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# Communications in Mathematical Sciences

## Volume 11 (2013)

### Number 3

### On the Doi model for the suspensions of rod-like molecules: global-in-time existence

Pages: 831 – 850

DOI: http://dx.doi.org/10.4310/CMS.2013.v11.n3.a8

#### Authors

#### Abstract

The Doi model for the suspensions of rod-like molecules in a dilute regime describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained. The orientation distribution of the rods on the microscopic level is described by a Fokker-Planck-type equation on the sphere, while the fluid flow is given by the Navier-Stokes equations, which are now enhanced by an additional macroscopic stress $σ$ reflecting the orientation of the rods on the molecular level. Prescribing arbitrarily the initial velocity and the initial orientation distribution in suitable spaces we establish the global-in-time existence of a weak solution to our model defined on a bounded domain in three dimensional space. The proof relies on a quasi-compressible approximation of the pressure, the construction of a sequence of approximate solutions, and the establishment of compactness.

#### Keywords

Doi model, suspensions of rod-like molecules, global-in-time existence, fluid-particle interaction model, polymeric fluids, Navier-Stokes equations, incompressible and viscous flows, Smoluchowski equation

#### 2010 Mathematics Subject Classification

35B40, 35B45, 35Dxx, 76N10