Theoretical Studies of Cross-Stream Migration Non-Spherical Particles in an Arbitrary Quadratic Flow of a General Second-Order Fluid
Shiyan Wang,
Purdue University, West Lafayette, USA
Abstract
Cross-stream particle migration in viscoelastic
suspensions is essential in many biological as well as industrial processes,
yet the effect of particle shape [1] on this phenomenon is not well understood.
In this work, we perform a theoretical study on the dynamics of an
arbitrary-shaped particle in a quadratic flow of a general second-order fluid
[2,3]. In this analysis, we solve for the polymeric
force and torque on a particle by performing a perturbation expansion on the
fluid velocity and pressure field using the Weissenberg number (Wi) as a small parameter. The force and torque are evaluated
to O(Wi) using the Lorentz reciprocal theorem. The
total polymeric force and torque can be summed into two parts: (i) an
analytical solution where 
(
and 
being
the first and second normal stress coefficients), and (ii) a remaining part
where 
that requires numerical evaluation. The total
solution compares well to previous studies on spherical particles. We then
apply the derived solutions to investigate the migration and orientation
dynamics of ellipsoidal particles in different quadratic
flow profiles (e.g., slit-like flows, tube-like flows). We will identify the
key factors that governs the migration dynamics of the particles in different
non-Newtonian fluids (dilute polymer solutions, emulsions, and colloidal
dispersions) based on the ratio of normal stress coefficients. We finally
compare the results of the general model and to previous models that made the
co-rotational assumption of ,
and comment on the effectiveness of the latter model in describing qualitative physics.
References:
[1] Martin, C.P., Wang, S. and Kim, S., 2019. Surface
tractions on an ellipsoid in Stokes flow: Quadratic ambient fields. Physics of
Fluids, 31(2), p.021209.
[2] Tai, C.W., Wang, S. and Narsimhan,
V., 2020. Cross-stream migration of non-spherical particles in a second-order
fluid–theories of particle dynamics in arbitrary quadratic flows. J. Fluid Mech, 895(A6).
[3] Wang, S., Tai, C.W. and Narsimhan,
V., 2020. Dynamics of spheroids in an unbound quadratic flow of a general
second-order fluid. Physics of Fluids, 32(11), p.113106.
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