350
5 Variational Problems of Conditional Extrema
Fig. 5.5 Capillary tube
model diagram
R
O
x
y
θ
r
h
of the liquid and air, σ SL is the surface tension coefficient of the solid and liquid,
σ S A is the surface tension coefficient of the solid and air. Taking the intersection
point of the liquid level and the capillary tube axis as the origin of coordinates,
the coordinate system is established, as shown in Fig. 5.5. The distance from the
origin to the level of the liquid surface is y 0 . The quality of infinitesimal volume is
dm = 2πx(y 0 + y)ρdx, the height of the center of mass of infinitesimal volume is
y c = (y 0 + y)/2. Find the approximate formula of capillarity.
Solution The gravitational potential energy of the liquid is
J 1 =
r
0
2πx y(ρ L − ρ A )gy c dx =
r
0
πx(ρ L − ρ A )g(y 0 + y)
2 dx
(5.1)
The curved liquid surface energy is
J 2 =
r
0
2πxσ L A ds =
r
0
2πxσ L A
1 + y dx
(5.2)
The energy of liquid interface in the inner wall of the capillary is
J 3 = 2πr (σ SL − σ S A )y 1 = 2πr y 1 = 2πr h = Φ
(5.3)
where, y 1 = h is the height of the capillary liquid from the origin of coordinates, at
this time, it should be regarded as unknown quantity.
The total energy functional of the liquid in the capillary tube is
J = J 1 + J 2 + J 3 =
r
0
πx[(ρ L − ρ A )g(y 0 + y) 2 + 2σ L A
1 + y ]dx + 2πR y 1
(5.4)
The Euler equation of the functional is