In this work, fiber suspension rheology in any orthogonal system of coordinates is described in terms of the second- and fourth-order orientation tensors. At present, the theoretical basis of the dilute and semi-dilute fiber suspensions are reviewed generally. Due to the symmetry properties of the orientation tensors and normalization condition of the orientation distribution function the independent components of the second-order tensor are reduced from original “9” to “5” while the independent components of the fourth-order tensor are reduced from “81” to “13” components in any orthogonal system of coordinates. Moreover in contracted notation, these components are reduced to “3” and “5” for the second and fourth-order tensors; respectively. The closure approximation method, i.e. to approximate the even higher order tensors in terms of the lower order one; herein the fourth-order tensor in terms of the second-order tensor, is used to solve the equation of state. Various closure approximations are reviewed. Basis of some constitutive models that describe the rheological stresses in fiber suspensions are outlined. The transition from macrostructure state to the microscopic one are briefly investigated through the Folgar-Tucker as well as Advani-Tucker models. Finally, exact solutions of simple shearing flow for dilute and semi-dilute rigid fiber suspensions are performed. |
