Abstract
Clebsch has shown that the components of the velocity of a fluid
u
,
v
,
w
, parallel to rectangular axes
x
,
y
,
z
, may always be expressed thus
u
=
dχ
/
dx
+ λ
dΨ
/
dx
,
v
=
dχ
/
dy
+ λ
dΨ
/
dy
,
w
=
dχ
/
dz
+ λ
dΨ
/
dz
; where
λ, Ψ
are systems of surfaces whose intersections determine the vortex lines; and the pressure satisfies an equation which is equivalent to the following
p
/
ρ
+ V = –
dχ
/
dt
–½{(
dχ
/
dx
)
2
+ (
dχ
/
dy
)
2
+(
dχ
/
dz
)
2
} + ½
λ
2
{(
dΨ
/
dx
)
2
+(
dΨ
/
dy
)
2
+(
dΨ
/
dz
)
2
} where
p
is the pressure,
ρ
the density, and V the potential of the forces acting on the liquid. It is shown in this paper that an equation of a complicated nature in
λ
only can be obtained in the following cases (that is to say, as in cases of irrotational motion, the determination of the motion depends on the solution of a single equation only):— (1.) Plane motion, referred to rectangular coordinates
x
,
y
. The equation is somewhat simpler when the vortex surfaces are of invariable form, and move parallel to one of the axes of coordinates with arbitrary velocity.
Cited by
7 articles.
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