we report on our recent efforts on the formulation and the evaluation of a non-overlapping domain decomposition method for the parallel solution of two-dimensional compressible viscous flows. This work extends a previous study [11] which was concerned with the design of a domain decomposition solver for the Euler equations discretized on unstructured triangular meshes. As in [11], the method relies on the formulation of an additive Schwarz type algorithm where the interface conditions express the continuity of the normal flux components. The starting point is a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes for the spatial approxima- tion. Time integration of the resulting semi-discrete equations is performed by using a linearized backward Euler implicite scheme. As a result, each pseudo time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical fluxes.