scholarly journals Generalized Scale-Invariant Solutions to the Two-Dimensional Stationary Navier--Stokes Equations

2015 ◽  
Vol 47 (1) ◽  
pp. 955-968 ◽  
Author(s):  
Julien Guillod ◽  
Peter Wittwer
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
Stationary Navier Stokes Equations

On nonhomogeneous boundary value problems for the stationary Navier–Stokes equations in two-dimensional symmetric semi-infinite outlets

2015 ◽  
Vol 15 (04) ◽  
pp. 543-569 ◽  
Author(s):  
M. Chipot ◽  
K. Kaulakytė ◽  
K. Pileckas ◽  
W. Xue
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Domain Boundary ◽  
Boundary Value ◽  
Symmetric Domain ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Dirichlet Integral ◽  
Navier Stokes Equations ◽  
Stationary Navier Stokes Equations

We study the stationary nonhomogeneous Navier–Stokes problem in a two-dimensional symmetric domain with a semi-infinite outlet (for instance, either paraboloidal or channel-like). Under the symmetry assumptions on the domain, boundary value and external force, we prove the existence of at least one weak symmetric solution without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value [Formula: see text] over the inner and the outer boundaries may be arbitrarily large. Only the necessary compatibility condition (the total flux is equal to zero) has to be satisfied. Moreover, the Dirichlet integral of the solution can be finite or infinite depending on the geometry of the domain.

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.

Singularities

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Stokes Equations ◽  
Negative Answer ◽  
Three Dimensions ◽  
Picard Iteration ◽  
Navier Stokes ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
L2 Norm ◽  
Two Dimensional Flow ◽  
Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

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