A Note on the Kinematics of Rigid Molecules in Linear Flow Fields and Kinetic Theory for Biaxial Liquid Crystal Polymers Jun Li ∗ , Sarthok Sircar † , and Qi Wang ‡ Abstract We present a systematic derivation of the extended Jefferys’ orbit for rigid ellipsoidal and V-shaped polymer molecules in linear incompressible viscous flows using a Lagrange multiplier’s method based on a constraining force argument [5]. It reproduces the well-known Jefferys’ orbit for rotating ellipsoids [12]. The method is simple and applicable to any rigid body immersed in a linear flow field so long as a discrete set of representative points on the rigid body can be identified that possess the same rotational degrees of freedom as the rigid body itself. The kinematics of a single V-shaped rigid polymer driven by a linear flow field are discussed, where steady states exist along with time-periodic states in limited varieties. Finally, we show how the kinematics of the rigid V-shaped polymer can be used in the derivation a kinetic theory for the solution of rigid biaxial liquid crystal polymers. Keywords: Kinematics, kinetic theory, linear flows, ellipsoids, biaxial liquid crystal polymers, V-shaped polymer. 1 Introduction The configurational space kinetic theory for rigid polymers or solid suspensions in another fluid is built upon two basic ingredients, the interaction potential to each point in the configurational phase space and the kinematics of the point under imposed flow fields, where each pint in the configurational space represents the full configuration of a polymer or the suspension particle [1, 5]. When the host fluid is viscous and incompressible, the kinematics of the phase point in the configurational space is often derived with respect to an imposed linear flow field which is an exact solution of the Stokes equation. Jefferys studied the kinematics of an ellipsoid immersed in a viscous fluid (Stokes fluid) and derived the well-known Jefferys’ orbit for the rotating ellipsoid about its own center of mass [12]. This was later used in many theory development and applications [2, 13, 11, 16]. Eshelby [9] and recently Wetzel and Tucker [17] examined the kinematics of an ellipsoidal inclusion in elastic and viscous media. They derived the explicit formula for the kinematics of the three major axes of an ellipsoidal inclusion using the Eshelby tensor. More recently, a number of theories for polymer blends have been developed based on deformable ellipsoidal droplets, whose constitutive equation also relies on the kinematics of the deformable ellipsoids in imposed linear flow fields [7, 6]. With the surging interest in modeling dynamics of suspensions and/or rigid polymer molecules of biaxial symmetry using kinetic theories, there is the need to provide an on-the-fly method to derive the kinematics of the representative axes of a rotating rigid molecule or particle in imposed * School of Mathematics, Nankai University, Tianjin, P. R. China. † Department of Mathematics, University of South Carolina, Columbia, SC 29208. ‡ Department of Mathematics and NanoCenter at USC, University of South Carolina, Columbia, SC 29208. Tel: (803)-777-6268, qwang@math.sc.edu. 1 electronic-Liquid Crystal Communications January 22, 2009 http://www.e-lc.org/docs/2009_01_21_22_53_30