Local and Global Existence of Solutions to Equations for Flows of Fibre Suspensions

Abstract

We establish the existence and uniqueness, locally and globally in time, of solutions to the governing equations for fibre suspension flows, for sufficiently small data. The linear and quadratic closure rules are considered, and the rotary diffusivity is assumed to be constant. The existence of a unique classical solution, local in time, is proven for the cases of both linear and quadratic closure rules. By restricting consideration to the physically significant case of constant rotary diffusivity it is possible to obtain the first results on global existence for this problem. In particular, the existence and uniqueness of a classical solution, globally in time, is established for sufficiently small data. The solution is shown to be stable in the absence of a body force. Existence and uniqueness of solutions to the steady problem are also established.

A Stability Result on the Orientation of Red Blood Cells in the Venule Network

“…The use of only the second-order tensor amounts to an approximation of Ψ (p) by truncating the series. Thus, a tensor representation is always an approximate description [1,11,[16][17][18]. However, in many theories the second-and fourth-order tensors give exactly the information needed to determine many physical properties of the suspension [1,11,[16][17][18].…”

“…Equations (27) and (28) are know as the "evolution equation" for the orientation tensor A [1,11,[16][17][18].…”

“…The extra stress E is found by solving for the stress field around a single massless particle. Existing theories which predict the viscosity of suspensions of particles in a Newtonian fluid, all lead to expressions of the form [1,11,17,18]:…”

“…The model studied in this article was used by many authors to model fibre suspension flows, polymers,... [2,3,14,[16][17][18]. This model will be suitable to model blood in the region under consideration as the primary rouleaux formations are assumed rigid and short and contact between rouleaux is not frequent.…”

“…Munganga et al [16][17][18] studied the thermodynamic stability, existence and uniqueness of solutions in the absence of body forces. Existence and uniqueness, both locally and globally in time, of solutions to the governing equations for suspension flows, were established for sufficiently small data.…”