Regularity of the Free Boundary for the One-Dimensional Flow of Gas in a Porous Medium

In this paper we deal with the one-dimensional Stefan problemut−uxx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ+, u(x, 0) =u0(x)with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u)[ges ] Ceγ1[mid ]u[mid ] for some γ1 large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u)[ges ] Ce√2πe[mid ]u[mid ].

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Numerical Integration of the Equations Governing the One-Dimensional Flow of a Chemically Reactive Gas

The Physics of Fluids ◽

10.1063/1.1692345 ◽

1969 ◽

Vol 12 (11) ◽

pp. 2292 ◽

Cited By ~ 16

Author(s):

H. E. Bailey

Keyword(s):

Numerical Integration ◽

One Dimensional ◽

Dimensional Flow ◽

The One ◽

Reactive Gas

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Theoretical and Computational Analysis of an Auxiliary Power Unit (APU) Installation Cooling System

Volume 1: Aircraft Engine; Ceramics; Coal, Biomass and Alternative Fuels; Controls, Diagnostics and Instrumentation ◽

10.1115/gt2012-68643 ◽

2012 ◽

Author(s):

Parthiv N. Shah ◽

Tricia Waniewski Sur ◽

R. Scott Miskovish ◽

Albert Robinson

Keyword(s):

Network Model ◽

Cooling System ◽

Dimensional Model ◽

Ambient Conditions ◽

Modeling Framework ◽

Flow Network ◽

One Dimensional ◽

Dimensional Flow ◽

System Pressure ◽

The One

This paper presents a theoretical one-dimensional model and computational fluid dynamics (CFD) simulations of a tailcone-installed APU cooling system. The work is motivated by the need to deliver sufficient cooling airflow to critical components within an aircraft tailcone compartment. The cooling system considered herein utilizes (1) an eductor system at the APU exhaust and (2) a ram air scoop near an upstream inlet to the compartment to induce the necessary cooling flow during ground and in-flight APU operation. A one-dimensional flow network model provides a framework for the quantification and matching of eductor pumping and system pressure drop characteristics. Detailed CFD models that simulate internal tailcone compartment flows driven by ambient conditions external to the aircraft in ground or flight operation support the one-dimensional model and are used to characterize component performance and assess different scoop and eductor designs. The one-dimensional flow network model is calibrated to the CFD results to predict system cooling performance under known APU loads at points on the ground and in the flight envelope. The agreement between the models is encouraging and suggests the modeling framework and CFD techniques discussed will be applicable to future designs and improvements of eductor-driven aircraft compartment cooling systems.

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