Flow normal to the helical axis

Consider a cholesteric film between two plane parallel plates, one of which is moving with constant velocity V in its own plane. The plates occupy the planes z = h. We examine solutions of the form 

From the symmetry of the problem it is clear that 9 and v should be even 

It is seen that r + 0 even though the shear is confined to the zx plane in other words, secondary flow occurs. 

P = (Ag—Aj)sin0cos0/Hj. Leslie assumed the following boundary conditions  

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Fig. 4.5.5. Theoretical variation of the apparent viscosity with pitch P = 2n/q for flow normal to the helical axis of a cholesteric (or twist nematic) at low shear rates. Plot of versus P for twisted PAA. The separation between the...

The general theory of shear flow normal to the helical axis has been discussed by Leslie. This basically uses concepts similar to those described by the Ericksen-Leslie-Parodi theory of nematics. The main difference is that the twist term of the deformation free energy will be ... 

Most screws of SSEs are single flighted, with Ls = Ds, referred to as square-pitched screws. The radial distance between the root of the screw and the barrel surface is the channel depth, H. The main design variable of screws is the channel depth profile that is H(z), where z is the helical, down-channel direction, namely, the direction of net flow of the material. The angle formed between the flight and the plane normal to the axis is called the helix angle, 0, which, as is evident from Fig. 6.8, is related to lead and diameter... 

For waterworks applications, decanters normally will be operated with polymer addition facilities, at alternative admission points. They will need full erosion protection for the flights, using tiles. They will have some kind of cake baffle, possibly a cone, and a variable speed back-drive, with good differential and torque control. They will be operated with deep neutral ponds, with axial flow (i.e. with flight windows, such that the liquid flows parallel to the axis, rather than around the helical space), and with provision for wash-out prevention at start-up. 

Figure 30. Apparent viscosity, tj pp, in Poiseuille flow as a function of the shear rate for different chiral nematic pitch lengths (in micrometers) 1.9 (O), 2.6 (x), 3 (+), 3.9 ( ), 6 (A), 9.1 ( ), and °° (V) (redrawn from [122]). The helical axis was normal to the flow.

The above situation is true for weak stress gradients. When the stress gradient becomes larger than that needed to realign the helical axis, the flow will be normal to the helix, and the apparent viscosity becomes similar to that of nematics (see Figure 4.8). 

The explanation of the flow along the helix axis is much more challenging. This was done by Helfrich, based on a new concept called "permeation." The idea is the following. Imagine that we have a normal Poiseuille flow in a tube, i.e., the speed is larger inside than at the walls. This flow would evidently lead to a distortion of the helix. This distortion, however, requires a threshold torque, so Helfrich assumed that at low shear rates the flow occurs so that the helical structure remains intact. The velocity profile is flat and not parabolic (plug flow) (see Figure 4.7). 

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