The Boundary Layer Equations and a Dimensional Split Method for Navier-Stokes Equations in Exterior Domain of a Spheroid and Ellipsoid

In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.

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Su, J. , Fan, H. , Feng, W. , Chen, H. and Li, K. (2015) The Boundary Layer Equations and a Dimensional Split Method for Navier-Stokes Equations in Exterior Domain of a Spheroid and Ellipsoid. International Journal of Modern Nonlinear Theory and Application, 4, 48-87. doi: 10.4236/ijmnta.2015.41005.

 [1] Li, K.T. and Huang, A.X. (2013) Boundary Shape Control of Navier-Stokes Equations and Dimensional Splitting Methods and Its Applications. Science Press, Beijing (in Chinese). [2] Li, K.T., Chen, H. and Yu, J.P. (2013) An New Boundary Layer Equations and Applications to Shape Control. Scientia Sinica (Mathematics), 43, 965-1021 (in Chinese). http://dx.doi.org/10.1360/012012-428 [3] Li, K.T., Su, J. and Huang, A.X. (2010) Boundary Shape Control of the Navier-Stokes Equations and Applications. Chinese Annals of Mathematics, 31B, 879-920. [4] Ciarlet, P.G. (2000) Mathematical Elasticity, Vol. III: Theory of Shells. North-Holland, Amsterdam. [5] Li, K.T., Zhang, W.L. and Huang, A.X. (2006) An Asymptotic Analysis Method for the Linearly Shell Theory. Science in China, Series A, 49, 1009-1047. [6] Temam, R. and Ziane, M. (1997) Navier-Stokes Equations in Thin Spherical Domains. Contemporary Mathematics, 209, 281-314. http://dx.doi.org/10.1090/conm/209/02772 [7] Li, K.T., Yu, J.P. and Liu, D.M. (2012) A Differential Geomety Methods for Rational Navier-Stokes Equations with Complex Boundary and Two Scales Paralelell Algorithms. Acta Mathematicae Applicae Sinica, 35, 1-41. [8] Li, K.T., Huang, A.X. and Zhang, W.L. (2002) A Dimension Split Method for the 3-D Compressible Navier-Stokes Equations in Turbomachine. Communications in Numerical Methods in Engineering, 18, 1-14. [9] Li, K.T. and Liu, D.M. (2009) Dimension Splitting Method for 3D Rotating Compressible Navier-Stokes Equations in the Turbomachinery. International Journal of Numerical Analysis and Modeling, 6, 420-439. [10] Li, K.T., Yu, J.P., Shi, F. and Huang, A.X. (2012) Dimension Splitting Method for the Three Dimensional Rotating Navier-Stokes Equations. Acta Mathematicae Applicatae Sinica—English Series, 28, 417-442. http://dx.doi.org/10.1007/s10255-012-0161-7 [11] Temam, R. (1984) Navier-Stokes Equations, Theorem and Numerical Ananlysis. North Holland, Amsterdam, U.S.A., New York. [12] Giraut, V. and Raviart, P.A. (1985) Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin. [13] Alassar, R.S. and Badr, H.M. (1999) Oscillating Flow over Oblate Spheroids. Acta Mechanica, 137, 237-254. http://dx.doi.org/10.1007/BF01179212 [14] Rimon, Y. and Cheng, S.I. (1969) Numerical Solution of a Uniform Flow over a Sphere at Intermediate Reynolds Numbers. Physics of Fluids, 12, 949-959. http://dx.doi.org/10.1063/1.2163685