A new idea is suggested to extend the viscous vector equations in non-orthogonal curvilinear coordinates.In the expansion equation obtained with the help of this new method

the number of terms is greatly decreased

so that it is easy to apply to engineering calculation.Meanwhile the expanded forms of compressible and incompressible equations are also presented and a generalized expression is given.Finally

by the aid of this new method

the numerical calculations have been carried out for 2-dimensional compressible viscous cascade flow and 3-dimensional compressible viscous cascade flow.It is shown that the new expansion equation is of great value to engineering practice.