Codimension bounds and rigidity of ancient mean curvature flows by the tangent flow at −∞

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, by adapting the work of Colding–Minicozzi [11], we prove codimension bounds for ancient mean curvature flows by their tangent flow at [Formula: see text]. In the case of the [Formula: see text]-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at [Formula: see text].

The Influence of Material Properties on the Hydrodynamic Performance of Propeller

Fluid-solid coupling numerical method of propellers is applied to make numerical calculations of propellers made of different materials without considering the material anisotropic. Comparative analysis of hydrodynamic performance and wake flow field of the propellers are carried on under typical working conditions. The influence of the key parameters of material on the efficiency of marine propellers is analyzed. The results show that the material with greater elasticity makes the magnitude of speed increase,especiously the axial velocity. On the other hand, compared with the strong rigidity material propeller, the small rigidity material propeller increases a little in the highest value of propeller efficiency curve.