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# Communications in Mathematical Sciences

## Volume 5 (2007)

### Number 1

### Steady States and Dynamics of $2-D$ Nematic Polymers Driven by an Imposed Weak Shear

Pages: 113 – 132

DOI: http://dx.doi.org/10.4310/CMS.2007.v5.n1.a5

#### Authors

#### Abstract

We study the $2-D$ Smoluchowski equation governing the evolution of orientational distribution of rodlike molecules under an imposed weak shear. We first recover the well-known isotropic-to-nematic phase transition result: in the absence of flow the isotropic-nematic phase transition occurs at $U =2$ where $U$ is the normalized polymer concentration, representing the intensity of the Maier-Saupe interaction potential. Then we show that in the presence of an imposed weak shear there is a threshold $(U\sb 0\approx 2:41144646)$ for $U$: When $U < U\sb 0$, steady state solution exists; otherwise there is no steady state. Furthermore, we carry out multi-scale asymptotic anlaysis to study the slow time evolution driven by the weak shear. It is revealed that, to the leading order, the order parameter of the orientational distribution is invariant with respect to time whereas the angular velocity of the director is position-dependent. When $U < U\sb 0$, the director of the orientational distribution converges to a stable steady state position; when $U > U\sb 0$, the angular velocity of the director is always positive and the orientational distribution will not reach a steady state. Finally, the effect of weak shear on the phase diagram is investigated. It is found that the phase relation under weak shear can be obtained from the pure nematic phase relation through a simple algebraic transformation.

#### Keywords

Nematic polymers; Kinetic theory; Maier-Saupe potential; Shear flow; Perturbation analysis

#### 2010 Mathematics Subject Classification

35Kxx, 70Kxx, 82D60