SUMMARYVarious adaptation techniques for the computation of viscous flows have been employed. This work describes the combination of adaptive grid-embedding and equation adaptation methods on semi-structured grids for compressible viscous flows. In this combination, Navier-Stokes equations are solved within appreciably viscous macro-cells, whereas for the remaining macro-cells the equations are reduced to the Euler equations. Meanwhile, subdividing is used in macro-cells from the beginning of the adaptation procedure without any restriction. This combination is used to solve three internal and external flow model problems. The combination of adaptive grid-embedding and equation adaptation is complex, but it is shown in this paper that it is efficient and worth to be used in CPU time-and memory space-consuming problems.