266
P. Liu
Because the liquid is incompressible, the volume of the microelements does
not change at any time. Based on that, we can obtain
∂ x
∂a
∂z
∂a
∂ x
∂b
∂z
∂b
= 1
This is the uncompressed Lagrange continuous equation. Usually, we use
∂
∂t
∂ x
∂a
∂z
∂a
∂ x
∂b
∂z
∂b
= 0
Using Newton’s second law, the equation of motion established under
gravity is
∂ 2 x
∂t 2 = −
1
ρ
∂ p
∂ x
∂ 2 z
∂t 2 = g −
1
ρ
∂ p
∂ y
Based on the derivation of compound functions
∂ p
∂a
=
∂ p
∂ x
∂ x
∂a
+
∂ p
∂z
∂z
∂a
The equation of motion can be expressed as
−
∂ 2 x
∂t 2
∂ x
∂a
+
g −
∂ 2 z
∂t 2
∂z
∂a
−
1
ρ
∂ p
∂a
= 0
−
∂ 2 x
∂t 2
∂ x
∂b
+
g −
∂ 2 z
∂t 2
∂z
∂b
−
1
ρ
∂ p
∂b
= 0
Substituting a and b with x 0 and z 0 , respectively, we can obtain the
Lagrangian-type continuous equation and motion equation that characterizes