<p>This study investigates the rectilinear oscillations of two coaxially aligned spherical particles in an unbounded couple stress fluid at low Reynolds numbers, addressing a fundamental problem in microfluidics and biomechanics where microstructure effects dominate. The importance lies in applications such as drug delivery and material processing, where understanding particle-fluid interactions is critical. The unsteady Stokes equations were solved using a superposition of fundamental solutions in spherical coordinates, centered on each particle, with no-slip boundary conditions enforced via a collocation method. Key results include the quantification of in-phase and out-of-phase drag force coefficients, revealing that increasing the couple stress parameter (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_11707_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar{\eta }\)</EquationSource> </InlineEquation>) enhances drag forces by up to 50% for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_11707_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar{\eta } = 0.9\)</EquationSource> </InlineEquation> compared to Newtonian cases (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_11707_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\bar{\eta } = 0\)</EquationSource> </InlineEquation>). Numerical simulations demonstrated robust convergence across dimensionless parameters (e.g., separation distance <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_11707_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta\)</EquationSource> </InlineEquation>, frequency <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_11707_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation>), with tabulated data showing agreement within % of established solutions for steady-state and single-sphere oscillations. Novelty arises from extending prior work on viscous fluids to couple stress fluids, uncovering how microstructural effects amplify drag and alter oscillation dynamics. For instance, at <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_11707_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha = 60\)</EquationSource> </InlineEquation>, drag forces increased by 30% for closely spaced spheres (<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_11707_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta = 1.05\)</EquationSource> </InlineEquation>), highlighting the interplay between frequency and microstructure. This work advances predictive models for complex fluids and provides design insights for microfluidic systems.</p>

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Oscillatory interactions of two spheres in an unbounded couple stress fluid

  • Shreen El-Sapa,
  • Azza M. Algatheem

摘要

This study investigates the rectilinear oscillations of two coaxially aligned spherical particles in an unbounded couple stress fluid at low Reynolds numbers, addressing a fundamental problem in microfluidics and biomechanics where microstructure effects dominate. The importance lies in applications such as drug delivery and material processing, where understanding particle-fluid interactions is critical. The unsteady Stokes equations were solved using a superposition of fundamental solutions in spherical coordinates, centered on each particle, with no-slip boundary conditions enforced via a collocation method. Key results include the quantification of in-phase and out-of-phase drag force coefficients, revealing that increasing the couple stress parameter ( \(\bar{\eta }\) ) enhances drag forces by up to 50% for \(\bar{\eta } = 0.9\) compared to Newtonian cases ( \(\bar{\eta } = 0\) ). Numerical simulations demonstrated robust convergence across dimensionless parameters (e.g., separation distance \(\delta\) , frequency \(\alpha\) ), with tabulated data showing agreement within % of established solutions for steady-state and single-sphere oscillations. Novelty arises from extending prior work on viscous fluids to couple stress fluids, uncovering how microstructural effects amplify drag and alter oscillation dynamics. For instance, at \(\alpha = 60\) , drag forces increased by 30% for closely spaced spheres ( \(\delta = 1.05\) ), highlighting the interplay between frequency and microstructure. This work advances predictive models for complex fluids and provides design insights for microfluidic systems.