284
L. V. Shmeleva et al.
be taken into account that from the point of view of the local reference system, the
velocity v 0s looks like the speed of “flowing through” of the solid phase via the interface. Thus, its component v
(ζ )
0s , as well as the gas component v
(ζ )
s , are negative. In
this case, the transformations (9) for v
(ζ )
0s can be used immediately in the form:
v
(ζ )
0s =
S x
N
v
(x)
0s +
S y
N
v
(y)
0s −
1
N
v
(z)
0s ,
(14)
where v
(i)
0s are the components of the velocity vector of surface points.
Since z(t) = S(t, x(t), y(t)), then the velocity component v
(z)
0s is equal to
dz
dt
= v
(z)
0s = S t + S x
dx
dt
+ S y
dy
dt
= S t + S x v
(x)
0s + S y v
(y)
0s .
Substituting the result obtained in (14) we have
v
(ζ )
0s = −S t /N .
(15)
Substituting this into (13) and taking into account v
(ζ )
s = −
P
σ
s
σ
√
κa
and ρ s = a P
η ,
as well as the explicit form of the coefficient N defined after formula (7), we obtain
S t =
2
√
aκ
γρ 0s
P
β
s
1 + S 2
x + S 2
y
(16)
or in dimensionless form:
∂∂
∂θ
=
β
1 +
∂∂
∂ X
2
+
∂∂
∂Y
2
.
(17)
Here γ ≡ κ − 1, β ≡
κ+1
2κ
, = P/P 0 is the dimensionless pressure, θ = t/t 0 is
the dimensionless time, Σ = S/S 0 is the dimensionless function of the crater form
and dimensionless coordinates X = x/S 0 , Y = y/S 0 . During the dimensionless
procedure, it was found that
S 0 =
bγ
4 L
κ 5/2 ρ 0s α 2 L d
, P 0 =
γ
2 L d q in
2α 1/2 κ
√
ϕ 0
t 0 =
bγ
4
ϕ 0 L
q in α 2 κ 5/2 L d
, a =
κα
ϕ 0 γ 2 P
2σ
0
where α ≡ κ + 1, and parameter b, which includes important characteristics of the
vaporized substance, such as the R g gas constant, μ a is the molecular mass of the
gas, d a is the effective diameter of the molecule, m a is the mass of the atom of the
vaporized substance, k B is the constant Boltzmann, and has the form:
L. V. Shmeleva et al.
be taken into account that from the point of view of the local reference system, the
velocity v 0s looks like the speed of “flowing through” of the solid phase via the interface. Thus, its component v
(ζ )
0s , as well as the gas component v
(ζ )
s , are negative. In
this case, the transformations (9) for v
(ζ )
0s can be used immediately in the form:
v
(ζ )
0s =
S x
N
v
(x)
0s +
S y
N
v
(y)
0s −
1
N
v
(z)
0s ,
(14)
where v
(i)
0s are the components of the velocity vector of surface points.
Since z(t) = S(t, x(t), y(t)), then the velocity component v
(z)
0s is equal to
dz
dt
= v
(z)
0s = S t + S x
dx
dt
+ S y
dy
dt
= S t + S x v
(x)
0s + S y v
(y)
0s .
Substituting the result obtained in (14) we have
v
(ζ )
0s = −S t /N .
(15)
Substituting this into (13) and taking into account v
(ζ )
s = −
P
σ
s
σ
√
κa
and ρ s = a P
η ,
as well as the explicit form of the coefficient N defined after formula (7), we obtain
S t =
2
√
aκ
γρ 0s
P
β
s
1 + S 2
x + S 2
y
(16)
or in dimensionless form:
∂∂
∂θ
=
β
1 +
∂∂
∂ X
2
+
∂∂
∂Y
2
.
(17)
Here γ ≡ κ − 1, β ≡
κ+1
2κ
, = P/P 0 is the dimensionless pressure, θ = t/t 0 is
the dimensionless time, Σ = S/S 0 is the dimensionless function of the crater form
and dimensionless coordinates X = x/S 0 , Y = y/S 0 . During the dimensionless
procedure, it was found that
S 0 =
bγ
4 L
κ 5/2 ρ 0s α 2 L d
, P 0 =
γ
2 L d q in
2α 1/2 κ
√
ϕ 0
t 0 =
bγ
4
ϕ 0 L
q in α 2 κ 5/2 L d
, a =
κα
ϕ 0 γ 2 P
2σ
0
where α ≡ κ + 1, and parameter b, which includes important characteristics of the
vaporized substance, such as the R g gas constant, μ a is the molecular mass of the
gas, d a is the effective diameter of the molecule, m a is the mass of the atom of the
vaporized substance, k B is the constant Boltzmann, and has the form: