  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.
