Appl. Math. Mech. -Engl. Ed. 31(7), 827–838 (2010) DOI 10.1007/s10483-010-1317-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 2010
Applied Mathematics and Mechanics (English Edition)
Simple waves for two-dimensional compressible pseudo-steady Euler system∗ Geng LAI (
),
Wan-cheng SHENG ()
(Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China) (Communicated by Xing-ming GUO)
Abstract A simple wave is defined as a flow in a region whose image is a curve in the phase space. It is well known that “the theory of simple waves is fundamental in building up the solutions of flow problems out of elementary flow patterns” see Courant and Friedrichs’s chassical book “Supersonic Flow and Shock Waves”. This paper mainly concerned with the geometric construction of simple waves for the 2D pseudo-steady compressible Euler system. Based on the geometric interpretation, the expansion or compression simple wave flow construction around a pseudo-stream line with a bend part are constructed. It is a building block that appears in the global solution to four contact discontinuities Riemann problems. Key words self-similar Euler system, simple wave, generalized characteristic analysis, pseudo-stream line, sonic circle, sonic edge Chinese Library Classification O175.27, O175.29, O351.3 2000 Mathematics Subject Classification 35L65, 35J70, 35R35
1
Introduction The two dimensional isentropic Euler system takes the form ⎧ ⎪ ⎨ ρt + (ρu)x + (ρv)y = 0, (ρu)t + (ρu2 + p)x + (ρuv)y = 0, ⎪ ⎩ (ρv) + (ρuv) + (ρv 2 + p) = 0, t x y
(1)
where the variables (u, v), ρ, and p represent the velocity, the density, and the pressure, respectively. For polytropic gases, the equation of state is p(ρ) = Aργ ,
(2)
where the adiabatic exponent γ is a constant between 1 and 53 , and A is a positive constant. We are interested in the so-called self-similar (or pseudo-steady) flow, that means the flow depends only on the self-similar variables ξ = xt and η = yt . By self-similar transformation, ∗ Received Mar. 20, 2010 / Revised May 26, 2010 Project supported by the National Natural Science Foundation of China (No. 10971130) and the Shanghai Leading Academic Discipline Project (No. J50101) Corresponding author Wan-cheng SHENG, Professor, Ph. D., E-mail:
[email protected]
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system (1) can be written into the form ⎧ ⎪ ⎨ (ρU )ξ + (ρV )η = −2ρ, (ρU 2 + p)ξ + (ρU V )η = −3ρU, ⎪ ⎩ (ρU V )ξ + (ρV 2 + p)η = −3ρV,
(3)
where (U, V ) = (u − ξ, v − η) is called pseudo-flow velocity[1–3] . For smooth flow, system (3) can be reduced to ⎧ (ρU )ξ + (ρV )η + 2ρ = 0, ⎪ ⎪ ⎪ c2 ⎪ ⎨ = −U, U Uξ + V Uη + (4) γ−1 ξ ⎪ ⎪ 2 ⎪ c ⎪ ⎩ U Vξ + V Vη + = −V, γ−1 η γp where c = ρ denotes the speed of sound. If the flow is also irrotational, i.e., uy = vx ⇒ uη = vξ , we can introduce a potential function φ(ξ, η), such that φξ = U,
φη = V.
(5)
Hence, from the last two equations of system (4), we obtain pseudo-Bernoulli’s law U2 + V 2 c2 + + φ = const. 2 γ−1
(6)
Therefore, the isentropic irrotational self-similar flow can be governed by (c2 − U 2 )uξ − U V (uη + vξ ) + (c2 − V 2 )vη = 0, uη − vξ = 0,
(7)
supplemented by pseudo-Bernoulli’s law (6). For pseudo-steady flow, the pseudo-stream line is defined as a continuous line drawn through the pseudo-steady flow so that it has the direction of the pseudo-flow velocity (U, V ) at every point. By computation, we have x˙ = ξ˙ = t y˙ η˙ = = t
x˙ x 1 − 2 = (u − ξ) = t t t y˙ y 1 − 2 = (v − η) = t t t
U , t V , t
(8) (9)
where a dot “ · ” denotes the differentiation with respect to the time t in Lagrange’s representation. From (8) and (9), we have that in the (ξ, η) plane particles move in the direction of the pseudo-flow velocity. Therefore, if in a flow region the density is increasing (or decreasing) along each pseudo-stream line, then the flow is called compressed (or expanded). A simple wave is defined as a flow in a region whose image is a curve[4] . It was shown that two-dimensional steady simple wave flow region is covered by a one-parametric family of straight wave characteristic lines, along each of which u, v and consequently c, p, ρ remain constant, and a nonconstant state of flow adjacent to a region of constant state is always a simple wave[5]. In a recent paper[6] , the authors proved that for two-dimensional self-similar flow, a nonconstant state of flow adjacent to a region of constant state is always a simple. Simple wave interactions were considered extensively recently[7–13] .
Simple waves for two-dimensional compressible pseudo-steady Euler system
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In this paper, we are mainly concerned with two-dimensional pseudo-steady simple waves. Our main results are as follows: Geometrically, if a two-dimensional pseudo-steady simple wave and its images are represented in the same coordinate, the images of the simple wave can be represented by a hodograph curve
Λ: ξ = u(s), η = v(s), s1 < s < s2 equipped with sonic circles Cs centered at u(s), v(s) with radius c(s). By differentiating pseudo-Bernoulli’s law
2
2 we found that c(s) satisfies 4 c (s) = (γ − 1)r (s) in which r(s) denotes the arc-length of the hodograph from some point on the hodograph curve. The simple wave flow region is covered by a one-parametric of straight wave characteristic lines C(s) along each of which u, v and c are constant: u = u(s), v = v(s), c = c(s). Each straight wave characteristic line C(s) is tangent to the sonic circle Cs , and its direction is perpendicular to the direction of the hodograph curve Λ at the corresponding point. We also prove that the straight characteristics of an expansion simple waves can extend up to its sonic edge without intersecting with each other, but the straight characteristics of a compression simple wave will intersect with each other as they extend toward its sonic edge which may cause the formation of shock wave. Based on the geometric interpretation, we construct simple wave flow construction around a pseudo-stream line with a bend part, this flow pattern appears as a building block in the global solution to four contact discontinuities Riemann problems[14] .
2
Generalized characteristic analysis
System (7) can be written into the following matrix form: 2
c − U 2 −U V −U V c2 − V 2 0 u u + = . (10) 0 0 −1 v ξ 1 0 v η 2
c − U 2 −U V −U V c2 − V 2 Let B = and D = , the eigenvalues of system (7) 0 −1 1 0 are determined by | λB − D |= 0, that is, (V − λU )2 − c2 (1 + λ2 ) = 0.
(11)
Then, we have
c2 (U 2 + V 2 − c2 ) . (12) λ = λ± = U 2 − c2 Thus, system (7) is hyperbolic if and only if the flow is pseudo-supersonic, i.e., U 2 + V 2 > c2 . From now on, we shall confine ourselves to pseudo-supersonic flow. Then, there are two families of wave characteristics defined by dη = λ± , (13) dξ which are also called C± characteristics. Therefore, by simple computation system (7) can be written into the following characteristic form: ∂+ u + λ− ∂+ v = 0, (14) ∂− u + λ+ ∂− v = 0, UV ±
where ∂± = ∂ξ + λ± ∂η . Once a fixed pseudo-supersonic flow of (6) and (7) is given, then the C+ and C− characteristics may be represented in the form β(ξ, η) = const. and α(ξ, η) = const., respectively, and form a curvilinear net. By introducing characteristic parameters α and β we have the following characteristic equations for system (14): C+ : ηα = λ+ ξα ,
C− : ηβ = λ− ξβ ,
Γ+ : uα + λ− vα = 0,
Γ− : uβ + λ+ vβ = 0,
(15) (16)
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where Γ± characteristic lines are the projections in the (u, v)-plane of the images of the C± characteristic lines. From (15) and (16), we have uα ξβ + vα ηβ = 0,
uβ ξα + vβ ηα = 0,
(17)
which express an important fact that: If (u, v) and (ξ, η) are represented in the same coordinate plane, the directions of the C-characteristic lines of one kind are perpendicular to Γ-characteristic lines of the other kind. Or more precisely, the directions of C+ and Γ− and of C− and Γ+ through corresponding points (ξ, η) and (u, v) are perpendicular. By (11), we have (U, V ) · (λ, −1) 2 c2 = , (18) |(λ, −1)| which means that the component of the pseudo-flow velocity normal to the direction of the C characteristic line is equal to the speed of sound. From (18), we can immediately have if along a C characteristic line u, v and c remain constant, then this characteristic line is a straight line. Since in the (ξ, η) plane there is not time axis, in order to interpret the property of wave propagation it is necessary to introduce the concept of characteristic direction. The direction of the C± characteristic lines is defined as the tangent direction which forms with the pseudo-flow velocity vector (U, V ) an acute angle ω. Geometrically, the C+ characteristic direction forms with the pseudo-flow direction the angle ω in the positive sense, or in the clockwise direction, and the C− characteristic direction forms with the pseudo-flow direction the angle ω in the negative sense, or in counterclockwise directions (see Fig. 1).
Fig. 1
C characteristic and characteristic direction in the (ξ, η) plane
By computation, we have 2
2
c2 = q 2 sin2 ω,
(19)
2
in which q = U + V . The angle ω is called the pseudo-Mach angle, and the quantity M=
q 1 = c sin ω
(20)
is called the pseudo-Mach number of the pseudo-flow. For pseudo-sonic flow, which q = c or M = 1, we have ω = 90◦ and the direction of the pseudo-flow velocity is perpendicular to the characteristic direction. By differentiating pseudo-Bernoulli’s law (6), we have (u − ξ)∂± u + (v − η)∂± v + ∂±
c2 = 0. γ−1
(21)
By using characteristic parameters, we have (u − ξ)uα + (v − η)vα +
c2 =0 γ−1 α
(22)
Simple waves for two-dimensional compressible pseudo-steady Euler system
and (u − ξ)uβ + (v − η)vβ +
831
c2 = 0. γ−1 β
(23)
We now introduce two sign functions s+ (α, β) and s− (α, β) such that ⎧ ⎨ 1, if (u − ξ, v − η) · (uα , vα ) > 0, s+ (α, β) = 0, if (u − ξ, v − η) · (uα , vα ) = 0, ⎩ −1, if (u − ξ, v − η) · (uα , vα ) < 0
(24)
⎧ ⎨ 1, if (u − ξ, v − η) · (uβ , vβ ) > 0, s− (α, β) = 0, if (u − ξ, v − η) · (uβ , vβ ) = 0, ⎩ −1, if (u − ξ, v − η) · (uβ , vβ ) < 0.
and
(25)
Therefore, by (16) and (18) we have cα = −
γ−1 2 s+ uα + vα2 , 2
Hence, we have
cβ = −
γ−1 2 s− uβ + vβ2 . 2
(26)
2 (γ − 1)2 2 r (s) , c (s) = 4
(27)
where r(s) denotes the arc-length of Γ characteristic line.
3
Simple wave solutions A simple wave is defined as a flow in a region whose image is curve u(ξ, η) = u(s(ξ, η)),
v(ξ, η) = v(s(ξ, η)),
c(ξ, η) = v(s(ξ, η)),
s 1 < s < s2 .
(28)
Inserting it into (14) and (21), we have (u (s) + λ∓ v (s))∂± s = 0,
(29)
which expresses that the simple wave flow region is covered by a one parametric family of straight characteristics, along each of which u, v, and c remain constant. It is of interests to consider the following question: In order to obtain a simple wave solution what data are suitable to be prescribed along a given curve l: ξ = ξ(s), η = η(s). One possibility is that let u = u(s), v = v(s), c = c(s) along this curve, such that (u(s) − v(s))2 + (v(s) − η(s))2 > c2 ,
(30)
u (s) + λ− (s)v (s) = 0,
(31)
(u(s) − ξ(s))u (s) + (v(s) − η(s))v (s) + where λ− =
(u − ξ)(v − η) −
2c(s) c (s) = 0, γ−1
c2 (u − ξ)2 + (v − η)2 − c2 (u − ξ)2 − c2
(32)
.
(33)
Let transformation T : (s, σ) → (ξ, η) be ξ = ξ(s) + σ, η = η(s) + σλ− (s). If η (s) = ξ (s)λ− (s), ∂(ξ,η) = ξ (s)λ− (s) − η (s) − σλ− (s) does not vanish on the curve l. Thus, then the Jacobin ∂(s,σ)
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the transformation T is smoothly invertible in a neighborhood N (l) of l. For any (ξ, η) ∈ N (l), we define (34) (u, v, c)(ξ, η) = (u, v, c)(s(ξ, η)), which means that u, v, and c are constant along the straight line C− (s) which issuing from the point (ξ(s), η(s)) with slop dη dξ = λ− (s) (see Fig. 2). We shall show that (u, v, c)(ξ, η) defined by (34) is a simple wave solution of (6) and (7).
Fig. 2
Simple wave adjacent to an arbitrary curve l
Theorem 3.1 (u, v, c)(ξ, η) defined by (34) is a simple wave solution of (6) and (7). Proof It suffices to prove (14) and (21). From (33) we have that the straight line C− (s) is tangent to the circle Cs centered at (u(s), v(s)) with radius c(s), and λ− ≡ λ− (s) along each straight line C− (s). So, we have ∂− u = ∂− v = ∂− c = 0. Therefore, we obtain ∂− u + λ+ ∂− v = 0,
(u − ξ)∂− u + (v − η)∂− v + ∂−
c2 = 0. γ−1
By computation, we have ∂+ u + λ− ∂+ v = uξ + λ+ uη + λ− (vξ + λ+ vη )
= u (s) + λ− (s)v (s) (sξ + λ+ sη ) = 0.
From (31), we see that the straight line C− (s) is perpendicular to the vector u (s), v (s) . Thus, (u − ξ)u (s) + (v − η)v (s) is a constant along each straight line C− (s). Hence, we have (u − ξ)∂+ u + (v − η)∂+ v +
2c ∂+ c γ−1
= (u − ξ)u (s) + (v − η)v (s) +
2c c (s) (sξ + λ+ sη ) γ−1 2c = (u(s) − ξ(s))u (s) + (v(s) − η(s))v (s) + c (s) (sξ + λ+ sη ) γ−1 = 0. The proof of this theorem is complete.
Simple waves for two-dimensional compressible pseudo-steady Euler system
833
Similarly, if (31) is replaced by u (s) + λ+ (s)v (s) = 0, where λ+ =
(u − ξ)(v − η) +
c2 (u − ξ)2 + (v − η)2 − c2 (u − ξ)2 − c2
(35)
,
(36)
we can obtain a simple wave solution in a neighborhood of l, and the straight characteristics are C+ characterisitcs.
4
Geometric construction of simple waves
From the preceding discussions, we know that the image of a simple wave is a curve u = u(s), v = v(s), c = c(s), s1 < s < s2 , and along the curve there holds γ−1 c (s) = u (s)2 + v (s)2 (37) 2 or γ−1 c (s) = − u (s)2 + v (s)2 . (38) 2 Each straight characteristic line is mapped into one point of this curve. If a simple wave and its image are represented in the same coordinate the images of a simple wave by the curve Λ: ξ = u(s), η = v(s), s1 < s < s2 equipped with sonic circles Cs centered at (u(s), v(s)) with radius c(s). Each straight characteristic C(s) is tangent to the sonic circle Cs , and its direction is perpendicular to the direction of Λ at the corresponding point. If along each straight characteristic line of a simple wave (u(s)− ξ)u (s)+ (v(s)− η)v (s) > 0, then by computation we know that the straight characteristic line C− (s) (or C+ (s)) can be described in the form: (39) ξ = ξ(s) + tv (s), η = η(s) − tu (s), where
c(s)u (s) ξ(s) = u(s) − , u (s)2 + v (s)2
c(s)v (s) η(s) = v(s) − , u (s)2 + v (s)2
(40)
and t < 0 (or t > 0) is an abscissa along the straight line s = const. We call the curve ξ = ξ(s), η = η(s), s1 < s < s2 the sonic edge of the simple wave. On the contrary, if along each straight characteristic line of a simple wave (u(s) − ξ)u (s) + (v(s) − η)v (s) < 0, then the straight characteristics C− (s) (or C+ (s)) can be described in the form: ξ = ξ(s) − tv (s), where
c(s)u (s) ξ(s) = u(s) + , u (s)2 + v (s)2
η = η(s) + tu (s),
(41)
c(s)v (s) η(s) = v(s) + u (s)2 + v (s)2
(42)
and t < 0 (or t > 0). In Fig. 3, we show an expansion simple wave (along each pseudo-stream line c, and consequently p and ρ are decreasing) whose straight characteristics are C− characteristic. In Fig. 4, we show a compression simple wave (along each pseudo-stream line c, and consequently p and ρ are increasing). Next, we shall show that along the direction of straight characteristics the straight characteristics of an expansion simple can extend up to the sonic edge without intersecting with each other, but the straight characteristics of a compression simple wave will intersect with each other before the sonic edge which may cause the formation of shock wave[15].
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Fig. 3
An expansion simple wave flow pattern in the (ξ, η) plane
Fig. 4
Compression simple wave flow pattern in the (ξ, η) plane
Theorem 4.1 Along the direction of straight characteristics the straight characteristics of an expansion simple wave can extend up to the sonic edge without intersecting with each other. Proof It suffices to consider the case that the straight characteristics converge as they extend toward the sonic edge. Therefore, the curve Λ: ξ = u(s), η = v(s), s1 < s < s2 is a convex or concave curve. Without loss of generality, we assume that the straight characteristic lines are C− (s), s1 s s2 , and along each straight characteristic line (u(s)−ξ, v(s)−η)·(u (s), v (s)) > 0 . Therefore, by the previous results we know that along each pseudo-stream line s is increasing. Let C− (s ) and C− (s ), s < s be any two given straight characteristics, and α be the angle that C− (s ) forms with C− (s ) in clockwise sense (see Fig. 5). It suffices to prove that if α is less than π then C− (s ) and C− (s ) do not intersect with each other before they reach the sonic edge. If we assume, contrary to it, that
C− (s ) and C− (s ) intersect
at a point A before the sonic edge. Then, let B be the point u(s ), v(s ) , C be the point u(s ), v(s ) , and E be a point on C− (s ) such that CE is perpendicular to C− (s ), and D and F be points on C− (s ) such that BD and CF are perpendicular to C− (s ). CE intersect with C− (s ) at a point G. Since (u(s) − ξ, v(s) − η) · (u (s), v (s)) > 0 along the straight characteristic C− (s ), s < s and α < π, we have |BD| < |CF |. (43) Therefore, we have c(s ) = |CE| > |CG| > |CF | > |BD| = c(s ), which is contradict to the fact that c(s ) < c(s ). The theorem is proved.
(44)
Simple waves for two-dimensional compressible pseudo-steady Euler system
Fig. 5
835
The straight characteristics of an expansion wave extend up to the sonic edge
Theorem 4.2 For a simple wave solution, if at s0 (s1 < s0 < s2 ) we have (u (s0 ), v (s0 )) = (0, 0), then the straight characteristic line C(s0 ) does not tangent to the sonic edge. we assume that along the straight characteristic line
Proof Without
loss of generality, u(s0 ) − ξ u (s0 ) + v(s0 ) − η v (s0 ) > 0, then in a neighborhood of s0 the sonic edge can be prescribed by (40). By computation we have
γ − 1 (45) ξ (s0 ), η (s0 ) · u (s0 ), v (s0 ) = 1 + u (s0 )2 + v (s0 )2 . 2
vector ξ (s0 ), η (s0 ) is not perpendicular Because γ is a constant between 1 and 53 , so the
to the vector u (s0 ), v (s0 ) . Since the vector u (s0 ), v (s0 ) is perpendicular to the straight characteristic line C(s0 ), the straight characteristic line C(s0 ) does not tangent to the sonic edge. Theorem 4.3 Along the direction of straight characteristics the straight characteristics of a compression simple wave will intersect with each other before the sonic edge. Proof Without loss of generality, we assume that the straight characteristics are C+ and along each pseudo-stream line s is increasing. We assume, contrary to the theorem, that the straight characteristics can extend up to the sonic edge without intersecting with each other, that Since it is a compression simple wave, then there exists s0 ∈ (s1 , s2 ) such
see Fig. 6.
u (s ), v (s ) does not vanish, and along the straight characteristic line C (s ), u(s + 0 0) −
0 0 ξ u (s0 ) + v(s0 ) − η v (s0 ) < 0. By Theorem 4.2, we know that C+ (s0 ) does not tangent to the sonic edge, then they exists a small δ > 0 such that {(ξ(s), η(s)) | 0 < s0 − s < δ} lie outside the sonic circle Cs0 . Let A, B, C and D be the points u(s0 ), v(s0 ) , u(s1 ), v(s1 ) ,
ξ(s0 ), η(s0 ) and ξ(s
1 ), η(s1 ) , respectively. Since along the straight characteristic line C+ (s0 ) u(s0 ) − ξ u (s0 ) + v(s0 ) − η v (s0 ) < 0, if s0 − s1 is small enough there has u(s0 ) > u(s1 ), and hence c(s0 ) = |AD| < |BC| = c(s1 ), which is contradict to the fact that c(s0 ) > c(s1 ). We then conclude the theorem.
5
Simple waves around a pseudo-stream line with a bend part
In this section we shall consider the following question: For a given pseudo-stream line with a straight part up to the point A, then bends along a smooth bend. The flow is of constant state (u0 , v0 , p0 , ρ0 ) in a region adjacent to the straight part of the pseudo-stream line before A. From [6] we know that the flow adjacent to the bend part is a simple wave, then how to construct
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the simple wave flow pattern around the bend? We confine ourselves to the case of pseudosupersonic flow, so we assume that at the point A there holds (u0 − ξA )2 + (v0 − ηA )2 > c20 .
Fig. 6
The straight characteristics of a compression simple wave intersect with each other
Let the bend part of the pseudo-stream line be η = f (ξ), ηA = f (ξA ), and f be a smooth function. Suppose that f < 0, f < 0. We now construct the simple wave solution in detail. A A The constant state flow region can be terminated by a straight characteristic line C+ or C− , A A 2 2 2 C+ and C− can be determined by A and the sonic circle (ξ − u0 ) + (η − v0 ) = c0 . To start off with we discuss the first possibility. We assume that along the bend part pseudo-stream line u = u(ξ), v = v(ξ), c = c(ξ). Hence, by the preceding results we know that u(ξ), v(ξ) and c(ξ) satisfy ⎧ u (ξ) + λ+ (ξ)v (ξ) = 0, ⎪ ⎪ ⎪ ⎨ v(ξ) − f (ξ) = f (ξ), (46) u(ξ) − ξ ⎪ ⎪ ⎪ γ − 1 ⎩ c (ξ) = − u (ξ)2 + v (ξ)2 , 2 where
2
2 u(ξ) − ξ v(ξ) − f (ξ) + c2 (ξ) u(ξ) − ξ + v(ξ) − f (ξ) − c2 (ξ) . (47) λ+ (ξ) =
2 u(ξ) − ξ − c2 (ξ) Therefore, in order to specify the simple wave determined by the constant state and the pseudo-stream line, we only need to solve the ordinary system (46) with the initial value (u, v, c)(ξA ) = (u0 , v0 , c0 ).
(48)
From the second equation of the system (46) we have v − f = (u − ξ)f .
(49)
Inserting it into (47), we can get λ+ (ξ) =
(u − ξ)2 f +
c2 ((u − ξ)2 (1 + f 2 ) − c2 ) . (u − ξ)2 − c2
(50)
Differentiating (49) with respect to ξ, we then obtain v = u f + (u − ξ)f .
(51)
Simple waves for two-dimensional compressible pseudo-steady Euler system
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So, the first equation of system (46) becomes
u + λ+ u f + (u − ξ)f = 0,
(52)
i.e., u = −
(u − ξ)f λ+ = F1 (ξ, u, c). 1 + λ+ f
(53)
The third equation of system (46) becomes γ−1 c =− 2
(u − ξ)f λ 2 +
1 + λ+ f
(u − ξ)f f λ+ 2 + (u − ξ)f − = F2 (ξ, u, c). 1 + λ+ f
(54)
Since (u(ξA ) − ξA )2 + (v(ξA ) − f (ξA ))2 − c(ξA )2 > 0
(55)
at the point ξ = ξA , the vector (1, f (ξA )) is parallel to the vector (u0 − ξA , v0 − f (ξA )), so by the results of characteristic analysis we have 1 + λ+ (ξA )f (ξA ) > 0.
(56)
Hence, F1 and F2 are continuously differentiable in a small neighborhood of (ξA , u0 , c0 ). Therefore, according to the classical existence theorem for ordinary system we know that there exists a δ > 0 such that the initial value problem (46) and (48) is uniquely solvable in (ξA , ξA + δ). Obviously, by the extending theorem, we have that the solution can be extended to a point ξB at which u(ξB ), v(ξB ), and c(ξB ) satisfy (u(ξB ) − ξB )2 + (v(ξB ) − f (ξB ))2 − c(ξB )2 = 0
(57)
1 + λ+ (ξB )f (ξB ) = 0
(58)
and
(see Fig. 7). From the third equation of system (46), it is easy to see that the simple wave solution obtained is an expansion simple wave solution. Similarly, if the zone of constant state A , the simple wave around the bend part is a compressive simple wave. is terminated by C−
Fig. 7
Simple wave flow construction around a pseudo-stream line with a bend part
Acknowledgements Geng LAI thanks the department of mathematics at the Pennsylvania State University for hospitality during his one year visit as an exchange student. Both authors thank Prof. Tong ZHANG and Prof. Yu-xi ZHENG for sincere discussion.
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Geng LAI and Wan-cheng SHENG
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