Accompanying(4)with the dual relations (V,Vi-ViVi)=:6ji3w' ViVi the vorticity is defined as w= &VkVing = wns, where 6ji3: =Ig [no,", g are termed as co- and contra-variant components of Eddington tensor that as the same to metric tensor are insensitive to co- and contra- variant derivatives. and one has the ovel velocity decomposition Lemma 1(Velocity Decomposition v(x+bx,t)-v(x,t)=D·D∑+u×∑+V·K·6∑ whered: =2(ViVi+V,Vig@g can be defined as strain tensor, K: bijg'8g is termed as surface curvature tensor. The last appended term is due to the contribution of the surface curvature 2.3 Kinetics Basically, so termed intrinsic generalized Stokes formula of the second kind is introduced no-更dl 更+H where n: =T x ny is the normal vector of the boundary on the tangent plane in which T is the unit tangent vector of the boundary, p can be any kind of tensor field defined on the surface. It is worthy of note that most of the governing equations for fows on surfaces and the ones for thin enough shells and plates, as indicated by Aris(1962)and Synge Chien(1941) respectively, can be directly and readily deduced through(6). Its detailed proof can be referred to the paper y Xie et al.(2013) On the mass conservation, it can be represented and analysed (, t)do+ n(pv)dl=/, P(a, t)do+/v(pv)do where p denotes the surface density. Subsequently, the differential form for mass conservation is deduced (a, t)+V(pv)=at (x,t)+Vm(x,1)+pV=p+pB=0,:=VV(7) On the momentum conservation. it is considered that the rate of change of the momentum are contributed by the surface tension, inner pressure, inner friction and surface force, in other words it is keeping valid fA ot (z, )do+f,m (pv)v dl=/A pa do=Ften+ Fine + Fois Faur where Ften:=f: m dl, Fpre: =-5spm dl, Fvis: =S. un.[(V, Vi+Vi)gog] dl are represented originally as curve integrals that can be transformed directly to surface inte- grals through the intrinsic generalized Stokes formula of the second kind, and u denote the coefficients of surface tension and inner friction/ viscousity respectively. Fsur: =J fur do is represented directly as surface integral, fur can be the surface densities of friction, gravity, electromagnetic force and stochastic force et al. As reviewed by Boffetta Ecke(2012)and Bouchet Venaille(2012), the surface force usually plays the important role in dynamics ofAccompanying (4) with the dual relations (∇jVi − ∇iVj ) =: ϵji3ω 3 ⇔ ω 3 := ϵ 3ij∇iVj (5) the vorticity is defined as ω , ϵ 3kl∇kVl nΣ = ω 3nΣ, where ϵji3 := [gj , gi , nΣ] and ϵ 3kl := [nΣ, g k , g l ] are termed as co- and contra-variant components of Eddington tensor that as the same to metric tensor are insensitive to co- and contra-variant derivatives. And one has the novel velocity decomposition. Lemma 1 (Velocity Decomposition) V (x + δx, t) − V (x, t) = D · δ t Σ + 1 2 ω × δ t Σ + V · K · δ t Σ, where D := 1 2 (∇iVj + ∇jVi)g i ⊗ g j can be defined as strain tensor, K := bijg i ⊗ g j is termed as surface curvature tensor. The last appended term is due to the contribution of the surface curvature. 2.3 Kinetics Basically, so termed intrinsic generalized Stokes formula of the second kind is introduced I ∂Σ n ◦ −Φdl = ∫ Σ ( Σ ∇ ◦ −Φ + HnΣ ◦ −Φ ) dσ (6) where n := τ × nΣ is the normal vector of the boundary on the tangent plane in which τ is the unit tangent vector of the boundary, Φ can be any kind of tensor field defined on the surface. It is worthy of note that most of the governing equations for flows on surfaces and the ones for thin enough shells and plates, as indicated by Aris (1962) and Synge & Chien (1941) respectively, can be directly and readily deduced through (6). Its detailed proof can be referred to the paper by Xie et al. (2013). On the mass conservation, it can be represented and analysed 0 = ∫ t Σ ∂ρ ∂t(x, t) dσ + I ∂ t Σ n · (ρV ) dl = ∫ t Σ ∂ρ ∂t(x, t) dσ + ∫ t Σ ∇ · (ρV ) dσ where ρ denotes the surface density. Subsequently, the differential form for mass conservation is deduced ∂ρ ∂t(x, t) + ∇ · (ρV ) = [ ∂ρ ∂t(x, t) + V i ∂ρ ∂xi (x, t) ] + ρ∇iV i = ˙ρ + ρ θ = 0, θ := ∇iV i (7) On the momentum conservation, it is considered that the rate of change of the momentum are contributed by the surface tension, inner pressure, inner friction and surface force, in other words it is keeping valid ∫ t Σ ∂(ρV ) ∂t (x, t) dσ + I ∂ t Σ n · (ρV )V dl = ∫ t Σ ρ a dσ = Ften + F int pre + Fvis + Fsur where Ften := H ∂ t Σ γn dl, F int pre := − H ∂ t Σ pn dl, Fvis := H ∂ t Σ µn · [ (∇jVi + ∇iVj ) g i ⊗ g j ] dl are represented originally as curve integrals that can be transformed directly to surface inte￾grals through the intrinsic generalized Stokes formula of the second kind, γ and µ denote the coefficients of surface tension and inner friction/viscousity respectively. Fsur := ∫ t Σ fsurdσ is represented directly as surface integral, fsur can be the surface densities of friction, gravity, electromagnetic force and stochastic force et al. As reviewed by Boffetta & Ecke (2012) and Bouchet & Venaille (2012), the surface force usually plays the important role in dynamics of 4