A model of rod-like cholesteric liquid crystals subjected to slow steady simple shear and low frequency small amplitude oscillatory shear flow is formulated and applied to permeation flow, also known as Helfrich permeation flow [Helfrich (1969, 1970)], when the cholesteric helix is oriented along the velocity direction and the orientation distortions retain the original planar chiral structure. In permeation flow the elastic forces exactly balance viscous forces. The presence of a small internal length scale (pitch of the helix leads to boundary layer behavior in the velocity and orientation fields at any shear rate and at any driving frequency. The thickness of the boundary layer region is of the order of the pitch. The apparent viscosity is larger than typical nematic viscosities by a factor proportional to the ratio of the gap thickness to the pitch length. The effect is due to the Ericksen elastic stress. The response of cholesterics to small amplitude oscillatory shear corresponds to a purely viscous material. The out-of-phase stress component vanishes because the viscous force is exactly cancelled by the elastic force. In the terminal region, the loss modulus G" exhibits a classical frequency ω dependence (G" ∝ ω) but its magnitude is much greater than for other cholesteric flows by a factor proportional to the ratio of the gap thickness to the pitch length.
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