2023/10/01 by Arjun Sharma, Sharma, Arjun, Donald L. Koch +1
Chemical Engineering · Physics and Astronomy · #Rheology and Fluid Dynamics Studies #Experimental and Theoretical Physics Studies
paper · pdf · doi:10.48550/arxiv.2310.00792
The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio, κ. A regular perturbation expansion in the polymer concentration, c, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the O(c) correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviors qualitatively dependent upon c⋅ De (De is the imposed shear rate times the polymer relaxation time) and κ and quantitatively on c. At a small but finite c⋅ De, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing κ, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate c⋅ De, a particle starting near the flow-gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller c⋅ De) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing c⋅ De, the saddle-node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing c⋅ De, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.