Inertial migration of slender prolate and thin oblate spheroids in plane Poiseuille flow

A neutrally buoyant rigid spheroid suspended in a wall-bounded plane Poiseuille flow undergoes cross-stream migration, driven by inertia. We examine theoretically the migration of a spheroid of aspect ratio kappa <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>κ</mml:mi> </mml:math> $\kappa$ in the limit of small but finite particle Reynolds number ( italic Re Subscript p <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> ${\textit{Re}}_{\!p}$ ), and for small confinement ratios ( lamda <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>λ</mml:mi> </mml:math> $\lambda$ ), with the channel Reynolds number, italic Re Subscript c Baseline equals italic Re Subscript p Baseline divided by lamda squared <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:mrow> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Re</mml:mtext> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> ${\textit{Re}}_c = {\textit{Re}}_{\!p}/\lambda ^2$ , assumed arbitrary; here, lamda equals upper L divided by upper H <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:math> $\lambda =L/H$ with upper L <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>L</mml:mi> </mml:math> $L$ being the semimajor axis of the spheroid and upper H <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>H</mml:mi> </mml:math> $H$ denoting the separation between channel walls. For small <jats:alt-text content-type="machine-generated"