Specificity of Boundary Conditions for Laser-Stimulated …
297
along the z-axis, it can be assumed that the shear components of the stress tensor
are much smaller than the diagonal component of this tensor. It is as follows:
P
(0)s
ξζ
2 +
P
(0)s
ηζ
2
P s +q
(ζ )
s /c+P
(0)s
ζ ζ
2 1. But even if we assume that all three components of the stress
tensor are quantities of the same order, then we can estimate the order of magnitude of the term
P
(0)s
ξζ
2 +
P
(0)s
ηζ
2
P s +q
(ζ )
s /c+P
(0)s
ζ ζ
2 . It will be ~0.22. This allows us to neglect this
summand. Further, using the condition of balance of mass flux, we obtain v
(ζ )
s −v
(ζ )
0s =
1 −
ρ s
ρ 0s
v
(ζ )
s . The density of the solid phase is much higher than the density of the
gas
ρ s
ρ 0s
∼ 10
−3
1
, therefore realize the equality
i
v
i
s − v
i
0s
2 =
v
(ζ )
s
2 .
Consider the factor
1 −
ρ s
ρ 0s
(n·vs)
c
+ L d
n · q s
, which is part of the boundary
condition. Since the speed with which the gas can fly off the surface is of the order
of the thermal velocity, then quantity
1 −
ρ s
ρ 0s
(n·vs)
c
∼ 10
−5 . Taking it into account
makes sense when the coefficient of non-thermal dissipative losses L d does not
exceed this value, that is, in fact, when the radiation flux does not have losses on the
surface and completely passes into the solid medium. Such a situation is unlikely
when the surface is destroyed, so we will assume
1 −
ρ s
ρ 0s
(n · v s )
c
+ L d
n · q s
≈ L d
n · q s
.
Discarding all small summands in (11), we finally obtain
(n · v s )(ρ s U s + P s ) + (n · v s )ρ s
v
(ζ )
s
2
2
+ L d
n · q s
− Lλ s (n · gradT ) s − (n · v s )ρ s U 0s = 0,
where v
(ζ )
s is the normal component of the velocity vector.
In this equation, the first term can be interpreted as the energy flux of a substance
into the gas phase, the second as its convective outflow into the plasma-gas medium
from the interface, the third and fourth terms are the flows associated with the
transfer of stimulating (electromagnetic) energy (3-d summand) and thermal energy
(summand 4) across the interface. The last summand determines the energy flux
from the condensed phase to the interface between the phases, and U 0s is the internal
energy per unit mass of the condensed medium, which is related to the specific heat
of the condensate-gas phase transition ϕ 0 by the relation U 0s = −ϕ 0 [17]. This ratio
is due to the fact that internal energy is determined by the binding energy between
atoms in a solid, and by definition it is a negative value. Therefore, the condition for
the balance of the energy flux will be considered in the form:
297
along the z-axis, it can be assumed that the shear components of the stress tensor
are much smaller than the diagonal component of this tensor. It is as follows:
P
(0)s
ξζ
2 +
P
(0)s
ηζ
2
P s +q
(ζ )
s /c+P
(0)s
ζ ζ
2 1. But even if we assume that all three components of the stress
tensor are quantities of the same order, then we can estimate the order of magnitude of the term
P
(0)s
ξζ
2 +
P
(0)s
ηζ
2
P s +q
(ζ )
s /c+P
(0)s
ζ ζ
2 . It will be ~0.22. This allows us to neglect this
summand. Further, using the condition of balance of mass flux, we obtain v
(ζ )
s −v
(ζ )
0s =
1 −
ρ s
ρ 0s
v
(ζ )
s . The density of the solid phase is much higher than the density of the
gas
ρ s
ρ 0s
∼ 10
−3
1
, therefore realize the equality
i
v
i
s − v
i
0s
2 =
v
(ζ )
s
2 .
Consider the factor
1 −
ρ s
ρ 0s
(n·vs)
c
+ L d
n · q s
, which is part of the boundary
condition. Since the speed with which the gas can fly off the surface is of the order
of the thermal velocity, then quantity
1 −
ρ s
ρ 0s
(n·vs)
c
∼ 10
−5 . Taking it into account
makes sense when the coefficient of non-thermal dissipative losses L d does not
exceed this value, that is, in fact, when the radiation flux does not have losses on the
surface and completely passes into the solid medium. Such a situation is unlikely
when the surface is destroyed, so we will assume
1 −
ρ s
ρ 0s
(n · v s )
c
+ L d
n · q s
≈ L d
n · q s
.
Discarding all small summands in (11), we finally obtain
(n · v s )(ρ s U s + P s ) + (n · v s )ρ s
v
(ζ )
s
2
2
+ L d
n · q s
− Lλ s (n · gradT ) s − (n · v s )ρ s U 0s = 0,
where v
(ζ )
s is the normal component of the velocity vector.
In this equation, the first term can be interpreted as the energy flux of a substance
into the gas phase, the second as its convective outflow into the plasma-gas medium
from the interface, the third and fourth terms are the flows associated with the
transfer of stimulating (electromagnetic) energy (3-d summand) and thermal energy
(summand 4) across the interface. The last summand determines the energy flux
from the condensed phase to the interface between the phases, and U 0s is the internal
energy per unit mass of the condensed medium, which is related to the specific heat
of the condensate-gas phase transition ϕ 0 by the relation U 0s = −ϕ 0 [17]. This ratio
is due to the fact that internal energy is determined by the binding energy between
atoms in a solid, and by definition it is a negative value. Therefore, the condition for
the balance of the energy flux will be considered in the form: