On Boundary Conditions for Incompressible Navier-Stokes Problems

Abstract

We revisit the issue of finding proper boundary conditions for the field equations describing incompressible flow problems, for quantities like pressure or vorticity, which often do not have immediately obvious “physical” boundary conditions. Most of the issues are discussed for the example of a primitive-variables formulation of the incompressible Navier-Stokes equations in the form of momentum equations plus the pressure Poisson equation. However, analogous problems also exist in other formulations, some of which are briefly reviewed as well. This review article cites 95 references.

Authors' Reply

“…The result in [3], which is claimed to contradict, and therefore falsify, the result in [4] is established under the strong assumption of a smooth behavior of the solution {u, p} of the Navier-Stokes problem as t → 0. Hence there is actually no conflict at all between the two statements and the confusion is due to an improper notion of regularity for weak solutions of the Navier-Stokes problem used in [3]. To see clearly the difference in 'regularity' discussed above, see the 'Impulsive Start' discussion in [5, Section 3.19, p. 884].…”

“…This different behavior of normal and tangential components is the essential point of the regularity analysis in [4] and is missed in the arguments in [3]. The result in [3], which is claimed to contradict, and therefore falsify, the result in [4] is established under the strong assumption of a smooth behavior of the solution {u, p} of the Navier-Stokes problem as t → 0. Hence there is actually no conflict at all between the two statements and the confusion is due to an improper notion of regularity for weak solutions of the Navier-Stokes problem used in [3].…”

“…In Lemma 1 in the Appendix, the regularity result quoted from Heywood and Rannacher [4] does not appear in that paper in the form stated in [3]. Actually, it is known to be incorrect without additional compatibility conditions on the initial data.…”

“…The same proof as in our paper [1], with more details, is presented at the end of these remarks. He also claims that the results in [1] must be wrong because he had 'proved' in Corollary 1 of his earlier paper [3] that the S-CPPE is ill-posed. However, his proof of this corollary is not correct; it cannot be derived from Theorem 2 in [3] since a solution u of the modified problem with Equation (24) in [3] is not necessarily a solution of Equation (30) in [3] due to the fact that ∇ ·u is not necessarily zero.…”

“…This is due to the imprecise notion of regularity of solutions of the Navier-Stokes problem used in [3]. In the Heywood and Rannacher paper [4] it is shown that the corresponding weak solution has a limit p 0 (x) = lim t→0 p(x, t), which is the weak solution of the boundary value problem (for homogeneous boundary data u| * = 0)…”