A new nonlinear wave equation of a finite deformation elastic circular rod simultaneously introducing transverse inertia and shearing strain was derived by means of Hamilton principle. The nonlinear equation includes two nonlinear terms caused by finite deformation and double geometric dispersion effects caused by transverse inertia and transverse shearing strain. Nonlinear wave equation and corresponding truncated nonlinear wave equation were solved by the hyperbolic secant function finite expansion method. The solitary wave solutions of these nonlinear equations were obtained. The necessary condition of these solutions existence was given also.

In this paper, V.V. Golubef method is first extended to the diffuse laminar flow between two parallel spherical surfaces. With the boundary layer motion equation in spherical coordinates, we derive the momentum integral equation together with the energy integral equation for the laminar boundary layer of the entrance region between two parallel spherical surfaces. And then by applying Picards gradually approaching method for the momentum integral equation, we get the approximate expression which the entrance region length varies with the thickness of boundary layer. In the en~every coefficient of entrace region effect is analyzed and calculated.