Shearing of Flow-Aligning Nematics

The competition between and flow-induced and boundary-induced orientation is quantified by an Ericksen number, defined by 

Here is a typical Leslie viscosity, AT is a Frank constant, V is a flow velocity, /i is a length scale of the flow geometry, such as the tube diameter in Poiseuille flow, and ftn is the average shear rate. The Ericksen number is the ratio of the flow-induced viscous stress 6Ye.fi = /h to the Frank stress K/h. The appropriate Leslie viscosity or Frank constant 

In flow through a tube, therefore, the measured effective viscosity, which is defined to be proportional to the pressure drop, depends on the Ericksen number. Note that the Ericksen number is proportional to Vh, the velocity times the tube diameter, where we take h = 2R. Since the velocity V is proportional to QfR, Vh is proportional to Q/R. Thus, the data for various tube radii in Fig. 10-11a collapse onto a single line when plotted against AQfjtR (see Fig. 10-1 lb). This shows that the effective viscosity is a function of the Ericksen number, which is proportional to the velocity times the tube diameter. (For shear-thinning isotropic liquids, on the other hand, the viscosity depends on y ff, which is the velocity divided by the tube diameter.) Because of the orientation-dependence of the viscosity (illustrated in Fig. 10-9a), the wall layer is much more viscous than the core fluid and since the thickness 5 of the wall layer scales as it follows 

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