#### Document Type

Article

#### Publication Date

2006

#### DOI

10.1063/1.2359232

#### Publication Title

Physics of Fluids

#### Volume

18

#### Issue

10

#### Pages

103102 (15 pages)

#### Abstract

We establish reciprocity relations of the Doi-Hess kinetic theory for rigid rod macromolecular suspensions governed by the strong coupling among an excluded volume potential, linear flow, and a magnetic field. The relation provides a reduction of the flow and field driven Smoluchowski equation: from five parameters for coplanar linear flows and magnetic field, to two field parameters. The reduced model distinguishes flows with a rotational component, which map to simple shear (with rate parameter) subject to a transverse magnetic field (with strength parameter), and irrotational flows, for which the reduced model consists of a triaxial extensional flow (with two extensional rate parameters). We solve the Smoluchowski equation of the reduced model to explore: (i) the effect of introducing a coplanar magnetic field on each sheared monodomain attractor of the Doi-Hess kinetic theory and (ii) the coupling of coplanar extensional flow and magnetic fields. For (i), we show each sheared attractor (steady and unsteady, with peak axis in and out of the shearing plane, periodic and chaotic orbits) undergoes its own transition sequence versus magnetic field strength. Nonetheless, robust predictions emerge: out-of-plane degrees of freedom are arrested with increasing field strength, and a unique flow-aligning or tumbling/ wagging limit cycle emerges above a threshold magnetic field strength or modified geometry parameter value. For (ii), irrotational flows coupled with a coplanar magnetic field yield only steady states. We characterize all (generically biaxial) equilibria in terms of an explicit Boltzmann distribution, providing a natural generalization of analytical results on pure nematic equilibria [P. Constantin, I. Kevrekidis, and E. S. Titi, Arch. Rat. Mech. Anal. 174, 365 (2004); P. Constantin, I. Kevrekidis, and E. S. Titi, Discrete and Continuous Dynamical Systems 11, 101 (2004); P. Constantin and J. Vukadinovic, Nonlinearity 18, 441 (2005); H. Liu, H. Zhang, and P. Zhang, Comm. Math. Sci. 3, 201 (2005); C. Luo, H. Zhang, and P. Zhang, Nonlinearity 18, 379 (2005); I. Fatkullin and V. Slastikov, Nonlinearity 18, 2565 (2005); H. Zhou, H. Wang, Q. Wang, and M. G. Forest, Nonlinearity 18, 2815 (2005)] and extensional flow-induced equilibria [Q. Wang, S. Sircar, and H. Zhou, Comm. Math. Sci. 4, 605 (2005)]. We predict large parameter regions of bi-stable equilibria; the lowest energy state always has principal axis aligned in the flow plane, while another minimum energy state often exists, with primary alignment transverse to the coplanar field. *© 2006 American Institute of Physics*.

#### Original Publication Citation

Forest, M. G., Sircar, S., Wang, Q., & Zhou, R. (2006). Monodomain dynamics for rigid rod and platelet suspensions in strongly coupled coplanar linear flow and magnetic fields. II. Kinetic theory. *Physics of Fluids, 18*(10), 103102 doi:10.1063/1.2359232

#### Repository Citation

Forest, M. Gregory; Sircar, Sarthok; Wang, Qi; and Zhou, Ruhai, "Monodomain Dynamics for Rigid Rod and Platelet Suspensions in Strongly Coupled Coplanar Linear Flow and Magnetic Fields. II. Kinetic theory" (2006). *Mathematics & Statistics Faculty Publications*. 14.

http://digitalcommons.odu.edu/mathstat_fac_pubs/14