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In this paper, a new algorithm based on differential geometry viewpoint to solve the 3D rotating Navier-Stokes equations with complex Boundary is proposed, which is called Bi-parallel algorithm. For xample, it can be applied to passage flow between two blades in impeller and circulation flow through aircrafts with complex geometric shape of boundary. Assume that a domain in $R^3$ can be decomposed into a series sub-domain, which is called "flow layer", by a series smooth surface $\Im_k, k=1,...,M$. Applying differential geometry method, the 3D Navier-Stokes operator can be split into two kind of operator: the "Membrane Operator" on the tangent space at the surface $\Im_k$ and the "Bending Operator" along the transverse direction. The Bending Operators are approximated by the finite different quotients and restricted the 3D Naver-Stokes equations on the interface surface $\Im_k$, a Bi-Parallel algorithm can be constructed along two directors: "Bending" direction and "Membrane" directors. The advantages of the method are that: (1) it can improve the accuracy of approximate solution caused of irregular mesh nearly the complex boundary; (2) it can overcome the numerically effects of boundary layer, whic is a good boundary layer numerical method; (3) it is sufficiency to solve a two dimensional sub-problem without solving 3D sub-problem., Comment: 44 pages, 8 figures

In this paper A Geometrical Design Method for Blade's surface $\Im$ in the impeller is provided here $\Im$ is a solution to a coupling system consisting of the well-known Navier-Stokes equations and a four order elliptic boundary value problem . The coupling system is used to describe the relations between solutions of Navier-Stokes equations and the geometry of the domain occupied by fluids, and also provides new theory and methods for optimal geometric design of the boundary of domain mentioned above. This coupling system is the Eular-Lagrange equations of the optimal control problem which is describing a new principle of the geometric design for the blade's surface of an impeller. The control variable is the surface of the blade and the state equations are Navier-Stokes equations with mixed boundary conditions in the channel between two blades. The objective functional depending on the geometry shape of blade's surface describes the dissipation energy of the flow and the power of the impeller. First we prove the existence of a solution of the optimal control problem. Then we use a special coordinate system of the Navier-Stokes equations to derive the objective functional which depends on the surface $\Theta$ explicitly. We also show the weakly continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface., Comment: 27 pages