Solvability of a problem for transonic flow with a local supersonic region

Kuzmin, Alexander

doi:10.1007/pl00001450

Cited by 14 publications

(10 citation statements)

References 13 publications

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“…We note that there have been many studies on the transonic problems (e.g., [1], [3], [11], [12], [13] and references therein). In particular, we mention some works which are related to this paper.…”

Section: §1 Introduction and Main Resultsmentioning

confidence: 99%

Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

Xu¹,

Yin²

2010

Nagoya Math. J.

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Abstract. In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed. §1. Introduction and main resultsIf there is a uniform supersonic flow (ρ 0 , 0, 0, q 0 , P 0 ) which comes from minus infinity, and the flow hits the circular cone x 2 1 + x 2 2 = b 0 x 3 along the axis x 3 -direction, when the vertex angle 2 arctan b 0 of the cone is less than a critical value θ * , the following conic shock problems are illustrated in [6, pages 313-314, 414]: it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there will appear a weak or a strong self-similar shock attached at the vertex of the cone in terms of the different

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Section: §1 Introduction and Main Resultsmentioning

confidence: 99%

Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

Xu¹,

Yin²

2010

Nagoya Math. J.

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“…by virtue of (14). Therefore, the maximum of the function Ψ(x, y) is attained on the boundary of the domain D, i.e., either on the curve Γ or on the segment F 1 F 2 of the x-axis.…”

Section: Uniqueness Theoremmentioning

confidence: 98%

“…where The independence of the integral on the right-hand side in (16) on the integration path can be justified by a straightforward verification of the identity [14] Ψ xy ≡ Ψ yx .…”

Section: Uniqueness Theoremmentioning

confidence: 99%

“…Let ϕ(x, y) ∈ C 4 D be the potential of a transonic flow such that the function k(x, y) = (γ + 1) (1 − ϕ x ) satisfies inequalities (12)- (14) and inequality (15) is valid along the curve Γ with endpoints at the points F 1 and F 2 of the x-axis. Let boundary conditions (6)-(10) be given, where E 1 and E 2 are the points of intersection of the x-axis with the characteristic lines issuing from the point E of the sonic line, let the function k(x, 0) attain a minimum at the point C of the segment E 1 E 2 , and let the coefficient r(x) occurring in condition (9) be given by the expression r(x) = k x / k y + kk 2…”

Section: Uniqueness Theoremmentioning

confidence: 99%

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A modified Frankl-Morawetz problem on a transonic flow past an airfoil

Kuzmin

2004

Diff Equat

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“…The so-called polytropic flow means that the pressure P and the density ρ of the gas are described by the state equation P = Aρ γ with 1 < γ < 3. On the other hand, Z. Xin and H. Yin in [14] have established the global existence of a multidimensional conic shock for the uniform supersonic incoming flow with the large Mach number past a generally curved sharp cone under the suitable boundary condition on the conic surface (physically, this kind of boundary condition means that the body is perforated or porous; with respect to more explanations on the perforated boundaries, one can see [6,7]). In addition, by using Glimm's scheme, W.C. Lien and T.P.…”

Section: Introductionmentioning

confidence: 98%

Global supersonic conic shock wave for the steady supersonic flow past a cone: Polytropic gas

Cui

Yin

2009

Journal of Differential Equations

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In this paper, we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Based on the delicate asymptotic expansion of the background solution, one can verify that the boundary conditions on the shock and the conic surface satisfy the "dissipative" property. From this property, by use of the reflected characteristics method and the special form of the shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large. On the other hand, we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution, moreover, our proof approach is different from that in [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132].

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Solvability of a problem for transonic flow with a local supersonic region

Cited by 14 publications

References 13 publications

Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

A modified Frankl-Morawetz problem on a transonic flow past an airfoil

Global supersonic conic shock wave for the steady supersonic flow past a cone: Polytropic gas

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