Asymptotic behaviour of solutions to the Navier-Stokes equations of a two-dimensional compressible flow

2011 ◽  
Vol 27 (4) ◽  
pp. 697-712 ◽  
Author(s):  
Ying-hui Zhang ◽  
Zhong Tan
Keyword(s):  
Asymptotic Behaviour ◽  
Compressible Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Asymptotic Behaviour Of Solutions

scholarly journals Asymptotic behaviour of solutions to the stationary Navier–Stokes equations in two-dimensional exterior domains with zero velocity at infinity

2014 ◽  
Vol 25 (02) ◽  
pp. 229-253 ◽  
Author(s):  
Julien Guillod ◽  
Peter Wittwer
Keyword(s):  
Asymptotic Behaviour ◽  
Stokes Equations ◽  
Invariant Solution ◽  
The Body ◽  
Dimensional Case ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Scale Invariant ◽  
Rigorous Results ◽  
Navier Stokes Equations

We investigate analytically and numerically the existence of stationary solutions converging to zero at infinity for the incompressible Navier–Stokes equations in a two-dimensional exterior domain. Physically, this corresponds for example to fixing a propeller by an external force at some point in a two-dimensional fluid filling the plane and to ask if the solution becomes steady with the velocity at infinity equal to zero. To answer this question, we find the asymptotic behaviour for such steady solutions in the case where the net force on the propeller is nonzero. In contrast to the three-dimensional case, where the asymptotic behaviour of the solution to this problem is given by a scale invariant solution, the asymptote in the two-dimensional case is not scale invariant and has a wake. We provide an asymptotic expansion for the velocity field at infinity, which shows that, within a wake of width |x|2/3, the velocity decays like |x|-1/3, whereas outside the wake, it decays like |x|-2/3. We check numerically that this behaviour is accurate at least up to second order and demonstrate how to use this information to significantly improve the numerical simulations. Finally, in order to check the compatibility of the present results with rigorous results for the case of zero net force, we consider a family of boundary conditions on the body which interpolate between the nonzero and the zero net force case.

Remarks on the asymptotic behaviour of solutions to the compressible Navier–Stokes equations in the half-line

2002 ◽  
Vol 132 (3) ◽  
pp. 627-638 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Asymptotic Behaviour ◽  
Specific Volume ◽  
Stokes Equations ◽  
Global Solutions ◽  
Ideal Gas ◽  
Half Line ◽  
Energy Estimates ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Asymptotic Behaviour Of Solutions

We study the time-asymptotic behaviour of solutions to the Navier-Stokes equations for a one-dimensional viscous polytropic ideal gas in the half-line. Using a local representation for the specific volume, which is obtained by using a special cut-off function to localize the problem, and the weighted energy estimates, we prove that the specific volume is pointwise bounded from below and above for all x, t and that for all t the temperature is bounded from below and above locally in x. Moreover, global solutions are convergent as time goes to infinity. The large-time behaviour of solutions to the Cauchy problem is also examined.

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