scholarly journals The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$

2002 ◽  
Vol 1 (2) ◽  
pp. 221-235 ◽  
Author(s):  
Marcel Oliver ◽  
Keyword(s):  
Euler Equations ◽  
Stokes Equations ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Besov Class ◽  
Navier Stokes Equations ◽  
Short Time

scholarly journals Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

2021 ◽  
Vol 240 (1) ◽  
pp. 295-326
Author(s):  
Gennaro Ciampa ◽  
Gianluca Crippa ◽  
Stefano Spirito
Keyword(s):  
Euler Equations ◽  
Stokes Equations ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Vanishing Viscosity ◽  
Navier Stokes Equations ◽  
Vanishing Viscosity Limit ◽  
Lagrangian Solution ◽  
Viscosity Limit ◽  
2D Euler Equations

AbstractIn this paper we prove the uniform-in-time $$L^p$$ L p convergence in the inviscid limit of a family $$\omega ^\nu $$ ω ν of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ ω of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $$\omega ^{\nu }$$ ω ν to $$\omega $$ ω in $$L^p$$ L p . Finally, we show that solutions of the Euler equations with $$L^p$$ L p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

Hydrodynamic stability and the inviscid limit

1961 ◽  
Vol 10 (3) ◽  
pp. 420-429 ◽  
Author(s):  
K. M. Case
Keyword(s):  
Euler Equations ◽  
Hydrodynamic Stability ◽  
Stokes Equations ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linearized Euler Equations

It is shown that for appropriately posed problems, the solutions of the linearized Navier–Stokes equations approach those of the linearized Euler equations as the viscosity tends to zero.

Transonic nozzle flow of dense gases

1993 ◽  
Vol 247 ◽  
pp. 661-688 ◽  
Author(s):  
A. Kluwick
Keyword(s):  
Stokes Equations ◽  
Equations Of State ◽  
Weak Shock ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Perturbation Equation ◽  
Navier Stokes Equations ◽  
Sonic Point ◽  
Specific Heats ◽  
Dense Gases

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Navier–Stokes equations supplemented with realistic equations of state for gases which have relatively large specific heats a novel form of the viscous transonic small-perturbation equation is derived. Evaluation of the inviscid limit of this equation shows that three sonic points rather than a single sonic point may occur during isentropic expansion of such media, in contrast to the case of perfect gases. As a consequence, a shock-free transition from subsonic to supersonic speeds cannot, in general, be achieved by means of a conventional converging–diverging nozzle. Nozzles leading to shock-free flow fields must have an unusual shape consisting of two throats and an intervening antithroat. Additional new results include the computation of the internal thermoviscous structure of weak shock waves and a phenomenon referred to as impending shock splitting. Finally, the relevance of these results to the description of external transonic flows is discussed briefly.

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