The momentum equations for the two layers now read
μ 1
∂
2 u 1
∂z 2 = g
∂h
∂x
,
ρ 2 μ 2
∂
2 u 2
∂z 2 = gρ 1
∂h
∂x
− gðρ 2 − ρ 1 Þ
∂ξ
∂x
.
The continuity equations take the form
∂w 1
∂z
= −
∂u 1
∂x
,
∂w 2
∂z
= −
∂u 2
∂x
.
In this case, there are also as many as 6 boundary conditions:
ρ 1 μ 1
∂u 1
∂z
z = 0
= τ,
u 2 D
ð Þ= 0,
w 2 D
ð Þ= 0,
w 1 ξ
ð Þ
u 1 ξ
ð Þ
= ξ
′ x
ð Þ,
w 2 ξ
ð Þ
u 2 ξ
ð Þ
= ξ
′ x
ð Þ,
ρ 1 μ 1
∂u 1
∂z
z = ξ
= ρ 2 μ 2
∂u 2
∂z
z = ξ
.
The solution for Model 2 is given by the following formulas:
u 1 x, z
ð Þ=
S 1 x
ð Þ
2
z
2 + A 1 z + B 1 x
ð Þ,
u 2 x, z
ð Þ=
S 2 x
ð Þ
2
z
2 + A 2 z + B 2 x
ð Þ,
w 1 x, z
ð Þ= −
S
′
1 x
ð Þ
6
z
3
−
A
′
1 x
ð Þ
2
z
2
− B
′
1 x
ð Þz + C 1 x
ð Þ,
470
P. O. Zavialov et al.