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A continuum two-fluid theory for dilute fiber suspensions. (English) Zbl 0702.76008

Summary: A dilute two-phase suspension of fibers consisting of slender, flexible, viscous fibers dispersed in a Newtonian fluid is considered. The fibers are of arbitrary length, and they slip relative to the carrier fluid. The primary purpose of the present theory is to predict and compute the motion and orientation of a collection of fibers in response to the carrier fluid flow field. The problem is treated via the two-fluid continuum approach, that is, the suspension is considered as two interacting continua, each continuum having its own concentration and velocity fields, and having its own set of conservation equations. In addition, the fiber phase has a unit vector assigned to each point, which describes the local fiber orientation. An evolution equation for determining the fiber orientation is derived. The fiber phase momentum equation includes inertia on a macro scale, includes momentum exchange with the carrier fluid, and includes the fiber phase stress tensor.
The fiber phase stress tensor is obtained by first considering an individual fiber and then generalizing, through a statistical approach, to the continuum description of a collection of fibers. The fiber phase stress tensor can be related to an effective viscosity, the fiber orientation, and the fiber velocity gradient. The effective viscosity of the fiber phase is found to depend on the viscosity of the fiber material and on the length of the fiber relative to a characteristic length. For long fibers, the effective viscosity asymptotes to the viscosity of the fiber material. For short fibers the effective viscosity depends, among other quantities, on the square of the fiber aspect ratio. Numerical results are presnted for the two-dimensional motion and orientation of fibers entering a laminar simple shear flow. It is found that the development of the fiber variables depends significantly on the effective viscosity of the fiber phase, in addition to depending on the fiber response time.

MSC:

76A05 Non-Newtonian fluids
76T99 Multiphase and multicomponent flows
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