__ ABSTRACT:__
Cross stream migration and stable orientations of elliptic particles
falling in an Oldroyd-B fluid in a channel are studied. We show that
the normal component of the extra stress on a rigid body vanishes;
lateral forces and torques are determined by the pressure. Inertia
turns the longside of the ellipse across the stream and elasticity
turns it along the stream; tilted off-center falling is unstable.
There are two critical numbers; elasticity and Mach numbers. When the
elasticity number is smaller than critical the fluid is essentially
Newtonian with broadside-on falling at the centerline of the channel.
For larger elasticity numbers the settling turns the longside of the
particle along the stream in the channel center for all velocities
below a critical one, identified with a critical Mach number of order
one. For larger Mach numbers the ellipse flips into broadside-on
falling again. The critical numbers are functions of the channel
blockage ratio, the particle aspect ratio and the
retardation/relaxation time ratio of the fluid. Two ellipses falling
nearby, attract, line-up and straighten-out in a long chain of
ellipses with longside vertical, all in a row. Stable, off-center
tilting is found for ellipses falling in shear thinning fluids and for
cylinders with flat ends in which particles tend align their longest
diameter with gravity.

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- Part #1;
- Part #2;
- Part #3;
- Supplement: Normal stresses at the boundary of solid rigid in an Oldroyd-B fluid.

Mail to: huang@aem.umn.edu for any comment on this paper.