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This paper presents a fully consistent theoretical framework for liquid crystal (LC) channel waveguides that have been experimentally demonstrated in previous publications. We revise the optical design of the LC waveguides in silicon grooves and implement here a vectorial, fully consistent model of the LC waveguide electro-optical behaviour, based on the finite element method. The numerical investigation shows that LC waveguides demonstrate properties of propagation control and switching. They switch on and off with a low applied voltage. We discuss the major design parameters of the device and the effect of loss-inducing control electrodes.