Simple waves of the two dimensional compressible Euler equations in magnetohydrodynamics

Accepted Manuscript Simple waves of the two dimensional compressible Euler equations in magnetohydrodynamics Jianjun Chen, Wancheng Sheng PII: DOI: Reference:

S0893-9659(17)30215-X http://dx.doi.org/10.1016/j.aml.2017.05.023 AML 5288

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Applied Mathematics Letters

Received date : 6 May 2017 Revised date : 22 May 2017 Accepted date : 22 May 2017 Please cite this article as: J. Chen, W. Sheng, Simple waves of the two dimensional compressible Euler equations in magnetohydrodynamics, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.05.023 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Applied Mathematics Letters 00 (2017) 1–6

Simple waves of the two dimensional compressible Euler equations in magnetohydrodynamicsI Jianjun Chen, Wancheng Sheng∗ Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China

Abstract In this paper, we are concerned with the simple waves for the two dimensional (2D) compressible Euler equations in magnetohydrodynamics. By using the sufficient conditions for the existence of characteristic decompositions to the general quasilinear strictly hyperbolic systems, we establish that any wave adjacent to a constant state must be a simple wave to the two dimensional compressible magnetohydrodynamics system for a polytropic Van der Waals gas and a polytropic perfect gas. c 2011 Published by Elsevier Ltd.

Keywords: Simple wave, two dimensional Euler equations, magnetohydrodynamics, Van der Waals gas, polytropic gas 2010 MSC: Primary: 35L65, 35J70, 35R35, Secondary: 35J65.

1. Introduction

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A flow in a region is called simple wave if it depends on a single parameter. Thus, the images of characteristics in the physical plane fall on the same characteristic in the phase plane. It plays an important role in the theories of gas dynamics and fluid mechanics. It can be used to describe and build up solutions of flow problems [4, 11, 6, 16, 12, 13, 3, 5, 9]. In the famous book “supersonic flow and shock waves”, Courant and Friedrichs [4] showed that a flow in a region adjacent to a constant state is always a simple wave. By using the characteristic decomposition method, Li etc. [11] generalized the well-known result on 2D steady potential flow to the 2D Compressible Euler Equations. Recently, Hu and Sheng [7, 8] generalized this result to non-reducible equations. In this paper, our aim is to extend the the well-known result to the two-dimensional compressible magnetohydrodynamics system. The ideal magnetofluid is that the viscosity coefficients and heat conductivity are neglected, and the electric conductivity tends to infinity in the fluid. An ideal unsteady compressible magnetohydrodynamic system can be governed by the following Euler equations [1, 10] I Supported

by NSF of China (11371240) and Shanghai Municipal Education Commission of Scientific Research Innovation Project (11ZZ84). author Email addresses: [email protected] (Jianjun Chen), [email protected] (Wancheng Sheng)

∗ Corresponding

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J.J.Chen, W.C.Sheng / Applied Mathematics Letters 00 (2017) 1–6

                                      

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20

→ − ρt + div ρU = 0, → →   → − − − − → − ρU + div ρU ⊗ → u + pI − µ rot H × H = 0, t ! →  → 1 → −2 − → −  − → − → − ρE + µ| H| + div ρU E + U p − div µ U × H × H = 0, 2 t → → − − → − H t − rot U × H = 0, → − div H = 0,

which implies that H/ρ is a constant along each stream line. The above relation is essentially an expression of the frozen-in law. In other words, each magnetic field line moves with the particles on this line and magnetic fields lines seem to be ”frozen” on the fluid moving with them, see [10]. Thus, we assume H to be a known function defined by H = κ0 ρ, κ0 is a positive constant. The magnetohydrodynamic equations (1) can be transformed into the following form   ρt + (ρu) x + (ρv)y = 0,       µ     (ρu)t + ρu2 + p + H 2 + (ρuv)y = 0,   x   2 (2) µ    2  (ρv)t + (ρuv) x + ρv + p + H 2 = 0,    y 2    µ    (ρE + H 2 )t + (ρuE + up + µuH 2 ) x + (ρvE + vp + µvH 2 )y = 0. 2 In this paper, we consider a kind of more realistic gas, the polytropic Van der Waals gas, for which the equations of state and internal energy are respectively A(S ) b1 − , (τ − a1 )δ+1 τ2

e(S , τ) =

A(S ) b1 − , τ δ(τ − a1 )δ

(3)

where A(S ) only depends on the entropy, τ = ρ1 is the specific volume, a1 and b1 are the positive constants, δ is a constant between 0 and 1 [2, 9]. For the case of b1 = 0, the equation of state (3) can be also used to modelize dusty gases, seen as perfect polytropic gases with dust pollution [14, 15]. In particular, if a1 = 0 and b1 = 0, we have the following equations in the case of polytropic perfect gases p(ρ, S ) = A0 (S )ρδ+1 ,

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(1)

→ −2 → − where ρ is the fluid density, p is the pressure, U = (u, v, w) is the velocity vector of the fluid, E = e + |U2 | is the → − specific total energy, e = e(ρ, S ) is the specific internal energy, S is the specific entropy, H = (H1 , H2 , H3 ) is the magnetic field vector, and the constant µ is the magnetic permeability. → − −u = (u, v, 0) and the We assume that the magnetic filed is orthogonal to the fluid velocity, i.e., H = (0, 0, H) and → variable (u, v, p, ρ, H) is independent of the space variable z. Then, by the first and the fifth equations of (1), we have ! ! ! H H H +u +v = 0, ρ t ρ x ρ y

p(τ, S ) =

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2

e=

p . δρ

(4)

The main objective of this paper is to establish the results that any hyperbolic state adjacent to a constant state must be a simple wave to two dimensional magnetohydrodynamics system. We establish this result for the steady isentropic magnetohydrodynamics system in section 3 and the pseudo-steady magnetohydrodynamics system in section 4. Further, our result extends to the full magnetohydrodynamics system by using the fact that entropy and vorticity are constant along the pseudo-flow characteristics in section 5.

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2. preliminaries We consider the general 2 × 2 quasilinear strictly hyperbolic system ! ! ! u a11 a12 u + = 0, v x a21 a22 v y 35

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(5)

where the coefficients ai j = ai j (x, y, u, v), i, j = 1, 2. If the differential equation (5) is reducible, that is, the coefficients ai j , i, j = 1, 2 only depend on u and v, Courant and Friedrichs had given the result that if in a flow region a section of a characteristic carries constant values of u and v, then in regions adjacent to this section the flow is a simple wave, [4]. For the general case, Hu and Sheng established a more general sufficient condition for the existence of characteristic decompositions [7, 8]. By using these decompositions, they extended the result of Courant and Friedrichs on reducible equations to the non-reducible equations. If the coefficients ai j (i, j = 1, 2) satisfy

where

 a21 A2 − a12 A3 = 0,      (a22 − a11 )A3 + a21 (A1 − A4 ) = 0,      (a − a )A + a (A − A ) = 0, 22 11 2 12 1 4

A1 = a11x + a11 a11y + a21 a12y ,

A2 = a12x + a12 a11y + a22 a12y ,

A3 = a21x + a11 a21y + a21 a22y ,

A4 = a22x + a12 a21y + a22 a22y ,

(6)

(7)

u and v are parameters in the coefficients ai j = ai j (x, y, u, v), i, j = 1, 2, then the flow in a region adjacent to a constant must be a simple wave in which the variables (u, v) are constant along a family of characteristics. Here, the characteristics may be a set of the non-straight curves. In particular, if the coefficients ai j (i, j = 1, 2) satisfy A1 = 0, A2 = 0, A3 = 0, A4 = 0, 45

(8)

then the hyperbolic state adjacent to a constant state must be a simple wave in which the variables (u, v) are constant along a family of characteristic that are straight lines. 3. Steady Magnetohydrodynamics System For smooth flow, the steady isentropic magnetohydrodynamics system of (2) can be written as the following form   ρu x + uρ x + ρvy + vρy = 0,        ρ(uu x + vuy ) + pρ + µHHρ ρ x = 0,         ρ(uv x + vvy ) + pρ + µHHρ ρy = 0,

with the equation of state

p(τ) = 50

A b1 − 2, δ+1 τ (τ − a1 )

where A > 0 is a constant. If the flow is irrotational, the system (9) can be transformed into        2 2 2 2    w − u u x − uv v x + uy + w − v vy = 0,    v x − uy = 0,

(9)

(10)

(11)

supplemented by Bernoulli’s law

  δ δ+1 u2 + v2 A (δ + 1)ρ − a1 ρ + − 2b1 ρ + b2 = Const., 2 δ(1 − a1 ρ)δ+1 3

(12)

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q p A(δ+1)τ2 2b1 where w2 = c2 + b2 is the magneto-acoustic speed with c = −τ2 p0 (τ) = (τ−a δ+2 − τ as the local sound speed 1) q and b = µκ02 ρ as the Alfven speed. Because the system (12) is a reducible equation, by [4] or [7, 8], we directly obtain the following theorem. 55

Theorem 1. The flow in a region adjacent to a constant state for the 2-D isentropic irrotational steady magnetohydrodynamics system (11) with a Van der Waals gas (10) or a polytropic ideal gas (4) is a simple wave in which the physical variables (u, v, w) are constant along a family of wave characteristics which are straight lines. 4. Pseudo-Steady Magnetohydrodynamics System

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For two-dimensional unsteady isentropic magnetohydrodynamics system of (2) in the self-similar plane (ξ, η) = ( xt , yt ), it can be written as   (ρU)ξ + (ρV)η + 2ρ = 0,        µκ02 2    2  ρU + p + ρ + (ρUV)η + 3ρU = 0, (13)  ξ  2    2    µκ     (ρUV)ξ + ρV 2 + p + 0 ρ2 + 3ρV = 0, η 2 where (U, V) = (u − ξ, v − η) is called the pseudo-flow velocity. If the flow is also irrotational, i.e., uη = vξ , then there exists a potential function ϕ such that ϕξ = U and ϕη = V. Thus, by the last two equations of system (13) we have the pseudo-Bernoulli’s law   A (δ + 1)ρδ − a1 ρδ+1 1 2 (U + V 2 ) + − 2b1 ρ + b2 + ϕ = Const., (14) 2 δ(1 − a1 ρ)δ+1 q where b = µκ02 ρ is called the Alfven speed. For irrotational smooth flow, system (13) can be written as  2 2 2 2    (w − U )uξ − UV(uη + vξ ) + (w − V )vη = 0, (15)    vξ − uη = 0, √ supplemented by the pseudo-Bernoulli’s law (14), where w = c2 + b2 is the magneto-acoustic speed. The system (15) can be transformed into matrix form as follows     w2 − V 2   u   u   −2UV     +  w2 − U 2 w2 − U 2    (16)  v  = 0. v ξ  −1 0 η Taking u and v as parameters in ρ, by (14), we get ρξ = 0 and ρη = 0. Then we have ! ! !   2UV 2UV w2 − V 2 −2UV    + − 2 A1 =    w2 − U 2 ξ w2 − U 2 w2 − U 2 η w − U2 η           2(v − η) w2 − (u − ξ)2 + 4(u − ξ)2 (v − η) 4(u − ξ)2 (v − η)  2(v − η)    = − = 0,   2 2 − 2   w − (u − ξ)2  2 2 2 2  w − (u − ξ) w − (u − ξ)   ! !   2 2   w2 − V 2 2UV  A = w −V − 2    w2 − U 2 ξ w2 − U 2 w2 − U 2 η           −2(u − ξ) w2 − (u − ξ)2  w2 − (v − η)2 2(u − ξ)    = + · 2 = 0,     2 2 2  w − (u − ξ) w − (u − ξ)2 2 − (u − ξ)2   w      A3 = 0,       A4 = 0. 4

(17)

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Thus the coefficients of the equations (15) satisfy the relations (6). We have the following result. 70

Theorem 2. The flow in a region adjacent to a constant state in the self-similar plane of the pseudo-steady magnetohydrodynamics system (15) for a Van der Waals gas (10) or a polytropic ideal gas (4) is a simple wave in which the physical variables (u, v, w) are constant along a family of wave characteristics which are straight lines. 5. Full Magnetohydrodynamics System For smooth solutions, (2) and (3) in the self-similar plane we can be written as

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 ∂ s ρ + ρ(uξ + vη ) = 0,        ρ∂ s u + pξ + µHHρ ρξ = 0,      ρ∂ s v + pη + µHHρ ρη = 0,       1 − a1 ρ    ∂ s p + uξ + vη = 0,  p(δ + 1) + (2a1 + δ − 1)b1 ρ2

(18)

where ∂ s = (u − ξ)∂ξ + (v − η)∂η , which we call pseudo-flow characteristic direction, as opposed to pseudo-wave characteristic directions. By direct calculation, we find that    ∂ s p + b1 ρ2 ((1 − a1 ρ)/ρ)δ+1 = 0. (19) Thus we get that entropy S is constant along pseudo-flow characteristic direction. Denote $ = v x − uy , in view of (2), we have     $t + (u$) x + (v$)y + (py + µHHρ ρy )/ρ − (p x + µHHρ ρ x )/ρ = 0. x

It satisfies

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∂ s ($/ρ) = 0,

y

(20) (21)

which implies that the vorticity vanishes everywhere for a region whose pseudo-flow lines come from a constant state. So the region is irrotational and isentropic. Thus we can obtain the following theorem by using the relations in the section 4. Theorem 3. If pseudo-flow characteristics of the full magnetohydrodynamics system with a Van der Waals gas (3) or a polytropic ideal gas (4) extend into a constant state, and then adjacent to the constant state in the self-similar plane is a simple wave in which the physical variables (u, v, p, w) are constant along a family of wave characteristics which are straight lines. References

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