Flows with Director Axis Reorientation

The preceding section deals with the case where the director axis is fixed during fluid flow. In more general situations director axis reorientation often accompanies fluid flows and vice versa. Taking into account the moment of inertia I and the torque T = hXf, where / is the molecular internal elastic field defined in Equation (3.11), is the torque associated with an externally applied field, and f is is the viscous torque associated with the viseous forces, the equation of motion describing the angular acceleration dQJdt as the director axis may be written as 

The viscous torque consists of two components one arising from pure rotational effect (i.e., no coupling to the fluid flow) given by Yi X JV and another arising from coupling to the fluid motion given by 72 Therefore, we have 

Here N is the rate of change of the director with respect to the immobile background fluid, given by 

From Equation (3.66) the viscous torque along the y direction is given by 

In the steady state,from whichthe shear torque vanishes,astablediiectoraxis orientation is induced by the flow with an angle 0flow given by 

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Consider the interaction geometiy shown in Figure 3.13. A homeotropically aligned nematic liquid crystal film is acted on by an electric or a magnetic field in the X direction. Let c ) denote the director axis reorientation angle from the original alignment direction z. Assume hard-boimdaiy conditions at the two cell walls at z = 0 and at z = <7. The flow is in the x direction, with a z dependence. 

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