On the expansion of a wedge of van der Waals gas into a vacuum

Lai, Geng

doi:10.1016/j.jde.2015.02.039

Cited by 41 publications

(18 citation statements)

References 28 publications

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“…Here we provide first‐order characteristic decompositions of characteristic angles without proof. The proofs of these decompositions can be found in a previous article 30 . These first‐order characteristic decompositions are very important to obtain a priori estimates for the characteristic angles and physical variables.…”

Section: System In Two‐dimensional Self‐similar Flowmentioning

confidence: 99%

“…We assume

\alpha \in [\pi/2, \pi ]

and

\beta \in [-\pi/2, 0]

so that we can use the following formulas of pseudo‐velocity from Ref. 30:

\begin{equation} U=c\dfrac{\cos \sigma }{\sin \omega },\quad V=c\dfrac{\sin \sigma }{\sin \omega } \end{equation}

with the first‐order decompositions

\{\begin{matrix} left {\bar{\partial}}_{-} c = 0true \frac{μ^{2} false(τ false)}{\tan ω} (c {\overset{\partial}{true}}_{-} α - 2 \sin^{2} ω), \\ left c {\bar{\partial}}_{-} β = normalΩ (τ, ω) \cos^{2} (ω) (c {\overset{\partial}{true}}_{-} α - 2 \sin^{2} ω) = 0true \frac{Ω (τ, ω)}{2 μ^{2} false(τ false)} \sin 2 ω {\bar{\partial}}_{-} c, \\ left {\bar{\partial}}_{+} c = - 0true \frac{μ^{2} false(τ false)}{\tan ω} (() c {\overset{\partial}{true}}_{+} β + 2 \sin^{2} ω), \\ left c {\bar{\partial}}_{+} α = normalΩ (τ, ω) \cos^{2} (ω) ((), c, \overset{...}{true}) \end{matrix}

…”

Section: System In Two‐dimensional Self‐similar Flowmentioning

confidence: 99%

“…In this section we derive the characteristic decomposition form for the variables

\alpha

\beta

, and

c

, which is important and very useful for establishing a priori gradient estimates of solution. First we cite the following second‐order decompositions of

c

from Lai 30 Proposition The variable

c

satisfies the following characteristic decompositions:

\{\begin{matrix} left c {\bar{\partial}}_{+} {\bar{\partial}}_{-} c = {\bar{\partial}}_{-} c ({} \sin 2 ω + \frac{{\bar{\partial}}_{-} c}{2 μ^{2} (τ) \cos^{2} ω} + (1 + 0true \frac{Ω (τ, ω) \cos 2 ω}{2 μ^{2} false(τ false)} + τ κ^{'} (τ)) {\overset{\partial}{true}}_{+} c), \\ left c {\bar{\partial}}_{-} {\bar{\partial}}_{+} c = {\bar{\partial}}_{+} c ({} \sin 2 ω + \frac{{\bar{\partial}}_{+} c}{2 μ^{2} (τ) \cos^{2} ω} + (1 + 0true \frac{Ω (τ, ω) \cos 2 ω}{2 μ^{2} false(τ false)} + τ κ^{'} (τ)) {\overset{\partial}{true}}_{-}) \end{matrix}

…”

Section: System In Two‐dimensional Self‐similar Flowmentioning

confidence: 99%

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On the existence and regularity of solutions of semihyperbolic patches to 2‐D Euler equations with van der Waals gas

Barthwal

Sekhar

2021

Stud Appl Math

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This article is concerned in establishing the existence and regularity of solution of semihyperbolic patch problem for two‐dimensional isentropic Euler equations with van der Waals gas. This type of solution appears in the transonic flow over an airfoil and Guderley reflection and is very common in the numerical solution of Riemann problems. We use the idea of characteristic decomposition and bootstrap method to prove the existence of a global smooth solution, which is uniformly C1,12$C^{1, \tfrac{1}{2}}$ continuous up to the sonic curve. We also prove that the sonic curve is C1,12$C^{1, \tfrac{1}{2}}$ continuous. Further, we show the formation of shock as an envelope for positive characteristics before reaching their sonic points.

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Section: System In Two‐dimensional Self‐similar Flowmentioning

confidence: 99%

“…We assume

\alpha \in [\pi/2, \pi ]

and

\beta \in [-\pi/2, 0]

so that we can use the following formulas of pseudo‐velocity from Ref. 30:

\begin{equation} U=c\dfrac{\cos \sigma }{\sin \omega },\quad V=c\dfrac{\sin \sigma }{\sin \omega } \end{equation}

with the first‐order decompositions

\{\begin{matrix} left {\bar{\partial}}_{-} c = 0true \frac{μ^{2} false(τ false)}{\tan ω} (c {\overset{\partial}{true}}_{-} α - 2 \sin^{2} ω), \\ left c {\bar{\partial}}_{-} β = normalΩ (τ, ω) \cos^{2} (ω) (c {\overset{\partial}{true}}_{-} α - 2 \sin^{2} ω) = 0true \frac{Ω (τ, ω)}{2 μ^{2} false(τ false)} \sin 2 ω {\bar{\partial}}_{-} c, \\ left {\bar{\partial}}_{+} c = - 0true \frac{μ^{2} false(τ false)}{\tan ω} (() c {\overset{\partial}{true}}_{+} β + 2 \sin^{2} ω), \\ left c {\bar{\partial}}_{+} α = normalΩ (τ, ω) \cos^{2} (ω) ((), c, \overset{...}{true}) \end{matrix}

…”

Section: System In Two‐dimensional Self‐similar Flowmentioning

confidence: 99%

“…In this section we derive the characteristic decomposition form for the variables

\alpha

\beta

, and

c

, which is important and very useful for establishing a priori gradient estimates of solution. First we cite the following second‐order decompositions of

c

from Lai 30 Proposition The variable

c

satisfies the following characteristic decompositions:

\{\begin{matrix} left c {\bar{\partial}}_{+} {\bar{\partial}}_{-} c = {\bar{\partial}}_{-} c ({} \sin 2 ω + \frac{{\bar{\partial}}_{-} c}{2 μ^{2} (τ) \cos^{2} ω} + (1 + 0true \frac{Ω (τ, ω) \cos 2 ω}{2 μ^{2} false(τ false)} + τ κ^{'} (τ)) {\overset{\partial}{true}}_{+} c), \\ left c {\bar{\partial}}_{-} {\bar{\partial}}_{+} c = {\bar{\partial}}_{+} c ({} \sin 2 ω + \frac{{\bar{\partial}}_{+} c}{2 μ^{2} (τ) \cos^{2} ω} + (1 + 0true \frac{Ω (τ, ω) \cos 2 ω}{2 μ^{2} false(τ false)} + τ κ^{'} (τ)) {\overset{\partial}{true}}_{-}) \end{matrix}

…”

Section: System In Two‐dimensional Self‐similar Flowmentioning

confidence: 99%

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On the existence and regularity of solutions of semihyperbolic patches to 2‐D Euler equations with van der Waals gas

Barthwal

Sekhar

2021

Stud Appl Math

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“…This method was introduced by Li et al 17 in investigating interactions of two-dimensional rarefaction simple waves of the compressible Euler equations; see also other studies. [18][19][20][21][22][23][24] The rest of the paper is organized as follows. In Section 2, we derive a group of first-order and second-order characteristic equations of (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…We will use the method of characteristic decomposition to prove the main theorem. This method was introduced by Li et al 17 in investigating interactions of two‐dimensional rarefaction simple waves of the compressible Euler equations; see also other studies 18‐24 . The rest of the paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Existence of global bounded smooth solutions for the one‐dimensional nonisentropic Euler system

Lai

Zhao

2020

Math Methods in App Sciences

Self Cite

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We study the existence of global bounded smooth solutions to the one-dimensional (1D) nonisentropic Euler system with large initial data. We derive a group of characteristic decompositions for the 1D nonisentropic Euler system. Using these characteristic decompositions, we find some sufficient conditions on the initial data to obtain the existence of global bounded classical solutions to the Cauchy problem. By the method of characteristic decomposition, we also give a type of large initial data to show the formation of singularity for the 1D nonisentropic Euler system.

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The expansion of a non‐ideal gas around a sharp corner for 2‐D compressible Euler system

Chen

Shen

Yin

2022

Math Methods in App Sciences

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In this paper, we consider the problem of expansion of a non‐ideal gas turning around corresponding large or small sharp corner for 2‐D compressible Euler system. We focus on extending the results of this problem for polytropic gas to that for a more realistic gas, that is, van der Waals gas. In this case, rarefaction waves, shock wave, fan–jump composite waves, or fan–jump–fan composite waves may be appeared. The flow expands to vacuum state when the inclination angle θ$$ \theta $$ is small enough, that is, a zone of cavitation will come into being. Otherwise, the flow will arrive at a zone of constant state. The corresponding problem can be transformed into interaction of a planar rarefaction wave with a planar centered wave and interaction of a simple wave with rigid wall, which are actually characteristic boundary value problems for 2‐D self‐similar Euler system. The estimates of solution are obtained by making use of characteristic analysis, corresponding characteristic decompositions and invariant region of solution. Furthermore, by extending the local solution, and combining with those estimates and hyperbolicity, the existence of global classical solution up to the interface of non‐ideal gas with vacuum is obtained.

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On the expansion of a wedge of van der Waals gas into a vacuum

Cited by 41 publications

References 28 publications

On the existence and regularity of solutions of semihyperbolic patches to 2‐D Euler equations with van der Waals gas

On the existence and regularity of solutions of semihyperbolic patches to 2‐D Euler equations with van der Waals gas

Existence of global bounded smooth solutions for the one‐dimensional nonisentropic Euler system

The expansion of a non‐ideal gas around a sharp corner for 2‐D compressible Euler system

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