Fish and Lauder [4]. The vortex structures introduced by the manifold could be considered as a subspace in three dimen finite deformations of fins have been studied through com- sional Euclid space with a certain bulk. Thus, a group of putational fluid dynamics(CFD) by other researchers [5,6]. three numbers/ coordinates independently must be needed to In addition, the swim rules with respect to a single fish or describe its configuration. Certainly, the Cartesian coordi- a group of fishes have been studies based on CFD and con- nates is the naturally choose trol theory by Wu and Wang (1, 7]. Conversely, Du and Sun 8] discovered that the flying performances of insects could ensional motions be improved drastically by the finite deformations of wings as compared to the solid vibrations of wings. Lu and Yin Two dimensional motions are generally referred to as the mo- [9] pointed out that the separation could be effectively sup- tions on the plane or on a curved surface embedded in the pressed as the boundary of a two dimensional tunnel does three dimensional Euclid space. Physically, a two dimen travailing wave deformation. Wu et al. [10] put forward the sional motions should be considered as a model for the mo- concept of fluid roller bearing that could be utilized in the tion of continuous medium that is limited to a quite thin layer drag reduction of wings and so on. In addition, it is discov- as described by Irion [ 19] ered that the street of Karman vortices of a circular cylinder The typical two dimensional motion could be considered could be suppressed completely as some kind of travailing as the flowing soap film for fluid mechanics. Zhang et al. [20) wave deformations is trigged on its boundary [ll] reported the motions of flexible filaments with their wakes in On numerical studies, the popular methods of CFD utilized a flowing soap film in which the filaments should be consid to simulate and investigate the flows with deformable bound- ered as parts of the boundaries of the soap film. In addition aries could be concluded into three kinds. The first is termed the soap film flows around a circular cylinder generally po of the whole flow fields in which certain kind of additional flow visualizations using light interference techniques. It can body forces should be added to the general Navier-Stokes be noted that the discoveries of elegant vortical structures of equation(NSE). Correspondingly, the Cartesian coordinates the flows of soap films are accompanied with the considerable is commonly adopted to discrete NSE and boundaries. The surface density or film thickness variations. The amplitude of second is the transformation of the reference systems as usu- the density variation could be about 20% of its characteristic ally used in the system constituted by rigid bodies. Unfortu- or mean value but the mechanism of the density variation is nately, this kind of methods is only effective in few cases. The unreported. third is the transformation of coordinates that is widely used Two dimensional motions usually related to the motions in CFD that generally transform the differential equations of thin enough membranes but their amplitudes could take with respect to Cartesian coordinates to the curvilinear coor- finite values in regard to solid mechanics. Gutierrez [21]re- dinates in order to detail with the curved boundaries. Usually, ported the results from a series of numerical simulations that the constructed curvilinear coordinates are time-independent. put emphasis on the determination of the lower natural fre Luo and Bewley [13] studied the contra-variant form of nse quencies for the transverse vibration of annular membranes with respect to the time-dependent curvilinear coordinates in including the special case of a solid circular membrane when the point of view of the coordinate transformation. Compar- the mass per unit area varies linearly, quadratically, and cubi atively, the experimental studies are seldom reported due to cally with the radial coordinate. Buchanan[22]studied den the difficulties in the controls of the deformable boundaries sity which was assumed to vary linearly along the diameter and the measurements associated of the membrane and could vary circumferentially. Tsiatas On theoretical studies, the primary achievements usually and Katsikadelis [23] presented a solution for the problem originated from the point of view of vorticity and vortex dy- of initially non-flat membranes based on a new formulation namics could be concluded as three aspects, mainly devel- of the governing differential equations in terms of displace oped by Prof Wu J.Z. with his collaborators. Firstly is on the ments. They solved the problem by direct integration of the representation of the velocity gradient, subsequently one differential equations. In addition, the nonlinear membrane attain more advanced representation of the deformation rate model has been used to simulate the nonlinear vibrations of tensor on the deformable boundary [14]. Secondly is the in- single layer graphene sheets with large amplitudes [24]. The tegral representations of the resultant force and moment of related research indicates the promising applications of the momentum imposing on the deformable body surrounded by theory of membranes in nano-technology. fluids [15,16]. Thirdly is the coupling representations of the The vibration analysis of composite circular and annular boundary fluxes of the vorticity and dilation [17] membranes has been carried out with some results obtained Although a flow with deformable boundary can be pos- through exact solutions, energy methods and finite element essed of abundant attractive dynamics, its geometrical con- analysis. Although its variations are usually considered, the figuration is generally Euclidian manifold as viewed from surface density of a membrane is usually assumed as a cer- modern differential geometry [18]. The reputed Euclidian tain function of the position referred seldom to its governingXie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 433 Fish and Lauder [4]. The vortex structures introduced by the finite deformations of fins have been studied through com￾putational fluid dynamics (CFD) by other researchers [5,6]. In addition, the swim rules with respect to a single fish or a group of fishes have been studies based on CFD and con￾trol theory by Wu and Wang [1,7]. Conversely, Du and Sun [8] discovered that the flying performances of insects could be improved drastically by the finite deformations of wings as compared to the solid vibrations of wings. Lu and Yin [9] pointed out that the separation could be effectively sup￾pressed as the boundary of a two dimensional tunnel does travailing wave deformation. Wu et al. [10] put forward the concept of fluid roller bearing that could be utilized in the drag reduction of wings and so on. In addition, it is discov￾ered that the street of Karman vortices of a circular cylinder could be suppressed completely as some kind of travailing wave deformations is trigged on its boundary [11]. On numerical studies, the popular methods of CFD utilized to simulate and investigate the flows with deformable bound￾aries could be concluded into three kinds. The first is termed as immersed boundary method [12]. Its essential idea is to re￾gard the solid bodies embedded in the flow field as some parts of the whole flow fields in which certain kind of additional body forces should be added to the general Navier-Stokes equation (NSE). Correspondingly, the Cartesian coordinates is commonly adopted to discrete NSE and boundaries. The second is the transformation of the reference systems as usu￾ally used in the system constituted by rigid bodies. Unfortu￾nately, this kind of methods is only effective in few cases. The third is the transformation of coordinates that is widely used in CFD that generally transform the differential equations with respect to Cartesian coordinates to the curvilinear coor￾dinates in order to detail with the curved boundaries. Usually, the constructed curvilinear coordinates are time-independent. Luo and Bewley [13] studied the contra-variant form of NSE with respect to the time-dependent curvilinear coordinates in the point of view of the coordinate transformation. Compar￾atively, the experimental studies are seldom reported due to the difficulties in the controls of the deformable boundaries and the measurements associated. On theoretical studies, the primary achievements usually originated from the point of view of vorticity and vortex dy￾namics could be concluded as three aspects, mainly devel￾oped by Prof.Wu J.Z. with his collaborators. Firstly is on the representation of the velocity gradient, subsequently one can attain more advanced representation of the deformation rate tensor on the deformable boundary [14]. Secondly is the in￾tegral representations of the resultant force and moment of momentum imposing on the deformable body surrounded by fluids [15,16]. Thirdly is the coupling representations of the boundary fluxes of the vorticity and dilation [17]. Although a flow with deformable boundary can be pos￾sessed of abundant attractive dynamics, its geometrical con- figuration is generally Euclidian manifold as viewed from modern differential geometry [18]. The reputed Euclidian manifold could be considered as a subspace in three dimen￾sional Euclid space with a certain bulk. Thus, a group of three numbers/coordinates independently must be needed to describe its configuration. Certainly, the Cartesian coordi￾nates is the naturally choose. 1.2 Two dimensional motions Two dimensional motions are generally referred to as the mo￾tions on the plane or on a curved surface embedded in the three dimensional Euclid space. Physically, a two dimen￾sional motions should be considered as a model for the mo￾tion of continuous medium that is limited to a quite thin layer as described by Irion [19]. The typical two dimensional motion could be considered as the flowing soap film for fluid mechanics. Zhang et al. [20] reported the motions of flexible filaments with their wakes in a flowing soap film in which the filaments should be consid￾ered as parts of the boundaries of the soap film. In addition, the soap film flows around a circular cylinder generally po￾sitioned perpendicular to the planar soap have been studied. Currently, the flows of soap films are generally studied by the flow visualizations using light interference techniques. It can be noted that the discoveries of elegant vortical structures of the flows of soap films are accompanied with the considerable surface density or film thickness variations. The amplitude of the density variation could be about 20% of its characteristic or mean value but the mechanism of the density variation is unreported. Two dimensional motions usually related to the motions of thin enough membranes but their amplitudes could take finite values in regard to solid mechanics. Gutierrez [21] re￾ported the results from a series of numerical simulations that put emphasis on the determination of the lower natural fre￾quencies for the transverse vibration of annular membranes including the special case of a solid circular membrane when the mass per unit area varies linearly, quadratically, and cubi￾cally with the radial coordinate. Buchanan [22] studied den￾sity which was assumed to vary linearly along the diameter of the membrane and could vary circumferentially. Tsiatas and Katsikadelis [23] presented a solution for the problem of initially non-flat membranes based on a new formulation of the governing differential equations in terms of displace￾ments. They solved the problem by direct integration of the differential equations. In addition, the nonlinear membrane model has been used to simulate the nonlinear vibrations of single layer graphene sheets with large amplitudes [24]. The related research indicates the promising applications of the theory of membranes in nano-technology. The vibration analysis of composite circular and annular membranes has been carried out with some results obtained through exact solutions, energy methods and finite element analysis. Although its variations are usually considered, the surface density of a membrane is usually assumed as a cer￾tain function of the position referred seldom to its governing