Pressure-type artificial boundary conditions for steady Navier-Stokes flows in pipe systems beyond the small-data regime

Blazy, Nazarov and Specovius-Neugebauer (J. Math. Fluid Mech. 9, 2007) constructed an artificial boundary condition (ABC) of Robin type, with a coefficient singular at the edge of the artificial boundary, that allows solutions of the stationary Stokes and Navier-Stokes systems in a three-dimensional domain with several semi-infinite cylindrical outlets, with pressures prescribed at remote axial stations, to be approximated by boundary-value problems on truncated bounded subdomains: they proved an isomorphism in Kondratiev weighted spaces, inverse bounds uniform in the truncation position, and exponentially decaying truncation-error estimates. However, for the nonlinear Navier-Stokes system these results hold only under a small-data hypothesis, which forces the Poiseuille amplitudes of the asymptotic flow to be small. The problem is to remove this restriction: solvability of the truncated nonlinear problem with the pressure-type ABC for arbitrarily large data (at least conditional on the existence of an unbounded-domain solution with its detached Poiseuille asymptotics), the spectral analysis of the linearized operator about a Poiseuille flow of large amplitude needed to justify the weighted-space estimates in that regime, and truncation-error estimates in that regime.

Background

A steady flow of a viscous incompressible fluid (velocity $v$, pressure $p$, kinematic viscosity $\nu>0$) in a three-dimensional pipe system is modeled by the stationary Navier-Stokes system together with the no-slip condition on the walls. The flow domain $\Omega$ is assumed to consist of a bounded junction body to which $J$ semi-infinite cylindrical outlets (pipes) $\Pi_j=\omega_j\times(0,\infty)$ are attached, each with a bounded cross-section $\omega_j$. In such domains every solution develops a detached asymptotics at infinity: $v=\sum_j\chi_j(a_j u^{0j}+b_j u^{1j})+\tilde v$, where $\chi_j$ are cut-off functions, $u^{0j}$ is the constant-pressure mode with coefficient $a_j$, $u^{1j}$ is the Poiseuille mode (the classical parabolic profile, written as a velocity-pressure pair $u^{1j}=(0,0,\varpi_j\Psi_j,-2\nu\varpi_j z^j)$ whose last entry is the accompanying linear pressure, where $\Psi_j$ solves $-\Delta\Psi_j=2$ in $\omega_j$ with $\Psi_j=0$ on $\partial\omega_j$ and $\varpi_j$ normalizes the flux) with flux coefficient $b_j$, and the remainder $\tilde v$ decays exponentially. The pressure in outlet $j$ grows linearly, $p\sim a_j-2\nu\varpi_j b_j z^j$, so prescribing the remote pressure at an axial station $z^j=T_j$ is equivalent to fixing the linear combination $2\nu\varpi_j T_j b_j-a_j=\mathfrak{h}_j$.

Numerical simulation requires truncating $\Omega$ to a bounded domain $\Omega_R$ by cutting each outlet at $z^j=R$; on each artificial cross-section $\Gamma_{R,j}$ an artificial boundary condition (ABC) must then be imposed. Two classical choices fail in the pressure-prescribed setting. Dirichlet prescription of the Poiseuille profile needs the fluxes $b_j$, which are global unknowns of the problem. And the widely used 'do-nothing' Neumann condition $\nu\partial_n v-p\,n=0$ leaves the linearly growing pressure term uncancelled: in an appendix to the 2007 paper the authors demonstrate, by explicit asymptotic matching in a three-outlet geometry, that it produces pressure shifts of order $O(R)$ and velocity perturbations of order $O(1/R)$, so the computed solution drifts with the truncation position $R$.

Blazy, Nazarov and Specovius-Neugebauer constructed the ABC adapted to prescribed remote pressures: a Robin-type condition on $\Gamma_{R,j}$ whose normal (axial) component carries the coefficient $2\nu(T_j-R)\Psi_j^{-1}(y^j)$. Because $\Psi_j$ vanishes on $\partial\omega_j$, this coefficient is singular at the edge $\partial\Gamma_{R,j}$ (it behaves like inverse distance to the edge), and the analysis is carried out in Kondratiev weighted spaces $V^m_\gamma(\Omega_R)$, Sobolev spaces whose norms contain powers of the distance to the edge, which is the standard framework for elliptic problems on polyhedral domains with edge singularities. For the truncated Stokes problem they proved a Fredholm property and, via a uniqueness argument at a convenient weight index $\gamma$, an isomorphism, with an explicitly quantified admissible interval of $\gamma$; the exponential decay rate $\beta^*>0$ of the remainder $\tilde v$ is a spectral threshold determined by the operator pencil (a polynomial family of boundary value problems in the spectral parameter generated by the Stokes system in the cylinder) associated with the cross-section $\omega_j$. Building an 'almost inverse' from solutions of a global problem on $\Omega$ and of model problems on the semi-infinite cylinders $\omega_j\times(-\infty,0)$, they obtained inverse bounds uniform in $R$ with explicit polynomial prefactors $1+R^{3/2}+\sum_j(T_j-R)^{l+1}$, where $l$ is a fixed nonnegative integer determined by the construction, and consequently truncation-error estimates: the truncated solution converges to the true solution exponentially in $R$ on interior subdomains. For the full nonlinear Navier-Stokes system, however, existence, uniqueness and error estimates of the same order were proved only under a small-data hypothesis $\|(f,g,\mathfrak{h})\|\le\rho$ on the problem data - the body force $f$, the boundary data $g$, and the prescribed remote pressures $\mathfrak{h}=(\mathfrak{h}_1,\ldots,\mathfrak{h}_J)$ - which in particular forces the Poiseuille amplitudes $b_j$ to be small: the contraction argument needs the convective term to be a small perturbation, and the invertibility argument for the linearized (Oseen-type) operator about the Poiseuille background was carried out in the weighted spaces only for a small background. This restriction is structural rather than technical: even on the unbounded domain itself, existence of steady Navier-Stokes flows in infinite pipe systems with arbitrary fluxes or pressure drops ('Leray's problem') is open in three dimensions and was settled by Korobkov, Pileckas and Russo only in plane and axially symmetric domains.

Problem Statement

Remove the small-data restriction from the pressure-type truncation theory of Blazy, Nazarov and Specovius-Neugebauer (2007) for the stationary Navier-Stokes system in a three-dimensional domain $\Omega$ with finitely many cylindrical semi-infinite outlets $\Pi_j=\omega_j\times(0,\infty)$ and prescribed remote pressures, i.e. prescribed values $\mathfrak{h}_j$ of the linear combinations $2\nu\varpi_j T_j b_j-a_j$ at remote axial stations $T_j$. For data $(f,g,\mathfrak{h})$ of arbitrary size (equivalently, Poiseuille amplitudes $b_j$ outside any fixed smallness range), establish:

(i) solvability of the truncated boundary-value problem on $\Omega_R$ with the Robin-type pressure artificial boundary condition whose axial coefficient is $2\nu(T_j-R)\Psi_j^{-1}(y^j)$ - an isomorphism in the Kondratiev weighted spaces $V^m_\gamma(\Omega_R)$ - with bounds uniform in the truncation position $R$, and with the dependence of all constants on the remote-pressure stations $T_j$ and on the data norm tracked explicitly, as in the settled small-data theory; this sub-goal may be posed conditionally on the existence of a solution of the unbounded-domain Navier-Stokes problem with its detached Poiseuille asymptotics, that existence being its sole hypothesis;

(ii) the spectral analysis needed to justify the weighted-space estimates outside the perturbative regime: the dependence of the decay threshold $\beta^*$ and of the admissible weight interval $\gamma$ on the geometry of the cross-sections $\omega_j$, and the behaviour of the spectrum of the linearized (Oseen-type) operator pencil about a Poiseuille flow of large amplitude $b_j$, in particular whether eigenvalues can approach or cross the real axis, thereby affecting the Fredholm property, the admissible weight window, or the exponential weights on which (i) and (iii) rest;

(iii) truncation-error estimates for the nonlinear problem of the same exponential order in $R$ as the small-data theory, with explicit dependence of all constants on the data norm.

All three parts (i)-(iii) are required; part (ii) is the enabling spectral analysis on which the weighted-space estimates in (i) and (iii) rest. A proof of (i) and (iii) that leaves (ii) open, or a partial spectral answer that does not suffice to justify those estimates, does not resolve the problem.

The objective is a quantitative truncation theory for pressure-driven steady viscous flow in infinite pipe systems, extending the small-data theory of 2007 to data $(f,g,\mathfrak{h})$ of arbitrary size - equivalently, to Poiseuille amplitudes $b_j$ outside any fixed smallness range - and to the full class of data and geometries of the Problem Statement.

References

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