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Volume 3, Issue 3
Asymptotic Analysis of Steady Viscous Shocks in a 1-D Finite Nozzle in the Small Viscosity Limit

Beixiang Fang & Ya-Guang Wang

DOI: 10.4208/cmaa.2024-0017

Commun. Math. Anal. Appl., 3 (2024), pp. 402-424.

Published online: 2024-09

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  • Abstract

In this paper we show that, as the viscosity is properly small, there exists a viscous transonic shock solution for the steady 1-D Navier-Stokes system with prescribed pressure at the exit, and it converges to a transonic shock solution to the 1-D steady Euler system as the viscosity goes to zero. Moreover, the position of the shock front is also derived. The key step is to reduce the pressure condition at the exit into a nonlinear boundary condition on the velocity, such that the boundary value problem for the Navier-Stokes system can be reformulated as a boundary value problem for an ODE with an unknown parameter.

  • Keywords

Asymptotic analysis, steady 1-D viscous shocks, 1-D finite nozzle, small viscous limits, Euler system, Navier-Stokes system.

  • AMS Subject Headings

35A02, 35L65, 35L67, 35Q31, 76L05, 76N10, 76N17

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMAA-3-402, author = {Fang , Beixiang and Wang , Ya-Guang}, title = {Asymptotic Analysis of Steady Viscous Shocks in a 1-D Finite Nozzle in the Small Viscosity Limit}, journal = {Communications in Mathematical Analysis and Applications}, year = {2024}, volume = {3}, number = {3}, pages = {402--424}, abstract = {

In this paper we show that, as the viscosity is properly small, there exists a viscous transonic shock solution for the steady 1-D Navier-Stokes system with prescribed pressure at the exit, and it converges to a transonic shock solution to the 1-D steady Euler system as the viscosity goes to zero. Moreover, the position of the shock front is also derived. The key step is to reduce the pressure condition at the exit into a nonlinear boundary condition on the velocity, such that the boundary value problem for the Navier-Stokes system can be reformulated as a boundary value problem for an ODE with an unknown parameter.

}, issn = {2790-1939}, doi = {https://doi.org/10.4208/cmaa.2024-0017}, url = {http://global-sci.org/intro/article_detail/cmaa/23382.html} }
TY - JOUR T1 - Asymptotic Analysis of Steady Viscous Shocks in a 1-D Finite Nozzle in the Small Viscosity Limit AU - Fang , Beixiang AU - Wang , Ya-Guang JO - Communications in Mathematical Analysis and Applications VL - 3 SP - 402 EP - 424 PY - 2024 DA - 2024/09 SN - 3 DO - http://doi.org/10.4208/cmaa.2024-0017 UR - https://global-sci.org/intro/article_detail/cmaa/23382.html KW - Asymptotic analysis, steady 1-D viscous shocks, 1-D finite nozzle, small viscous limits, Euler system, Navier-Stokes system. AB -

In this paper we show that, as the viscosity is properly small, there exists a viscous transonic shock solution for the steady 1-D Navier-Stokes system with prescribed pressure at the exit, and it converges to a transonic shock solution to the 1-D steady Euler system as the viscosity goes to zero. Moreover, the position of the shock front is also derived. The key step is to reduce the pressure condition at the exit into a nonlinear boundary condition on the velocity, such that the boundary value problem for the Navier-Stokes system can be reformulated as a boundary value problem for an ODE with an unknown parameter.

Beixiang Fang & Ya-Guang Wang. (2024). Asymptotic Analysis of Steady Viscous Shocks in a 1-D Finite Nozzle in the Small Viscosity Limit. Communications in Mathematical Analysis and Applications. 3 (3). 402-424. doi:10.4208/cmaa.2024-0017
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