Isentropic Gas Flow for the Compressible Euler Equation in a Nozzle

“…In [4], Lu and author investigate the global existence and decay estimate of solutions for the Laval nozzle. Moreover, in [9] and [11], the same author studied the general nozzle. Next we recall results for the nozzle with friction.…”

Section: Introductionmentioning

confidence: 99%

Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle

Tsuge

2012

Arch Rational Mech Anal

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In this paper, we study the motion of isentropic gas in the Laval nozzle. The Laval nozzle is the most important type of nozzle utilized in some turbines. In particular, we consider unsteady flows, including transonic gas flows, and prove the existence of global solutions for the Cauchy problem. In spite of its importance, this problem has received little attention until now. The most difficult point is to obtain bounded estimates for approximate solutions. To overcome this, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation and yield a sharper bounded estimate. As a result, we find an invariant region for our solutions. Finally, in order to prove their convergence, we use the compensated compactness framework.

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“…For the case of nozzle flow without the friction, namely b(x, t) = 0 and α(x, t) = 0, the global solution of the Cauchy problem was well studied (cf. [1,2,7,9] and the references cited therein); When a(x) = 0 and α(x, t) = 0, the source term b(x, t) in System ( 1) is corresponding to an outer force [3,8], and when b(x, t) = 0, a(x) = 0, α(x, t)u|u| in (1) corresponds physically to a friction term [5].…”

Section: Introductionmentioning

confidence: 99%

Soluciones globales para sistema isotérmico con fuente

Wang¹

2021

Rev Integr temas mat

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In this short note, we are concerned with the global existence of solutions to the isothermal system with source, where the inhomogeneous terms f(x,t,ρ,u)=b(x,t)ρ+(a′(x)/a(x))*ρu^2+α(x,t)ρu|u| are appeared in the momentum equation. Our work extended the results in the previous papers “Resonance for the Isothermal System of Isentropic Gas Dynamics” (Proc. A.M.S.139(2011),2821-2826), “Global Existence and Stability to the Polytropic Gas Dynamics with an Outer Force” (Appl. Math. Let-ters, 95(2019), 35-40) and “Existence of Global Solutions for Isentropic GasFlow with Friction” (Nonlinearity, 33(2020), 3940-3969), where the global solution was obtained for the source f(x,t,ρ,u)=(a′(x)/a(x))*ρu^2, f(x,t,ρ,u)=b(x,t)ρ, f(x,t,ρ,u)=α(x,t)ρu|u| respectively.

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Isentropic Gas Flow for the Compressible Euler Equation in a Nozzle

Cited by 27 publications

References 18 publications

Existence and stability of solutions to the compressible Euler equations with an outer force

Existence and stability of solutions to the compressible Euler equations with an outer force

Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle

Soluciones globales para sistema isotérmico con fuente

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