Taylor dispersion in sedimentation of an axisymmetric Brownian particle with centre offset

When a non-spherical particle sediments, its velocity generally changes in time as the particle orientation changes in time. This gives extra dispersion of the particle position in addition to the thermal Brownian motion. Brenner ( J. Colloid Interf. Sci. , vol. 71, 1979, pp. 189–208) studied this effect and formulated how to calculate the gravity-induced dispersion (called Taylor dispersion in sedimentation). However, he conducted the explicit calculation only for torque-free particles which keep an isotropic orientational distribution in the steady-state. In this paper, we study the effect of the gravitational torque on the Taylor dispersion. We limit the analysis to particles having uniaxial symmetry. In this case, the gravitational torque is caused by the offset l Subscript c <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:mi>l</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $l_{{c}}$ , the distance between the hydrodynamic centre and the gravitational force centre. The effect of the gravitational torque is represented by a dimensionless parameter alpha <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>α</mml:mi> </mml:math> $\alpha$ (called the Langevin parameter by Brenner) which is proportional to l Subscript c <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:mi>l</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $l_{{c}}$ . We obtain analytical expressions for the Taylor diffusivity for the two limits alpha much less than 1 <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>α</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:math> $\alpha \ll 1$ and alpha much greater than 1 <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>α</mml:mi> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn> </mml:math> $\alpha \gg 1$ . We show that the offset gives a significant effect on the diffusivity and changes the classical scaling of the Taylor dispersion at a large Péclet number. We also analyse the transient regime of the mean square displacement and show how the crossing time from the ballistic regime to the diffusive regime depends on the gravitational torque.