Bifurcations in the dynamics of a dipolar spheroid in a shear flow subjected to an external field
Abstract
When a dipolar spheroid is subjected to a shear flow in the presence of an external field, there is a torque due to the dipolefield interaction which tends to align the spheroid in the direction of the field, and a torque due to the shear flow which tends to rotate the particle in closed "Jeffrey orbits." The presence of an external field is known to result in complex phase portraits containing a limit cycle along with multiple stationary points [Almog and Frankel, J. Fluid Mech. 289, 243 (1995), 10.1017/S0022112095001327; Sobecki et al., Phys. Rev. Fluids 3, 084201 (2018), 10.1103/PhysRevFluids.3.084201]. When the external field is in the flow plane, depending on the strength and orientation of the field, the phase portrait in orientation space could have two to six stationary nodes and/or a limit cycle. When the external field strength is low, there are two stationary points off the flow plane which are both stable/unstable and an unstable/ stable limit cycle on the flow plane. When the external field strength is high, there is one stable node where the particle orientation is parallel to the field, and one unstable node where the particle orientation is antiparallel to the field. As the external field strength is increased, the manner in which the phase portrait evolves depends on the external field orientation with respect to the flow direction and the particle shape factor. It is shown that complex phase portraits result from the relatively simple variations in the location of the stationary points in the orientation space as the parameter Σ is varied. Here Σ is the dimensionless ratio of the torques due to the external field and the shear flow. Depending on the aspect ratio of the particle, there are up to two saddlenode bifurcations, two subcritical bifurcations on the flow plane, and one reverse saddlenode merger of two stationary points off the flow plane. The limits of an ideal thin rod/disk are shown to be singular limits, where there are no stationary points off the flow plane even for a low external field, and the orientation of the particle changes discontinuously as the crossstream component of the external field changes sign.
 Publication:

Physical Review Fluids
 Pub Date:
 March 2020
 DOI:
 10.1103/PhysRevFluids.5.033701
 Bibcode:
 2020PhRvF...5c3701K