SOME STUDIES ON THEORETICAL FRAMEWORK FOR TWO DIMENSIONAL FLOWS ON GENERAL FIXED SMOOTH SURFACES XIE XiLin, CHEN Yu SHI Qian Department of Mechanics Engineering Science, Fudan University, Shanghai 200433, China Original Manuscript updated on April 20, 2013 Abstract Two dimensional flows on fixed smooth surfaces have been studied in the point of view of vorticity dynamics. Firstly, the related deformation theory including kinematics and kinetics is developed. Secondly, some primary relations in vorticity dynamics have been extended to two dimensional fows on fixed smooth surface through which a theoretical framework of vorticity dynamics have been set up, mainly including governing equation of vorticity Lagrange theorem on vorticity, Caswell formula on strain tensor and stream function vorticity algorithm with pressure Possion equation for incompressible flows. The newly developed theory is characterized by the appearances of surface curvatures in some primary relations and governing equation Keywords: Two dimensional flows on fixed surfaces; Surface curvatures; Vorticity dynamics Riemannian geometry 1 Introduction Two dimensional Flows/deformations are naturally referred to Flows/deformations of con- tinuous mediums whose geometrical configurations can be taken as two dimensional surfaces embedded in three dimensional Euclidian space. Physically, the values of the thickness of con- tinuous mediums are quite smaller than the ones of character scales along flow directions. Such Hows/deformations may be divided into two groups. The first one is termed as flows on fired sur- faces that can be considered as models of long wandering rivers on the earth, real foods spreading over plains, depressions and valleys, atmosphere motions on planets, creeping flows on curved surfaces, fat soap films and so on. The second one is termed as self-motions deformations as models of bubble, cell and capsule deformations, vibrations of solid membranes, oil contamina- tions on sea surfaces et al The ways of attaining governing equations of natural conservation laws can also be separat into two kinds. The first kind is to do approximations based on general full dimensional govern- ing equations. For example, Roberts Li(2006)attained models of thin fuid flows with inertia on curved substrates based on central manifold approximation. The second kind is to do analysis in the point of view of general continuum mechanics. The equations governing fuid motion in a surface or interface have been studied by Aris(1962). Very recently, a novel kind of finite de- formation theory of continuous mediums whose geometrical configurations are two dimensional smooth surfaces have been developed by Xie et al.(2013). It mainly includes the definitions of initial and current configurations, deformation gradient tensor with its primary properties deformation descriptions, transport theories and governing equations corresponding to conser- vation laws. The general theory is suitable for any kind of two dimensional flows/deformations Particularly, the present study puts focus on flows on fixed smooth surface in the point of view of vorticity dynamics(see Wu et al., 2005)SOME STUDIES ON THEORETICAL FRAMEWORK FOR TWO DIMENSIONAL FLOWS ON GENERAL FIXED SMOOTH SURFACES ∗ XIE XiLin, CHEN Yu & SHI Qian Department of Mechanics & Engineering Science, Fudan University, Shanghai 200433, China. Original Manuscript updated on April 20, 2013 Abstract Two dimensional flows on fixed smooth surfaces have been studied in the point of view of vorticity dynamics. Firstly, the related deformation theory including kinematics and kinetics is developed. Secondly, some primary relations in vorticity dynamics have been extended to two dimensional flows on fixed smooth surface through which a theoretical framework of vorticity dynamics have been set up, mainly including governing equation of vorticity, Lagrange theorem on vorticity, Caswell formula on strain tensor and stream function & vorticity algorithm with pressure Possion equation for incompressible flows. The newly developed theory is characterized by the appearances of surface curvatures in some primary relations and governing equations. Keywords: Two dimensional flows on fixed surfaces; Surface curvatures; Vorticity dynamics; Riemannian geometry 1 Introduction Two dimensional flows/deformations are naturally referred to flows/deformations of con￾tinuous mediums whose geometrical configurations can be taken as two dimensional surfaces embedded in three dimensional Euclidian space. Physically, the values of the thickness of con￾tinuous mediums are quite smaller than the ones of character scales along flow directions. Such flows/deformations may be divided into two groups. The first one is termed as flows on fixed sur￾faces that can be considered as models of long wandering rivers on the earth, real floods spreading over plains, depressions and valleys, atmosphere motions on planets, creeping flows on curved surfaces, flat soap films and so on. The second one is termed as self-motions/deformations as models of bubble, cell and capsule deformations, vibrations of solid membranes, oil contamina￾tions on sea surfaces et al. The ways of attaining governing equations of natural conservation laws can also be separated into two kinds. The first kind is to do approximations based on general full dimensional govern￾ing equations. For example, Roberts & Li (2006) attained models of thin fluid flows with inertia on curved substrates based on central manifold approximation. The second kind is to do analysis in the point of view of general continuum mechanics. The equations governing fluid motion in a surface or interface have been studied by Aris (1962). Very recently, a novel kind of finite de￾formation theory of continuous mediums whose geometrical configurations are two dimensional smooth surfaces have been developed by Xie et al. (2013). It mainly includes the definitions of initial and current configurations, deformation gradient tensor with its primary properties, deformation descriptions, transport theories and governing equations corresponding to conser￾vation laws. The general theory is suitable for any kind of two dimensional flows/deformations. Particularly, the present study puts focus on flows on fixed smooth surface in the point of view of vorticity dynamics (see Wu et al., 2005). 1