<p>In this paper, the dynamic problem of a V-notch at the lower interface covered by the viscous fluid coating is studied. Firstly, the expression for the incident SH wave in the viscous fluid coating is obtained by Navier–Stokes equations. Then, the analytical expression of standing wave is established by the fractional Bessel function expansion method and Graf addition theorem. Finally, large-arc assume method is applied, the elastic half space base and viscous fluid coating are divided into two strips along the horizontal interface, the straight boundaries are converted into curved boundaries, and the expressions of scattering waves caused by curved boundaries are obtained. The integral equations are set up through boundary conditions and solved by applying orthogonal function expansion technique and effective truncation. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article. The innovation in research methods of this article is the introduction of the “large-arc assume method” and “Navier–Stokes equations.” After calculation, it can be concluded that: When <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4335_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta_{1} = {{3\pi } \mathord{\left/ {\vphantom {{3\pi } 4}} \right. \kern-0pt} 4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mfenced open="/"> <mphantom> <mpadded width="0pt"> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mpadded> </mphantom> </mfenced> </mrow> <mn>4</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the value of DSCF reaches the maximum 6.61 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4335_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( {\theta = - 180^\circ } \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mrow> <mi>θ</mi> <mo>=</mo> <mo>-</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> </mrow> </mfenced> </math></EquationSource> </InlineEquation>.</p>

Dynamic performance of V-notch at the lower interface covered by the viscous fluid coating: Navier–Stokes equations

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发表日期：

2025年4月25日

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Xi-meng Zhang,
Hui Qi,
Yi-ning Wu

摘要

In this paper, the dynamic problem of a V-notch at the lower interface covered by the viscous fluid coating is studied. Firstly, the expression for the incident SH wave in the viscous fluid coating is obtained by Navier–Stokes equations. Then, the analytical expression of standing wave is established by the fractional Bessel function expansion method and Graf addition theorem. Finally, large-arc assume method is applied, the elastic half space base and viscous fluid coating are divided into two strips along the horizontal interface, the straight boundaries are converted into curved boundaries, and the expressions of scattering waves caused by curved boundaries are obtained. The integral equations are set up through boundary conditions and solved by applying orthogonal function expansion technique and effective truncation. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article. The innovation in research methods of this article is the introduction of the “large-arc assume method” and “Navier–Stokes equations.” After calculation, it can be concluded that: When $\beta_{1} = {{3\pi } \mathord{\left/ {\vphantom {{3\pi } 4}} \right. \kern-0pt} 4}$ $β_{1} = 3 π /) 4$ , the value of DSCF reaches the maximum 6.61 $\left( {\theta = - 180^\circ } \right)$ $(θ = - 180^{\circ})$ .