PDF《gas》

2 Basic Equations We consider an unsteady one-dimensional motion in a vibrationally relaxing gas with Van der Waals equation of state. The gas molecules have only one lagging internal mode (i.e., vibrational relaxation) and the various transport e?ects are negligible. The basic equations can be written as (see [7]) ?ρ ?t + ρ ?u ?x + u ?ρ ?x + mρu x = 0, ?u ?t + u ?u ?x +

1 ρ ?p ?x = 0, (2.1) ?p ?t + u ?p ?x + γp

1 ? bρ ?u ?x + mu x + (γ ? 1)ρQ = 0, ?σ ?t + u ?σ ?x ? Q = 0, where ρ is the density, u the particle velocity, p the pressure, σ the vibrational energy, t the time, x the spatial coordinate, b the Van der Waals excluded vol- Asymptotical Solutions for a Vibrationally Relaxing Gas

425 ume, m=0,

1 and

2 correspond, respectively, to planar, cylindrical and spheri- cal symmetry. The quantity Q is the rate of change of vibrational energy that depends on the states ρ, p and σ, and is given by Q = ? σ(p, ρ) ? σ /τ, where ? σ is the equilibrium value of σ de?ned as ? σ = σ0 + cR(T ? T0), T is the translational temperature, R is the speci?c gas constant, su?x

0 refers to the initial rest condition, and the quantities τ and c, which are respectively the relaxation time and the ratio of vibrational speci?c heat to the speci?c gas constant, are assumed to be constant. The Van der Waals equation of state is taken to be of the form p(1 ? bρ) = ρRT, where b is the Van der Waals excluded gas volume. We denote a = (γp/(ρ(1 ? bρ)))1/2 as the speed of sound, and γ as the speci?c heat ratio of the gas. The equations (2.1) may be written in the matrix form as ?U ?t + A(U) ?U ?x + B(U) = 0, (2.2) where U and B(U) are the column vectors de?ned by U=(ρ, u, p, σ)T , B(U)= mρu x , 0, mγpu (1?bρ)x +(γ?1)ρ ? (σ?σ) τ , ? ? (σ?σ) τ T , where ? σ = σ0 + c p(1 ? bρ) ρ ? p0(1 ? bρ0) ρ0 . A(U) is the

4 x

4 matrix having the components Aij , and the nonzero ones are: A11 = A22 = A33 = A44 = u, A12 = ρ, A23 = 1/ρ, A32 = γp/(1 ? bρ). System (2.2) being strictly hyperbolic admits four families of characteristics, among them two represent waves propagating in ± x directions with the speed u ± a. The remaining two families form a set of double characteristics repre- senting entropy waves or particle paths propagating with velocity u. We consider waves propagating into an initial background state U0 = (ρ0, 0, p0, σ0)T . The characteristic speeds at U = U0 are given by λ1 = 0, λ2 = 0, λ3 = a0 and λ4 = ?a0. The subscript

0 refers to evaluation at U = U0, and is synonymous with the state of equilibrium. Math. Model. Anal., 14(4):423C434, 2009.

426 R. Arora

3 Interaction of High Frequency Waves We denote the left and right eigenvectors of A0 associated with the eigenvalue λi by L(i) and R(i) . These eigenvectors satisfy the normalization condition L(i) R(j) = δij,

1 ≤ i, j ≤ 4, where δij represents the Kronecker delta. These eigenvectors are obtained as L(1) = 1, 0, ?

1 a2

0 ,

0 , R(1) = (1, 0, 0, 0)T , L(2) = 0, 0, 0,

1 , R(2) = (0, 0, 0, 1)T , L(3) = 0, ρ0 2a0 ,

1 2a2

0 ,

0 , R(3) = (1, a0 ρ0 , a2 0, 0)T , (3.1) L(4) = 0, ? ρ0 2a0 ,

1 2a2

0 ,

0 , R(4) = (1, ? a0 ρ0 , a2 0, 0)T . We look for asymptotic solution for (2.2) as ? →

0 of the form U ? U0 + ? U1(x, t, ? θ) + ?2 U2(x, t, ? θ) + O(?3 ), (3.2) where U1 = (U11, U12, U13, U14)T is a smooth bounded vector, and vector U2 is bounded in (x, t) in a certain bounded region of interest having at most sub- linear growth in θ as θ → ±∞. Here ? θ = (θ1, θ2, θ3, θ4) represents the fast variables characterized by the functions φi as θi = φi/?, where φi,