  The paper develops a continuum theory of weak viscoelastic nematodynamics of
Maxwell type. It may describe the molecular elasticity effects in mono-domain
flows of liquid crystalline polymers as well as the viscoelastic effects in
suspensions of uniaxially symmetric particles in polymer fluids. Along with
viscoelastic and nematic kinematics, the theory employs a general form of
weakly elastic thermodynamic potential and the Leslie-Ericksen-Parodi type
constitutive equations for viscous nematic liquids, while ignoring inertia
effects and the Frank (orientation) elasticity in liquid crystal polymers. In
general case, even the simplest Maxwell model has many basic parameters.
Nevertheless, recently discovered algebraic properties of nematic operations
reveal a general structure of the theory and present it in a simple form. It is
shown that the evolution equation for director is also viscoelastic. An example
of magnetization exemplifies the action of non-symmetric stresses. When the
magnetic field is absent, the theory is simplified to the symmetric, fluid
mechanical case with relaxation properties for both the stress and director.
Our recent analyses of elastic and viscous soft deformation modes are also
extended to the viscoelastic case. The occurrence of possible soft modes
minimizes both the free energy and dissipation, and also significantly
decreases the number of material parameters. In symmetric linear case, the
theory is explicitly presented in terms of anisotropic linear memory
functionals. Several analytical results demonstrate a rich behavior predicted
by the developed model for steady and unsteady flows in simple shearing and
simple elongation.
