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Astron. Astrophys. 335, 370-378 (1998)

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4. Solution

To develop the solution we have to make a short excursus into the Lie group analysis of differential equations. A reader not familiar with this theory is referred to the textbooks (Ovsiannikov 1982; Ibragimov 1983; Olver 1986).

It is known that the system of n-dimensional hydrodynamic equations for a perfect fluid (1)-(3) with the arbitrary adiabatic index [FORMULA] admits the 4-parametric group of the point symmetry and can be represented by the following set of the infinitesimal operators of symmetry (Ovsiannikov 1982)

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

These operators constitute the linear space with the definite properties (commutator of any pair of operators can be expressed as a linear combination of the entire set) which is called the Lie algebra.

The existence of a group of symmetry of the system (1)-(3) in other words means that the system is invariant under the action of infinitesimal transformations given by (7)-(13). It is worthwile to note that a group generally defines the global transformations given by finite algebraic equations. In case of the finite transformations, finding a group of symmetry is not an easy problem since this procedure involves solving the nonlinear equations. It happened to be more convenient to deal with the infinitesimal transformations because they lead to linear determining equations which are always solvable. In this stage the appearance of algebra which determines the structure of a group is natural. A Lie group and its associated Lie algebra are intimately related so that any one of these allows to recover the other one and vice versa. In what follows we make no difference between them unless it is specially emphasized.

Each transformation we are dealing here with can be assigned a conventional physical meaning. In particular, the operators (7)-(8) describe the invariance of Eqs. (1)-(3) under translations in time and in space, the operators (9) describe invariance under rotations, the operators (10) present the Galilean transformations. The transformations allow to breed solutions once a particular one is known and, what is more important, to construct new classes of particular, so-called invariant, solutions reflecting the corresponding particular symmetrical properties of equations. The invariant solution meeting to X is, say, just a steady state solution, the solution invariant under rotations is a spherically symmetric one and so on. The self-similar solutions always arise as invariant solutions corresponding to the operators of stretching (Ovsiannikov 1982). In our case there are three operators of stretching, Z and Z, which reflect the possibility of scaling the physical independent and dependent variables. It is remarkable that due to the linear nature of algebra any particular linear combination of operators (7)-(13) belongs to this algebra too. This means that in the most general case one can construct the superposition of all 4 operators with 4 arbitrary constants and thus to obtain the 4-parametric class of particular invariant solutions.

If [FORMULA], the group is extended by an additional projective symmetry first found by Ovsiannikov (1958)

[EQUATION]

The invariant solution corresponding to X describes the asymptotic free expansion of fluid into vacuum (Ibragimov 1983).

Obviously, the solution we are interested in must coincide with the standard Sedov solution at early stages (t) when the overall expansion of background is not yet pronounced. The Sedov solution can be constructed as invariant solution meeting a certain superposition of Z, Z and Z. On the other hand, at late times (t) the background density [FORMULA] drops rapidly with time so that the blast wave dynamics asymptotically resembles expansion into a vacuum and thus should transit to the invariant solution generated by X. Both asymptotics must be continuously joined at intermediate times. It would then appear natural to search for a solution satisfying the described condition in the form

[EQUATION]

with some constants a, b and c to be determined later. Close inspection of the structure of Z shows that our suggestion may really justify itself because the coefficients of X are proportional to t or x and at t and x the contribution of X to Z is negligible while at large times, conversely, X dominates. Let us recall that the operator X exists only for [FORMULA]. Henceforward we assume n, [FORMULA] which corresponds to the monatomic ideal gas.

We are looking for the spherically symmetric solution. In spherical coordinates Z takes the form

[EQUATION]

The invariant solution is derived as follows. First we find the invariants J of the group solving the equation

[EQUATION]

Eq. (17) is a linear first order partial differential equation that can be solved with the help of the method of characteristics. It has four functionally independent solutions

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

To draw a parallel between the invariant and self-similar solutions it is worthwile to turn from the invariants J to the more habitual non-dimensional invariant variables [FORMULA], R, V and Z in the following way:

[EQUATION]

Here [FORMULA] is the dimensional constant to be determined later.

Any function of invariant is also invariant, so we can construct invariant solution suggesting that R, V and Z are functions of [FORMULA]. Expressing the physical variables in an explicit form we get

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where we have introduced the sound velocity c. In order that invariant solution (23)-(26) coincide with the self-similar Sedov solution at early times 0, we have to set

[EQUATION]

[EQUATION]

[EQUATION]

after which Eqs. (23)-(26) finally transform to

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The variable [FORMULA] in (30)-(33) plays the role of the independent invariant variable. The constant [FORMULA] is traditionally defined in such a way that [FORMULA] at the shock front; its rigorous definition is deferred until Eq. (60). Hence the range of the invariant variable is [FORMULA], where [FORMULA] is either a positive constant in case of a shell-like solution or zero if a solution extends up to the center of symmetry.

After the substitution of Eqs. (30)-(33) into the system (1)-(3) the latter one is reduced to the system of ordinary differential equations

[EQUATION]

[EQUATION]

[EQUATION]

which completely coincide with the system derived by Sedov for the self-similar solution (1959). This is not surprising because at small times t expressions (30)-(33) are consistent with the self-similar solution, therefore the reduced systems must coincide as well. But since the reduced system does not depend on time, it preserves its form for all moments t.

The boundary conditions for the problem are the standard conservation laws at the strong adiabatic shock jump

[EQUATION]

[EQUATION]

[EQUATION]

where the subscript 1 denotes the values taken immediately behind the shock front. The shock front velocity [FORMULA] can be found by setting [FORMULA] in Eq. (30) and differentiating this with respect to time:

[EQUATION]

Substituting [FORMULA], the preshock and postshock values taken from Eqs. (4), (6) and (31)-(33) into (37)-(39) we find that the boundary conditions are compatible with the invariant form and can be eventually written as

[EQUATION]

[EQUATION]

[EQUATION]

We again find that the boundary conditions coincide with those of Sedov for [FORMULA] (Sedov 1959). Since both the dynamic equations and the boundary conditions are the same as in the Sedov case we conclude that the solution of the problem (34)-(36), (40)-(42) is exactly the Sedov one and hence we may draw on the known solution from (Sedov 1959):

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Here the variable V for [FORMULA] varies in the range

[EQUATION]

In case 2 the solution bifurcates to a hollow density distribution and then V varies in the range

[EQUATION]

Altogether, the invariant solution is given by the formulas (30)-(33), (43)-(53).

An inspection of expressions (31), (33) shows that the profiles of density and pressure remain similar throughout the whole lifetime of the blast wave, their form coincides with the profiles of the Sedov self-similar solution. At the same time according to Eq. (32) the profile of velocity changes with time (Figs. 1 and 2) and tends to the homologous one, v, at t.


[FIGURE] Fig. 1. The normalized velocity profiles at different stages for a filled blast wave in a uniform ambient density

[FIGURE] Fig. 2. The same as in Fig. 1 but for a hollow blast wave in a nonuniform ambient density with [FORMULA]. Interior to r is void

It is important that in the matter reference frame the blast wave decelerates and eventually comes to rest: the relative jump of velocity

[EQUATION]

and the mass flux through the discontinuity

[EQUATION]

asymptotically vanish. This means that the blast wave asymptotically encloses the finite mass

[EQUATION]

However, at any finite moment t the mass flux is non-zero, and since the sound speed of preshock gas is identically zero by suggestion, the discontinuity remains a strong shock wave the whole time of expansion.

Let us consider how the energy of explosion E is redistributed in the flow with time. The total energy of the flow E occupied by a blast wave of radius R at any moment t is equal to a sum of the injected energy E and the kinetic energy E of unperturbed flow:

[EQUATION]

[EQUATION]

At the same time the total energy is combined as the kinetic, E, plus thermal, E, energy:

[EQUATION]

[EQUATION]

[EQUATION]

At the initial moment t the injected energy is allocated between the kinetic and thermal energies (in a uniform medium their initial fractions are E, E) while E. Thereafter the fraction of the thermal energy monotonically decreases with time down to zero; the energy of explosion E is thus expended for work done on the fluid transfer from the central area outwards:

[EQUATION]

One may wonder how Eqs. (54) and (56) containing the terms with different time dependencies can hold. It would not be difficult to show that this is the case. Substituting expressions (55), (57) and (58) into Eqs. (54) and (56), equating (54) to (56) and multiplying this by [FORMULA] gives us an equation quadratic in time. This equation must be an identity in t, so we obtain three different equations equating terms of equal order in t. The first equation for constant terms determines [FORMULA]

[EQUATION]

where

[EQUATION]

In case of the uniform medium, [FORMULA], Eq. (60) is reduced to the well-known (Sedov 1959)

[EQUATION]

the subscript [FORMULA] hereafter denotes parameters of the ambient fluid taken at the moment of explosion.

The remaining two equations for t and t, which with the help of the first one can be reduced to

[EQUATION]

[EQUATION]

hold automatically.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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