Dynamics of spheroids in an unbound quadratic flow of a general second-order fluid

Abstract

This work employs the second-order fluid model to investigate the effect of first and second normal stresses on the motion of spheroidal particles in unbound parabolic flows, where particles migrate toward the flow center. We specifically examine the effects of fluid Weissenberg number Wi and the ratio of normal stress coefficients α = ψ2/ψ1. Previous works have considered the motion of spheroidal particles in the co-rotational limit (α = −0.5), where the effect of fluid viscoelasticity is to modify the fluid pressure but not the shear stresses. Here, we examine all ranges of α that are found for functional complex fluids such as dilute polymer solutions, emulsions, and particulate suspensions and determine how viscoelastic shear stresses alter particle migration. We use perturbation theory and the Lorentz reciprocal theorem to derive the O(Wi) corrections to the translational and rotational velocities of a freely suspended spheroid in an unbound tube or slit flow. Our results show that for both prolate and oblate particles, the viscoelasticity characterized by α significantly affects the particle cross-stream migration, but does not qualitatively change the trends seen in the co-rotational limit (α = −0.5). For a range of α (−0.9 ≤ α ≤ 0) investigated in this work, particles possess the largest mobility when α = −0.9 and smallest mobility when α = 0. Although α does not alter particle rotation at a given shear rate, we observe significant changes in particle orientation during migration toward the flow center because changes in migration speed give rise to particles experiencing different shear histories.

Experimental and theoretical studies of cross-stream migration of non-spherical particles in a quadratic flow of a viscoelastic fluid

“…In our previous analysis, 28,29 we identified that the particle mobility coefficient M D is directly proportional to the effective length spanned by the particle L sg in the shear gradient direction (y direction in the studied flow profile). 28 For a prolate particle that undergoes log-rolling (500 o t o 3000 in Fig.…”

“…Details are listed in our previous publication. 29 This strategy can in principle examine particle motion in all types of quadratic flows (tube and channel flow, flows with varying rotation rates, or far-field suction flow). 46,47 Below we summarize the main results for tube flow as well as quantify how particle shape influences the time scale for particle migration.…”

“…In our previous theoretical studies, [27][28][29] we utilized a secondorder fluid constitutive model 42 . We specifically studied the orientation and migration behavior of spherical and spheroidal particles (prolate and oblate), and provided scaling relations to quantify the effect of particle shape and orientation on the migration speed.…”

“…2 Long time orientation behavior of prolate and oblate spheroids in a weakly viscoelastic fluid under various flow profiles. [25][26][27][28][29] This paper is structured as follows: Section 2 reviews the theory of non-spherical particle migration in a general quadratic flow of a second-order fluid from our recent studies, [27][28][29] as well as the major observations on the particle migration behavior from the theory. We discuss the typical migration trajectories observed, and in particular introduce new scaling information on how the particle aspect ratio, particle orientation, and particle initial position affect the time scale for particle migration.…”

“…In this section, we briefly summarize the theory of nonspherical particle migration presented in our previous studies. [27][28][29] We also discuss the important observations from the theory and introduce new information on how particle geometry and initial orientation/position alter the time scale for particle migration.…”