第4期 谢锡麟等:有限变形理论的若干进展及其在流体力学中的相关应用 557 [21] Aris R. Vectors, tensors, and the basic equations of fluid mechanics [M]. New York:Dover [22] Xie X L. On two kinds of differential operators on general smooth surfaces[J]. arXiv: 1306. 3671vl [23] Xie X L. A theoretical framework of vorticity dynamics for two dimensional flows on fixed smooth [24] Wu J Z, Ma H Y, Zhou M D. Vorticity and vortex dynamics[M]. New York: Spring-Verlag, 2005 Some developments of Finite Deformation Theories with Some Applications in Fluid Mechanics XIE XIlin, CHEN Yu. SHI Qian ( Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, China) Abstract: Some recent developments of finite deformation theories termed as "finite deformation theory with respect curvilinear coordinates corresponding to current physical configurations including time explicitly"and"finite deformation theory with respect to continuous mediums whose geometrical configurations are two dimensional surfaces"are narrated. The former and the latter theories correspond to continuous mediums whose geometrical configurations are Euclidian manifolds(bulk status)and Riemannian manifolds(surface status), respectively. As compared to general theories, both newly developed theories include constructions of physical and parametric configurations, definition of deformation gradient tensor with its primary properties, deformation descriptions transport theories and governing equations of conservation laws. As applications, stream function &. vorticity algorithm with respect to curvilinear coordinates including time explicitly, stream function & vorticity algorithm for two dimensional incompressible flows on fixed surfaces, and governing equations for oil diffusion on sea surfaces are presented with some results of tentative numerical studies. Keywords: finite deformation theory: stream function & vorticity algorithm: respect to curvilinear coordinates including time explicitly two dimensional flows on fixed surfaces: flows around cylinders with deformable boundaries: oil diffusion on the sea C1994-2013ChinaAcademicJOurnalElectronicPublishingHouse.Allrightsreservedhttp://www.cnki.net［２１］ Ａｒｉｓ Ｒ． Ｖｅｃｔｏｒｓ，ｔｅｎｓｏｒｓ，ａｎｄ ｔｈｅ ｂａｓｉｃ ｅｑｕａｔｉｏｎｓ ｏｆｆｌｕｉｄ ｍｅｃｈａｎｉｃｓ［Ｍ］． Ｎｅｗ Ｙｏｒｋ：Ｄｏｖｅｒ Ｐｕｂｌｉｃａｔｉｏｎｓ，ＩＮＣ，１９８９． ［２２］ ＸｉｅＸＬ．Ｏｎｔｗｏｋｉｎｄｓｏｆｄｉｆｆｅｒｅｎｔｉａｌｏｐｅｒａｔｏｒｓｏｎｇｅｎｅｒａｌｓｍｏｏｔｈｓｕｒｆａｃｅｓ［Ｊ］．ａｒＸｉｖ：１３０６．３６７１ｖ１ ［ｐｈｙｓｉｃｓ．ｆｌｕ－ｄｙｎ］，２０１３． ［２３］ ＸｉｅＸ Ｌ．Ａ ｔｈｅｏｒｅｔｉｃａｌｆｒａｍｅｗｏｒｋｏｆｖｏｒｔｉｃｉｔｙｄｙｎａｍｉｃｓｆｏｒｔｗｏｄｉｍｅｎｓｉｏｎａｌｆｌｏｗｓｏｎｆｉｘｅｄｓｍｏｏｔｈ ｓｕｒｆａｃｅｓ［Ｊ］．ａｒＸｉｖ：１３０４．５１４５ｖ１［ｐｈｙｓｉｃｓ．ｆｌｕ－ｄｙｎ］，２０１３． ［２４］ ＷｕＪＺ，ＭａＨ Ｙ，ＺｈｏｕＭ Ｄ．Ｖｏｒｔｉｃｉｔｙａｎｄｖｏｒｔｅｘｄｙｎａｍｉｃｓ［Ｍ］．ＮｅｗＹｏｒｋ：Ｓｐｒｉｎｇ－Ｖｅｒｌａｇ，２００５． ＳｏｍｅＤｅｖｅｌｏｐｍｅｎｔｓｏｆＦｉｎｉｔｅＤｅｆｏｒｍａｔｉｏｎＴｈｅｏｒｉｅｓｗｉｔｈ ＳｏｍｅＡｐｐｌｉｃａｔｉｏｎｓｉｎＦｌｕｉｄＭｅｃｈａｎｉｃｓ ＸＩＥＸｉ－ｌｉｎ，ＣＨＥＮＹｕ，ＳＨＩＱｉａｎ （ＤｅｐａｒｔｍｅｎｔｏｆＭｅｃｈａｎｉｃｓａｎｄＥｎｇｉｎｅｅｒｉｎｇＳｃｉｅｎｃｅ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ２００４３３，Ｃｈｉｎａ） Ａｂｓｔｒａｃｔ：Ｓｏｍｅｒｅｃｅｎｔｄｅｖｅｌｏｐｍｅｎｔｓｏｆｆｉｎｉｔｅｄｅｆｏｒｍａｔｉｏｎｔｈｅｏｒｉｅｓｔｅｒｍｅｄａｓ“ｆｉｎｉｔｅｄｅｆｏｒｍａｔｉｏｎｔｈｅｏｒｙｗｉｔｈｒｅｓｐｅｃｔ ｔｏｃｕｒｖｉｌｉｎｅａｒｃｏｏｒｄｉｎａｔｅｓｃｏｒｒｅｓｐｏｎｄｉｎｇｔｏｃｕｒｒｅｎｔｐｈｙｓｉｃａｌｃｏｎｆｉｇｕｒａｔｉｏｎｓｉｎｃｌｕｄｉｎｇｔｉｍｅｅｘｐｌｉｃｉｔｌｙ”ａｎｄ “ｆｉｎｉｔｅ ｄｅｆｏｒｍａｔｉｏｎｔｈｅｏｒｙ ｗｉｔｈｒｅｓｐｅｃｔｔｏｃｏｎｔｉｎｕｏｕｓ ｍｅｄｉｕｍｓ ｗｈｏｓｅｇｅｏｍｅｔｒｉｃａｌｃｏｎｆｉｇｕｒａｔｉｏｎｓａｒｅｔｗｏｄｉｍｅｎｓｉｏｎａｌ ｓｕｒｆａｃｅｓ”ａｒｅｎａｒｒａｔｅｄ．Ｔｈｅｆｏｒｍｅｒａｎｄｔｈｅｌａｔｔｅｒｔｈｅｏｒｉｅｓｃｏｒｒｅｓｐｏｎｄｔｏｃｏｎｔｉｎｕｏｕｓｍｅｄｉｕｍｓｗｈｏｓｅｇｅｏｍｅｔｒｉｃａｌ ｃｏｎｆｉｇｕｒａｔｉｏｎｓａｒｅＥｕｃｌｉｄｉａｎｍａｎｉｆｏｌｄｓ（ｂｕｌｋｓｔａｔｕｓ）ａｎｄＲｉｅｍａｎｎｉａｎｍａｎｉｆｏｌｄｓ（ｓｕｒｆａｃｅｓｔａｔｕｓ），ｒｅｓｐｅｃｔｉｖｅｌｙ．Ａｓ ｃｏｍｐａｒｅｄｔｏｇｅｎｅｒａｌｔｈｅｏｒｉｅｓ，ｂｏｔｈｎｅｗｌｙｄｅｖｅｌｏｐｅｄｔｈｅｏｒｉｅｓｉｎｃｌｕｄｅｃｏｎｓｔｒｕｃｔｉｏｎｓｏｆｐｈｙｓｉｃａｌａｎｄｐａｒａｍｅｔｒｉｃ ｃｏｎｆｉｇｕｒａｔｉｏｎｓ，ｄｅｆｉｎｉｔｉｏｎｏｆｄｅｆｏｒｍａｔｉｏｎｇｒａｄｉｅｎｔｔｅｎｓｏｒｗｉｔｈｉｔｓｐｒｉｍａｒｙｐｒｏｐｅｒｔｉｅｓ，ｄｅｆｏｒｍａｔｉｏｎｄｅｓｃｒｉｐｔｉｏｎｓ， ｔｒａｎｓｐｏｒｔｔｈｅｏｒｉｅｓａｎｄｇｏｖｅｒｎｉｎｇｅｑｕａｔｉｏｎｓｏｆｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓ．Ａｓａｐｐｌｉｃａｔｉｏｎｓ，ｓｔｒｅａｍｆｕｎｃｔｉｏｎ ＆ ｖｏｒｔｉｃｉｔｙ ａｌｇｏｒｉｔｈｍ ｗｉｔｈｒｅｓｐｅｃｔｔｏｃｕｒｖｉｌｉｎｅａｒｃｏｏｒｄｉｎａｔｅｓｉｎｃｌｕｄｉｎｇｔｉｍｅｅｘｐｌｉｃｉｔｌｙ，ｓｔｒｅａｍｆｕｎｃｔｉｏｎ ＆ ｖｏｒｔｉｃｉｔｙａｌｇｏｒｉｔｈｍ ｆｏｒｔｗｏｄｉｍｅｎｓｉｏｎａｌｉｎｃｏｍｐｒｅｓｓｉｂｌｅｆｌｏｗｓｏｎｆｉｘｅｄｓｕｒｆａｃｅｓ，ａｎｄｇｏｖｅｒｎｉｎｇｅｑｕａｔｉｏｎｓｆｏｒｏｉｌｄｉｆｆｕｓｉｏｎｏｎｓｅａｓｕｒｆａｃｅｓ ａｒｅｐｒｅｓｅｎｔｅｄｗｉｔｈｓｏｍｅｒｅｓｕｌｔｓｏｆｔｅｎｔａｔｉｖｅｎｕｍｅｒｉｃａｌｓｔｕｄｉｅｓ． Ｋｅｙｗｏｒｄｓ：ｆｉｎｉｔｅｄｅｆｏｒｍａｔｉｏｎｔｈｅｏｒｙ；ｓｔｒｅａｍｆｕｎｃｔｉｏｎ ＆ ｖｏｒｔｉｃｉｔｙａｌｇｏｒｉｔｈｍ；ｒｅｓｐｅｃｔｔｏｃｕｒｖｉｌｉｎｅａｒｃｏｏｒｄｉｎａｔｅｓ ｉｎｃｌｕｄｉｎｇｔｉｍｅｅｘｐｌｉｃｉｔｌｙ；ｔｗｏ ｄｉｍｅｎｓｉｏｎａｌｆｌｏｗｓｏｎｆｉｘｅｄｓｕｒｆａｃｅｓ；ｆｌｏｗｓａｒｏｕｎｄｃｙｌｉｎｄｅｒｓ ｗｉｔｈ ｄｅｆｏｒｍａｂｌｅ ｂｏｕｎｄａｒｉｅｓ；ｏｉｌｄｉｆｆｕｓｉｏｎｏｎｔｈｅｓｅａ 第４期 谢锡麟等：有限变形理论的若干进展及其在流体力学中的相关应用 ７５５