We present an application of the Residual-based variational multiscale (RBVMS) methodology to the computation of laminar eccentric annular pipe flow with eccentricities of 0.5 and 1. Isogeometric analysis is utilized for higher order approximation of the geometry and solution using Non-uniform rational B-splines (NURBS) functions. The ability of NURBS exactly representing curved circular geometries makes NURBS-based isogeometric analysis attractive for the application to the flow through the eccentric annuli. By using the exact representation of circular boundary, the limiting case of the eccentricity, i.e. the inner circular wall actually touches the outer circular wall, is successfully demonstrated.