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A Nitsche method for incompressible fluids with general dynamic boundary conditions

Pablo Alexei Gazca-Orozco, Franz Gmeineder, Erika Maringová Kokavcová, Tabea Tscherpel

Abstract

Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction boundary conditions. In particular, set-valued, non-monotone, noncoercive and dynamic relations may occur. For a unifying framework of such relations, we present a fully discrete numerical scheme for the time-dependent Navier-Stokes equations subject to impermeability and general slip type boundary conditions on polyhedral domains. Based on compactness arguments, we prove convergence of subsequences, finally ensuring the existence of a weak solution. The numerical scheme uses a general inf-sup stable pair of finite element spaces for the velocity and pressure, a regularisation approach for the implicit slip boundary condition and, most importantly, a Nitsche method to impose the impermeability and a backward Euler time stepping. One of the key tools in the convergence proof is an inhomogeneous Korn inequality that includes a normal trace term.

A Nitsche method for incompressible fluids with general dynamic boundary conditions

Abstract

Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction boundary conditions. In particular, set-valued, non-monotone, noncoercive and dynamic relations may occur. For a unifying framework of such relations, we present a fully discrete numerical scheme for the time-dependent Navier-Stokes equations subject to impermeability and general slip type boundary conditions on polyhedral domains. Based on compactness arguments, we prove convergence of subsequences, finally ensuring the existence of a weak solution. The numerical scheme uses a general inf-sup stable pair of finite element spaces for the velocity and pressure, a regularisation approach for the implicit slip boundary condition and, most importantly, a Nitsche method to impose the impermeability and a backward Euler time stepping. One of the key tools in the convergence proof is an inhomogeneous Korn inequality that includes a normal trace term.

Paper Structure

This paper contains 41 sections, 26 theorems, 295 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Impermeability boundary conditions and Nitsche's method
  3. Models and key difficulties
  4. Available results on numerical approximations
  5. Contributions of this paper and outline
  6. Outline
  7. Model and main results
  8. General slip type boundary conditions and assumptions
  9. Explicit noncoercive relations
  10. Implicit coercive relations
  11. Statement of main convergence result and context
  12. Examples of boundary conditions
  13. A Korn-type inequality with normal trace terms
  14. Approximation and discretisation
  15. Approximation of monotone graphs
  16. ...and 26 more sections

Key Result

theorem 1

Let $\Omega\subset \setR^d$ be an open and bounded Lipschitz domain, and suppose that $\Gamma\subset\partial\Omega$ satisfies Assumption assump:dom with some $1\leq q \leq\infty$. Moreover, let $1<p<\infty$ be such that the trace operator is continuous as a mapping $W^{1,p}(\Omega)^{d}\to L^{q}(\par In particular, we have Here, the underlying constants only depend on $\Omega$, $\Gamma$, $p$ and $

Figures (9)

  • Figure 1: Examples of relations covered by the framework in Section \ref{['sec:bc-slip']}.
  • Figure 2: Assumption \ref{['assump:dom']} and its geometric impact. Left-hand figure: An axisymmetric cone $\Omega$, for which $Ax\bot \boldsymbol{n}(x)$ holds for any $x\in\partial\Omega$, where $Ax\coloneqq x\times\xi$. Note that $\boldsymbol{n}(x)$ is always contained in the plane spanned by $\xi$ and $x$, and $Ax$ is orthogonal to this plane; see Example \ref{['ex:failureAssump']}. Right-hand figure: Polyhedral domains as the overall setting of the paper, and Corollary \ref{['cor:polyhedral']}. If $\Gamma\subset\partial\Omega$ is polyhedral and contains two non-collinear normals $\boldsymbol{n}(x),\boldsymbol{n}(\widetilde{x})$ as indicated for $x$ and $\widetilde{x}$, Assumption \ref{['assump:dom']} is satisfied.
  • Figure 3: Exact (red) and computed (blue) constitutive relation on $\Gamma_s$ for the smooth relation \ref{['eq:non-monotone-slip']}.
  • Figure 4: Wall stress and tangential velocity on $\Gamma_s$ for the smooth relation \ref{['eq:non-monotone-slip']}.
  • Figure 5: Exact (red) and computed (blue) constitutive relation on $\Gamma_s$ for the non-smooth problem \ref{['eq:nonsmooth_nonmonotone']}.
  • ...and 4 more figures

Theorems & Definitions (56)

  • remark 1: maximal monotonicity
  • theorem 1: Korn-type inequality with normal traces I
  • proof
  • corollary 1: Poincaré inequality
  • corollary 2: Korn-type inequality with normal traces II
  • proof
  • remark 2: optimality of Assumption \ref{['assump:dom']}
  • corollary 3: Assumption \ref{['assump:dom']} and polyhedral domains
  • proof
  • lemma 1: convergence lemma
  • ...and 46 more