of Helmholtz’s vortex
theorems:[5]
Let L be a closed curve within a frictionless fluid with the velocity components
uv. The line integral
(u1, u2, u3 = velocity components
x1, x2, x3 = coordinates)
is, according to Stokes’s theorem, equal to the surface integral of the vortex vector
over an arbitrary surface bounded by L. We want to get the temporal dependence
of the vortex quantity W under the condition that the curve participates in the liq-
uid’s flow. If the temporal derivative with reference to a liquid particle is denoted
as , the corresponding differential as D, then for any quantity Ψ one has
The Euler equations for the liquid’s motion then read
if one sets , which assumes that the liquid density ρ is solely a function
of the pressure p, and that the external forces are derivable from a unique potential
ϕ.
One then has
From this it follows
that[6]
.
Helmholtz’s vortex theorems are contained in this result .
W
L
³Σuvdxv
=
D
Dt
----- -
DΨ
Dt
-------- -
∂t
∂ψ ¦uv∂xv.∂ψ
+=
Duv
Dt
---------
∂xv
∂π
–
∂xv
∂ϕ
, –=
π
pd
ρ
³------- =
Duv dt©
∂xv
∂
π
ϕ)¹·
+ ( –
§
–=
Ddxv
dt¦∂xv
α
∂u
dxα. =
DW Duvdxv uvDxv) +
³Σ(
=
dt
xv©
π – ϕ –
uα2·
2
¦-----
+
¹
§
v
³¦dxv∂∂
0 = =
DW 0 =