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The Thorne-Roberts variable flavour number scheme (VFNS) for heavy quarks ispresented in detail for the specific case of charged current DIS. As in neutralcurrent DIS this provides a smooth extrapolation from the fixed flavour numberscheme (FFNS) appropriate at low $Q^2$ to the zero-mass variable flavour numberscheme (ZM-VFNS) appropriate as $Q^2 \to \infty$, and differs from alternativeversions of a VFNS by the definition of the coefficient functions at eachorder, and the strict ordering of the expansion in $\alpha_S$. However, thereare subtle differences from the neutral current case which are addressed here.We discuss both the LO and NLO expressions, the latter unfortunately requiringsome (minimal) modelling due to the current lack of some necessary ${\calO}(\alpha^2_S)$ FFNS coefficient functions.Comment: 16 pages, 5 figures. Revised version to be published in Eur. Phys.

A variable flavour number scheme for charged current heavy flavour structure functions