<p>In this paper we propose an effective approximation method for solving a mucociliary clearance model described by the volume-averaged Navier–Stokes equations with complex mix-boundary conditions and flow geometry interacted by the solid motion. The nonlinear equations are resolved using modified Chorin’s projection to tackle terms of inhomogeneous divergence and drag force, and this method discretizes the temporal variable with an implicit–explicit finite difference method and decouples the system to several elliptic equations and Poisson-type equations. Then, the spatial variable is discretized by the finite element methods, and the numerical experiments show that the fully discrete scheme is effective with the convergence order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3121_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(\Delta t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3121_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}(h^{1+s})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>s</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3121_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$0< s\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>). We also give a biomedical simulation with non-negative stochastic inflow conditions generated by Brownian excursions.</p>

A finite element approximation for the simulation of the flow impacted by metachronal coordination between beating cilia

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发表日期：

2025年2月17日

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Yiying Wang,
Yongkui Zou,
Shimin Chai

摘要

In this paper we propose an effective approximation method for solving a mucociliary clearance model described by the volume-averaged Navier–Stokes equations with complex mix-boundary conditions and flow geometry interacted by the solid motion. The nonlinear equations are resolved using modified Chorin’s projection to tackle terms of inhomogeneous divergence and drag force, and this method discretizes the temporal variable with an implicit–explicit finite difference method and decouples the system to several elliptic equations and Poisson-type equations. Then, the spatial variable is discretized by the finite element methods, and the numerical experiments show that the fully discrete scheme is effective with the convergence order $\mathcal {O}(\Delta t)$ $O (Δ t)$ and $\mathcal {O}(h^{1+s})$ $O (h^{1 + s})$ ( $0< s\le 1$ $0 < s \leq 1$ ). We also give a biomedical simulation with non-negative stochastic inflow conditions generated by Brownian excursions.