Thermodynamic Theory for Fiber Suspensions

In this paper three different approaches towards a continuum theory of fiber suspensions are discussed. The first one is the classical Thermodynamics of Irreversible Processes with internal variables. It derives constitutive equations for fiber suspensions on the basis of ONSAGERs phenomenological coefficients, which are related to the mechanical properties of the fibers. Secondly another method of exploiting the dissipation inequality, the method introduced by LIU is applied in order to derive results on constitutive equations. Finally a mesoscopic background theory is discussed. In this approach a distribution of different fiber orientations and fiber deformations is assumed. The fiber orientation and deformation are additional variables in the domain of field quantities. The new field quantities on the enlarged set of variables obey balance equations. The mesoscopic balance of mass results in an equation of motion for the distribution of fiber orientations and deformations. The usual macroscopic fields of continuum mechanics are averages of mesoscopic fields over the different orientations and deformations. Macroscopic quantities characterizing the orientational order, the alignment tensors are defined by the help of the orientation distribution function. They are internal variables in the macroscopic theory characterizing the distribution of fiber orientations. The result of the Irreversible Thermodynamics approach mainly concerns the stress tensor. Cauchy’s stress is usually assumed to be symmetric, but in fiber suspensions it has an anti-symmetric part, too. The skew-symmetric part of the stress shows up in the balance equation of angular momentum; it is related to the torque density, the rate of internal moment of momentum, and the couple stress. The latter is a result of the internal structure of the material. The fibers are assumed to be microscopic and the stress field smooth on macroscopic length scale. If the local structure of the flowing medium may be characterized by second order pseudo (axial) tensors as internal variables then in the linear Onsager equations there appears couple stress coupled to the gradient of the angular velocity. The results of the Liu-procedure are consistent with this observation. The couple stress can be calculated as a combination of derivatives of the free energy density (see equation (3.31)), which may have an anti-symmetric part, too. This result may be used to calculate the couple stress from an ansatz for the free energy density, taking into account the deformation of fibers. Another result of the Liu-exploitation of the dissipation inequality is that the assumption of the entropy flux being heat flux over temperature does not contradict the second law of thermodynamics, but it does not necessarily follow from the dissipation inequality. Date: February 2, 2008.