38
V. A. Gushchin et al.
is the spot radius, and s is the salinity perturbation (stratifying component), which
includes the salt compression ratio.
We consider the plane unsteady problem of the flow which occurs when there
is a collapse of the region A of homogeneous fluid, surrounded by a stably and
continuously density-stratified fluid (see Fig. 4.1).
The Navier–Stokes equations in the Boussinesq approximation describing the
flow of this type can be written as:
∂v
∂t
+ (v · ∇)v = −∇ p +
1
Re
v +
1
Fr
s
g
g
,
(4.2)
∇ · v = 0,
(4.3)
∂s
∂t
+ (v · ∇)s =
1
Sc · Re
s +
v
C
,
(4.4)
where v is the velocity vector with components u, v, respectively, along the x and
y axes of a Cartesian coordinate system selected as indicated in Fig. 4.1, ρ is the
density, p is the pressure minus hydrostatic one, s is the perturbation of salinity, the
Reynolds number Re = ρ 0 R 0
2 N/μ, the Froude number Fr = R 0 N
2 /g, the Schmidt
number Sc = μ/ρ 0 k s , k s is the diffusion coefficient of salts, μ is dynamic viscosity
coefficient, g = (0, –g), g is acceleration of free fall, ρ 0 is the density on the level y
= 0, and C = /R 0 is the scale ratio.
We assume that the initial time t = 0 the system on the plane R
2 is at rest, i.e.,
u = 0, v = 0, (x, y) ∈ R
2
,
(4.5)
density of fluid at the spot A is
ρ = 1, (x, y) ∈ A,
(4.6)
and outside of spot, i.e., in the area of R
2 \A,
ρ = 1 −
y
C
+ s, (x, y) ∈ R
2
\A,
(4.7)
the perturbation of salinity is defined by Eq. 4.8.
s =
y
C
if (x, y) ∈ A
0 if (x, y) ∈ R
2
\A
(4.8)
As an initial approximation for pressure, necessary in solving the equation for
pressure distribution is selected according to Eq. 4.9.