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 continuous mediums whose geometrical configurations can be taken as two dimensional surfaces embedded in three dimensional Euclidian space. Physically, the values of the thickness of continuous 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 surfaces 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 contaminations 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 governing 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 deformation 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 conservation 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
tial Physical Configuration Current Parametric Configuration Fixed Smooth Surface x=x(5,1) Current Phsical Configuration Initial Parametric Configuration →x =E(5) Figure 1: Sketch of the initial/ current and physical/parametric configurations, where ny and Ne denote the surface normal vectors associated to the current and initial physical con- figurations respectively, igi]_I and [Gala- are local co-variant bases. The tangent space denoted by T2 is span by the co- or contra-variant basis 2 Finite Deformation theory 2.1 Configurations, Coordinates, Curvatures &z Field Operators a two dimensional fow on a fixed smooth surface is sketched in Figure 1. The surface is represented by the vector valued mapping 2= 2(r)E R, and ail is the paramet. ic coordinates. The Hlow/ deformation can be represented by the smooth diffeomorphism a (S, t)E6(2c, 2x)in the parametric space, where [=]21 and (S_1 denote Eulerian and Lagrangian coordinates respectively, and 2e and >r are termed as the initial and current para- metric configurations respectively. Correspondingly, 2: = 2(28 and 2: 2(r) are as the initial and current physical configurations respectively. The smooth surface can be tak- en as a two dimensional Riemannian manifold with the fundamental quantity of the first kind Igi=(9i,9i)i=, and the one of the second kind (big=(ari(a), nE)12i=1.Gaussian curva- ture and mean curvature are defined as Kg= det(bil/ det[gi] and H= bs respectively. Einstein summation convention is adopted throughout the paper with subscripts and superscripts repre- senting co- and contra-variant quantities respectively, and quantities with capital/ small scripts associate with initial/current configurations Generally, the surface gradient operator V= l a is defined as,sy垂∈2x), 2)°-(y918g)g-0 =Vd ( go-gi)8g+d bi(go-n)g+p i(go-gi)8n where o-can be any available algebra tensor operator, VI denotes the co-variant deriva tive/differentation of the tensor component, e. g w(1)+吗-r町 Corresponding Author: XIE XiLin, Department of Mechanics &e Engineering Science, Fudan University HanDanRoad220,Shanghai200433,China.Tel:0086-21-55664283:Email:xiexilin@fudan.edu.cn
Figure 1: Sketch of the initial/current and physical/parametric configurations, where nΣ and NΣ denote the surface normal vectors associated to the current and initial physical con- figurations respectively, {gi} 2 i=1 and {GA} 2 A=1 are local co-variant bases. The tangent space denoted by T Σ is span by the co- or contra-variant basis. 2 Finite Deformation Theory 2.1 Configurations, Coordinates, Curvatures & Field Operators A two dimensional flow on a fixed smooth surface is sketched in Figure 1. The surface is represented by the vector valued mapping Σ = Σ(x) ∈ R 3 , and {x i} 2 i=1 is the parametric coordinates. The flow/deformation can be represented by the smooth diffeomorphism x = x(ξ, t) ∈ C ∞( ◦ Σξ, t Σx) in the parametric space, where {x i} 2 i=1 and {ξ A} 2 A=1 denote Eulerian and Lagrangian coordinates respectively, and ◦ Σξ and t Σx are termed as the initial and current parametric configurations respectively. Correspondingly, ◦ Σ := Σ( ◦ Σξ) and t Σ := Σ( t Σx) are termed as the initial and current physical configurations respectively. The smooth surface can be taken as a two dimensional Riemannian manifold with the fundamental quantity of the first kind {gij , (gi , gj )} 2 i,j=1, and the one of the second kind {bij , ( ∂gj ∂xi (x), nΣ)} 2 i,j=1. Gaussian curvature and mean curvature are defined as KG , det[bij ]/ det[gij ] and H , b s s respectively. Einstein summation convention is adopted throughout the paper with subscripts and superscripts representing co- and contra-variant quantities respectively, and quantities with capital/small scripts associate with initial/current configurations. Generally, the surface gradient operator Σ ∇ ≡ g l ∂ ∂xl is defined as, say Φ ∈ T 2 (T Σ), Σ ∇ ◦ −Φ ≡ ( g l ∂ ∂xl ) ◦ − ( Φ i ·jgi ⊗ g j ) , g l ◦ − ∂ ∂xl ( Φ i ·jgi ⊗ g j ) = ∇lΦ i ·j (g l ◦ −gi ) ⊗ g j + Φi ·j bli(g l ◦ −n) ⊗ g j + Φi ·j b j l (g l ◦ −gi ) ⊗ n where ◦− can be any available algebra tensor operator, ∇l denotes the co-variant derivative/differentation of the tensor component, e.g. ∇lΦ i ·j , ∂Φ i ·j ∂xl (x, t) + Γi lsΦ s ·j − Γ s ljΦ i ·s ∗Corresponding Author: XIE XiLin, Department of Mechanics & Engineering Science, Fudan University. HanDan Road 220, Shanghai 200433, China. Tel: 0086-21-55664283; Email: xiexilin@fudan.edu.cn. 2
here Tis denotes Christoffel symbol of the second kind. The contra-variant derivative relates generally to the co-variant one through V'2gVt. On the other hand, one defines the so termed Levi-Civita connection operator V=9V V。一重≡(gVg)。-(重189)会go-Va,(重18g3)=V重(go-9)8g where V_a denotes Levi-Civita connection defined on the surface/Riemannian manifold(see Durovin et al., 1992). It should be noted that the change of the order of co- and contra- variant derivatives must be related to Riemannian-Christoffel tensor that can be represented by Gaussian curvature and metric tensor for two dimensional riemannian manifolds as revealed by the relation =Vvp+KG(-99),+Kc(9p99-983) where R sp: 2 bibs -bspb"q denotes the component of Riemannian-Christoffel tensor It is worthy of note that Levi-Civita connection operator is just valid/ effective for surface tensor fields that just have components on the tangent plane, generally denoted by g(T2) where r E n denotes the order. However, the surface gradient operator can be applied to arbitrary tensor fields defined on the surface due to its definition is based on differential calculus of tensor normed spac 2.2 nematics Generally, the velocity of a fluid partial is determined as follows ar2 0∑ ()=a(s,1a()=vg Subsequently, the acceleration is 全(,t) (r, t)+Vvsv91+(vivi)n=agI+ann The material derivative of any surface tensor field is represented by ds9师 a(,t)=t(,)+Va(,1),V∈(T2) based on the velocity representation(1). The deformation gradient tensor denoted by F is introduced by the relation ∑(x+65,1,t)-x(x(:t),t)÷F·(∑(+6)-()),F:=(,tg()8G4() All kinds of deformation descriptions can be represented by the deformation gradient tensor with its properties as indicated in Xie et al.(2013 Similar to the familiar Helmholtz velocity decomposition, one has the relation v(x+6x,t)-v(x,t)=(V⑧V)·0∑ VV+V)g· i iVi-ViVi96g +(Vsbg)·∑,6∑=6x5gs(x)∈T∑
where Γi ls denotes Christoffel symbol of the second kind. The contra-variant derivative relates generally to the co-variant one through ∇l , g lt∇t . On the other hand, one defines the so termed Levi-Civita connection operator ∇ ≡ g l∇ ∂ ∂xl ∇ ◦ −Φ ≡ (g l∇ ∂ ∂xl ) ◦ −(Φi ·jgi ⊗ g j ) , g l ◦ −∇ ∂ ∂xl (Φi ·jgi ⊗ g j ) = ∇lΦ i ·j (g l ◦ −gi ) ⊗ g j where ∇ ∂ ∂xl denotes Levi-Civita connection defined on the surface/Riemannian manifold (see Durovin et al., 1992). It should be noted that the change of the order of co- and contravariant derivatives must be related to Riemannian-Christoffel tensor that can be represented by Gaussian curvature and metric tensor for two dimensional Riemannian manifolds as revealed by the relation ∇p∇qΦ i ·j = ∇q∇pΦ i ·j + R i·· q ·sp ·Φ s ·j + R ·s·q j·p ·Φ i ·s = ∇q∇pΦ i ·j + KG(δ i p δ q s − gspg iq)Φs ·j + KG(gjpg sq − δ s p δ q j )Φi ·s where R i·· q ·sp · , b i p b q s − bspb iq denotes the component of Riemannian-Christoffel tensor. It is worthy of note that Levi-Civita connection operator is just valid/effective for surface tensor fields that just have components on the tangent plane, generally denoted by T r (T Σ) where r ∈ N denotes the order. However, the surface gradient operator can be applied to arbitrary tensor fields defined on the surface due to its definition is based on differential calculus of tensor normed space. 2.2 Kinematics Generally, the velocity of a fluid partial is determined as follows V , ∂Σ ∂t (ξ, t) = ∂xi ∂t (ξ, t) ∂Σ ∂xi (x) =: V i gi (1) Subsequently, the acceleration is a , ∂V ∂t (ξ, t) = ( ∂V l ∂t (x, t) + V s∇sV l ) gl + ( b ijViVj ) n =: a l gl + ann (2) The material derivative of any surface tensor field is represented by dΦ dt ≡ Φ˙ , ∂Φ ∂t (ξ, t) = ∂Φ ∂t (x, t) + V s ∂Φ ∂xs (x, t), ∀ Φ ∈ T r (T Σ) (3) based on the velocity representation (1). The deformation gradient tensor denoted by F is introduced by the relation Σ(x(ξ + δξ, t), t) − Σ(x(ξ, t), t) .= F · ( ◦ Σ(ξ + δξ) − ◦ Σ(ξ) ) , F := ∂xi ∂ξA (ξ, t)gi (x) ⊗ GA(ξ) All kinds of deformation descriptions can be represented by the deformation gradient tensor with its properties as indicated in Xie et al. (2013). Similar to the familiar Helmholtz velocity decomposition, one has the relation V (x + δx, t) − V (x, t) .= (V ⊗ ∇) · δ t Σ = [ 1 2 (∇jVi + ∇iVj )g i ⊗ g j ] · δ t Σ + [ 1 2 (∇jVi − ∇iVj )g i ⊗ g j ] · δ t Σ +(V s bstg t ) · δ t Σ, δ t Σ = δxs gs (x) ∈ T Σ (4) 3
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 of
Accompanying (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 integrals 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
two dimensional flows. Furthermore the component equations of momentum conservation can be deduced 9vNV+V1(VV)+KG叼]+f Upan, =H(r-p)+p(26V, v/)+, where al and an are the components of acceleration on the tangent plane and in the normal direction respectively, as shown in(2). It is revealed by(8)that Gaussian curvature accompa- nying with the tangent velocity takes part in the momentum conservation on the tangent space, however mean curvature accompanying with the surface tension and inner pressure does the contribution to the momentum conservation in the surface normal direction On the moment of momentum conservation, one has the general conclusion Lemma 2(Moment of momentum Conservation) Generally, the mechanical action im- posed on the boundary can be represented by the so termed surface stress tensor t=tg;③g3+t39;8nx then the moment of momentum conservation takes the form g4(t×g)+mx=0∈R here my denotes the surface force couple Proof: Based on the surface stress tensor, the momentum conservation can be represented generally as tdl+fe de On the other hand, the moment of momentum conservation is represented as pax∑d=p,(m:t)×d+fs×∑d+l.mda with the differential form pax∑=v·(×∑)+fx∑+m=(v·t)×x+g·(txg)+fsx∑+mx(10) Substituting(9) into(10), the proof is completed. Furthermore, thanks to the representation g·(t×9)=-t")i3nx+√9(-t3g2+t3g2),g:= det[gijl it can be concluded that Corollary 1(On Representations of Surface Stress Tenor) The symmetry of the com- ponents of surface stress tensor on the tangent space, i. e. ti=tji, corresponds to the vanishing of the component of surface force couple in the surface normal direction. And the appearance of surface stress tensor in the surface normal direction, i. e. t 3#0, corresponds to the eristence of components of surface force couple on the tangent space As an application, the stress tensor corresponding to the actions of surface tension, inn oressure, inner friction as discussed previously takes the form t=(-p)I+H(ViVi+Vivi98g,I: =9ui98g that corresponds to the case of full zero surface force couple as usually satisfied by Newtonian fluid fows
two dimensional flows. Furthermore, the component equations of momentum conservation can be deduced ρal = − ∂p ∂xl (x, t) + µ [ g ij∇i∇jVl + ∇l (∇sVs) + KGVl ] + fsur,l ρan = H(γ − p) + µ [ 2b ij∇iVj ] + fsur,n (8) where al and an are the components of acceleration on the tangent plane and in the normal direction respectively, as shown in (2). It is revealed by (8) that Gaussian curvature accompanying with the tangent velocity takes part in the momentum conservation on the tangent space, however mean curvature accompanying with the surface tension and inner pressure does the contribution to the momentum conservation in the surface normal direction. On the moment of momentum conservation, one has the general conclusion Lemma 2 (Moment of momentum Conservation) Generally, the mechanical action imposed on the boundary can be represented by the so termed surface stress tensor t = t i ·jgi ⊗ g j + t i ·3gi ⊗ nΣ then the moment of momentum conservation takes the form g l · (t × gl ) + mΣ = 0 ∈ R 3 where mΣ denotes the surface force couple. Proof: Based on the surface stress tensor, the momentum conservation can be represented generally as ∫ t Σ ρ a dσ = I ∂ t Σ n · t dl + ∫ t Σ fΣ dσ ⇒ ρ a = Σ ∇ · t + fΣ (9) On the other hand, the moment of momentum conservation is represented as ∫ t Σ ρ a × Σ dσ = I ∂ t Σ (n · t) × Σ dl + ∫ t Σ fΣ × Σ dσ + ∫ t Σ mΣ dσ with the differential form ρ a × Σ = Σ ∇ · (t × Σ) + fΣ × Σ + mΣ = [ ( Σ ∇ · t ) × Σ + g l · (t × gl ) ] + fΣ × Σ + mΣ (10) Substituting (9) into (10), the proof is completed. Furthermore, thanks to the representation g l · (t × gl ) = −t ij ϵij3nΣ + √ g(−t 2 ·3g 1 + t 1 ·3g 2 ), g := det[gij ] it can be concluded that Corollary 1 (On Representations of Surface Stress Tenor) The symmetry of the components of surface stress tensor on the tangent space, i.e. tij = tji, corresponds to the vanishing of the component of surface force couple in the surface normal direction. And the appearance of surface stress tensor in the surface normal direction, i.e. t i ·3 ̸= 0, corresponds to the existence of components of surface force couple on the tangent space. As an application, the stress tensor corresponding to the actions of surface tension, inner pressure, inner friction as discussed previously takes the form t = (γ − p)I + µ (∇jVi + ∇iVj ) g i ⊗ g j , I := gijg i ⊗ g j (11) that corresponds to the case of full zero surface force couple as usually satisfied by Newtonian fluid flows. 5
On energy conservation, one has the identity 人(+)0+n0M(+2) (n·t)·vd+l:fs·vda+ where e denotes the surface density of internal energy and g> the heat flux. Furthermore, using 9). one can arrive at pe=g2t·(,t)=(-p+1)+5(VV+V)v+V)+ =(→P+7)0+2V8V+V8v+g where the last two identities are due to the adoption of the constitutive relation(11) 3 Vorticity Dynamics 3.1 General theories Firstly, the following identity has been derived Lemma 3 vx(b·F)·N=(V×b)·nx,vb∈T》,:= det aca(s,t)(12) where va andvsg'va are levi- Civita connection operators, Nz and nz are sur- face normal vectors corresponding to the initial and current physical configurations respectively FI denotes the determinant of F, VG: =[G1, G2, Nyl Proof 5×(N=(=3)(),Ns= a xt (5,t) BA3 B(bakx(,1)=243 axl 0x2 (2x)()-rh4(ax( 0x2 (b2a)(5,)=an(x,t)BA:(,)m(,t) G d<A(5, t)(ei3V,bi F|(V×b)·n As an application, the governing equation of vorticity can be derived Corollary 2(Vorticity Equation) +(V×a) Proof: Let b in the relation(12)be the velocity V, it reads v×(v:F)·N
On energy conservation, one has the identity ∫ t Σ ∂ ∂t ( e + |V | 2 2 ) (x, t) dσ + I ∂ t Σ n · (ρV ) ( e + |V | 2 2 ) dl = I ∂ t Σ (n · t) · V dl + ∫ t Σ fΣ · V dσ + ∫ t Σ qΣ dσ where e denotes the surface density of internal energy and qΣ the heat flux. Furthermore, using (9), one can arrive at ρ e˙ = g l · t · ∂V ∂xl (x, t) = (−p + γ)θ + µ 2 (∇iV j + ∇jV i )(∇iVj + ∇jVi) + qΣ =: (−p + γ)θ + µ 2 |V ⊗ ∇ + ∇ ⊗ V | 2 + qΣ where the last two identities are due to the adoption of the constitutive relation (11). 3 Vorticity Dynamics 3.1 General Theories Firstly, the following identity has been derived Lemma 3 [ ◦ ∇ × (b · F) ] · NΣ = |F|(∇ × b) · nΣ, ∀ b ∈ T Σ, |F| := √g √ G det [ ∂xi ∂ξA ] (ξ, t) (12) where ◦ ∇ , GL ◦ ∇ ∂ ∂ξL and ∇ , g l∇ ∂ ∂xl are Levi-Civita connection operators , NΣ and nΣ are surface normal vectors corresponding to the initial and current physical configurations respectively, |F| denotes the determinant of F, √ G := [G1, G2, NΣ]. Proof: [ ◦ ∇ × (b · F) ] · NΣ = [(GB ◦ ∇ ∂ ∂ξB ) × ( bi ∂xi ∂ξA (ξ, t)GA )] · NΣ = ◦ ∇B ( bi ∂xi ∂ξA (ξ, t)ϵ BA3 ) = ϵ BA3 ◦ ∇B ( bi ∂xi ∂ξA (ξ, t) ) = ϵ BA3 [ ∂ ∂ξB ( bi ∂xi ∂ξA ) (ξ, t) − Γ L BA ( bi ∂xi ∂ξL (ξ, t) )] = ϵ BA3 ∂ ∂ξB ( bi ∂xi ∂ξA ) (ξ, t) = ∂bi ∂xs (x, t) [ ϵ BA3 ∂xs ∂ξB (ξ, t) ∂xi ∂ξA (ξ, t) ] = 1 √ G det [ ∂xi ∂ξA ] (ξ, t)e si3 ∂bi ∂xs (x, t) = √g √ G det [ ∂xi ∂ξA ] (ξ, t)(ϵ si3∇sbi) = |F|(∇ × b) · nΣ As an application, the governing equation of vorticity can be derived Corollary 2 (Vorticity Equation) ω˙ 3 = −θω3 + (∇ × a) · nΣ (13) Proof: Let b in the relation (12) be the velocity V , it reads ω 3 = (∇ × V ) · nΣ = 1 |F| [ ◦ ∇ × (V · F) ] · NΣ 6
Furthermore. one can do the following deduction F NE+Flex(a F)+VX(V =-(V XV). ms+(Vxa).ng+(vxv(v)) 0u3+(V×a)·ny where the identities F=(VoV). F and d Fl/dt=8F(see Xie et al., 2013)are utilized As an appendant of the above proof, one has the conclusion Corollary 3(Lagrange Theorem on Vorticity) In the case of the acceleration field is ir rotational, a patch of continuous medium that is initially irrotational will keep irrotational at any time, conversely a patch of continuous medium with initially nonzero vorticity will possess vorticity at any time although the value can be changed Secondly, the following identity can be readily set up Lemma 4 V×(V×b)=V(V.b)+Kcb-△b,Vb∈T∑,△bV·(V×b) Proof V×(×b=Vx(oVa)×(g)=(r7m)×[)n e3qE3pi Vq(VPb)9k=( p 9-89)Vg(VPb)9k=Vi(Vb)9k-Vp(VPb)92 =V(Vb)9gk-△b Furthermore. one has Vi(V6)9k=V(Vib)+R. si-6'19k=V(Vb)+KG(6:8s-9sig)6'gk v(V·b)+Kcb where the change of the order of covariant or contra-variant derivatives should be related to Riemannian-Christoffel tensor. It is the end of the proof. It should be pointed out that this kind of identities is still keeping valid for any tensor field As an application, the governing equation of momentum conservation on the tangent plane (8)can be rewritten pag=VI-pV×u+2KGV+ fur. 1g,Ⅱ=-p+210 where V xw=E3 aF(a, t)g. Subsequently, the following coupling relations can be attained just by doing the dot and cross products by e on both sides of (14)respectively Corollary 4(Coupling Relations between Directional Derivatives of II and w pa·e+(Vxu)·e-2Kcv·e-f HDu's-lea, e, ns]+(VIL, e, ns]+(2uKGV, e, ns)+If>, e, nsl for alle st le=l. The coupling relations are valid at any point in the flow field Thirdly, the intrinsic decomposition is still valid for any surface tensor, i.e. there exists
Furthermore, one can do the following deduction ω˙ 3 = − θ |F| [ ◦ ∇ × (V · F) ] · NΣ + 1 |F| [ ◦ ∇ × (a · F) + ◦ ∇ × ( V · (V ⊗ Σ ∇) · F )] · NΣ = −θ(∇ × V ) · nΣ + (∇ × a) · nΣ + ( ∇ × ∇ ( |V | 2 2 )) · nΣ = −θω3 + (∇ × a) · nΣ where the identities F˙ = (V ⊗ Σ ∇) · F and d|F|/dt = θ|F| (see Xie et al., 2013) are utilized. As an appendant of the above proof, one has the conclusion Corollary 3 (Lagrange Theorem on Vorticity) In the case of the acceleration field is irrotational, a patch of continuous medium that is initially irrotational will keep irrotational at any time, conversely a patch of continuous medium with initially nonzero vorticity will possess vorticity at any time although the value can be changed. Secondly, the following identity can be readily set up Lemma 4 ∇ × (∇ × b) = ∇(∇ · b) + KGb − ∆b, ∀ b ∈ T Σ, ∆b , ∇ · (∇ × b) Proof: ∇ × (∇ × b) = ∇ × [(g p∇ ∂ ∂xp ) × ( big i ) ] = ( g q∇ ∂ ∂xq ) × [ (∇pbi)ϵ pi3n ] = ϵ 3kqϵ3pi∇q(∇p b i )gk = (δ k p δ q i − δ q p δ k i )∇q(∇p b i )gk = ∇i(∇k b i )gk − ∇p(∇p b i )gi = ∇i(∇k b i )gk − ∆b Furthermore, one has ∇i(∇k b i )gk = [∇k (∇ib i ) + R i··k ·si· b s ]gk = ∇(∇ · b) + KG(δ i i δ k s − gsig ik)b s gk = ∇(∇ · b) + KGb where the change of the order of covariant or contra-variant derivatives should be related to Riemannian-Christoffel tensor. It is the end of the proof. It should be pointed out that this kind of identities is still keeping valid for any tensor field. As an application, the governing equation of momentum conservation on the tangent plane (8) can be rewritten as ρ alg l = ∇Π − µ∇ × ω + 2µKGV + fsur,lg l , Π := −p + 2µθ (14) where ∇ × ω = ϵ k3l ∂ω3 ∂xk (x, t)gl . Subsequently, the following coupling relations can be attained just by doing the dot and cross products by e on both sides of (14) respectively. Corollary 4 (Coupling Relations between Directional Derivatives of Π and ω ) ∂Π ∂e = ρa · e + µ(∇ × ω) · e − 2µKGV · e − fΣ · e µ ∂ω3 ∂e = −[ρa, e, nΣ] + [∇Π, e, nΣ] + [2µKGV , e, nΣ] + [fΣ, e, nΣ] for all e ∈ T Σ s.t |e| = 1. The coupling relations are valid at any point in the flow field. Thirdly, the intrinsic decomposition is still valid for any surface tensor, i.e. there exists 7
Lemma 5 (Intrinsic Decomposition) es(e·重)-[e,e,亚] 更 ∈丌(T∑),e∈TΣs.t.|el=1 )e-[重,el,e where [e,重] denotes the cross product e x重 and so on Proof:Let重=動;918g1∈2(T∑) without lost of generality, and to calculate [睡重,eel]=[9189y,ekg,el]=[重e918n,e!=( ekei(e)g18g =配(ke一518g=(重e9,)8(ekg)一重 Evidently, one arrives at the conclusion. The other identity can be proved similarly Applied the intrinsic decomposition to the velocity gradient tensor, the strain tensor has the following representation, termed still as Caswell formula Corollary 5(Caswell Formula) On any fired solid boundary where the general viscous bound- ary condition is satisfied by the fluid, namely V=0 on the boundary, the strain tensor can be represented as following D=6n⑧n+(a×n)n+m⑧(a×n) Proof: Basically, a kind of local curvilinear coordinates denoted by a, corresponding to the fixed solid boundary can be set up such that the local co-variant basis gu,91 is orthogonal n the boundary with gu tangents to the boundary and 91=9= n. The details can be referred to the appendix. Subsequently, the intrinsic decomposition is utilized For the first term on the right hand. one has (V⑧V,n)=(vVV⑧V,n)+(VV,n) +(v⑧v,n) he va v thanks to the relations vav=Fugu+Vus ∈T due to the viscous boundary condition and v1vgn+V⊥v-g,g For the second term on the right hand, it is naught thanks to vV,n]=[(ⅴv)sg"+V1V)8g,n]=(VV)8g2,g]=(v⊥V)[g+,g+]=0 As a summary, one arrives at the relation v必V=(u×m)⑧n+0nn Accompanying with its conjugate relation, the proof is completed Readily, one has the following assertion Corollary 6 For any two dimensional incompressible viscous flow on any fired smooth surface, on any fired solid boundary, the directions corresponding to the mazimum or minimum rate of change of element material arc length with the same absolute value J 1/2 is T/4 or 37/4 with respect to the tangent direction of the boundary
Lemma 5 (Intrinsic Decomposition) Φ = e ⊗ (e · Φ) − [ e, [e, Φ] ] (Φ · e) ⊗ e − [ [Φ, e], e ] , ∀Φ ∈ T r (T Σ), e ∈ T Σ s.t. |e| = 1 where [e, Φ] denotes the cross product e × Φ and so on. Proof: Let Φ = Φi ·j gi ⊗ g j ∈ T 2 (T Σ) without lost of generality, and to calculate [ [Φ, e], e ] = [ [Φi ·jgi ⊗ g j , ekg k ], e ] = [ Φi ·j ekϵ jk3 gi ⊗ nΣ, e ] = Φi ·j (eke q )(ϵ jk3 ϵql3)gi ⊗ g l = Φi ·j (eke q )(δ j q δ k l − δ k q δ j l )gi ⊗ g l = (Φi ·j e j gi ) ⊗ (ekg k ) − Φ Evidently, one arrives at the conclusion. The other identity can be proved similarly. Applied the intrinsic decomposition to the velocity gradient tensor, the strain tensor has the following representation, termed still as Caswell formula. Corollary 5 (Caswell Formula) On any fixed solid boundary where the general viscous boundary condition is satisfied by the fluid, namely V = 0 on the boundary, the strain tensor can be represented as following D = θn ⊗ n + 1 2 (ω × n) ⊗ n + 1 2 n ⊗ (ω × n) Proof: Basically, a kind of local curvilinear coordinates denoted by {x q , x⊥} corresponding to the fixed solid boundary can be set up such that the local co-variant basis {gq , g⊥} is orthogonal on the boundary with gq tangents to the boundary and g⊥ = g ⊥ = n. The details can be referred to the appendix. Subsequently, the intrinsic decomposition is utilized V ⊗ ∇ = (V ⊗ ∇, n) ⊗ n − [ [V ⊗ ∇, n ], n ] For the first term on the right hand, one has (V ⊗ ∇, n) = (V ⊗ ∇ − ∇ ⊗ V , n) + (∇ ⊗ V , n) = ω × n + (∇ ⊗ V , n) where (∇ ⊗ V , n) = ( g q ⊗ ∇ ∂ ∂xq V + g ⊥ ⊗ ∇ ∂ ∂x⊥ V , n ) = θn thanks to the relations ∇ ∂ ∂xq V = ∇qv q gq + ∇qv ⊥g⊥ = 0 ∈ T Σ due to the viscous boundary condition and ( ∇ ∂ ∂x⊥ V , n ) = ( ∇⊥v q gq + ∇⊥v ⊥g⊥, g ⊥ ) = ∇⊥v ⊥ = θ − ∇qv q = θ For the second term on the right hand, it is naught thanks to [V ⊗ ∇, n] = [ (∇qV ) ⊗ g q + (∇⊥V ) ⊗ g ⊥, n ] = [ (∇⊥V ) ⊗ g ⊥, g ⊥ ] = (∇⊥V ) ⊗ [ g ⊥, g ⊥ ] = 0 As a summary, one arrives at the relation V ⊗ ∇ = (ω × n) ⊗ n + θn ⊗ n Accompanying with its conjugate relation, the proof is completed. Readily, one has the following assertion Corollary 6 For any two dimensional incompressible viscous flow on any fixed smooth surface, on any fixed solid boundary, the directions corresponding to the maximum or minimum rate of change of element material arc length with the same absolute value |ω 3 |/2 is π/4 or 3π/4 with respect to the tangent direction of the boundary. 8
3.2 Particular Theories For Incompressible Flows As generally, the incompressibility for two dimensional flows on fixed smooth surfaces is still defined as 6=0. Chomaz Cathalau(1990) revealed some velocity domains of soap films in which they can be considered as two dimensional incompressible flows. Based on the continuity equation(7), one has aV 0=Vsv t) d xs hen the stream function can be introduced through Vs=est3 an (a, t). Subsequently, the stream function vorticity algorithm can be derived Lemma 6( Stream Function Vorticity Algorithm for Incompressible Flows a-y (, au3 (x,t)+ (x,t) VSV V:(KGVi)+-EBvkfs Proof: The Stream function Possion equation is just from the definition w 3=E3tsV'vs.On vorticity equation, accompanying(13)with( 8 )and taking account of 0=0, one has n=E Thanks to the relation(see Xie et al., 2013 the proof is completed Finally, the pressure Possion equation can be derived Lemma 7(Pressure Possion Equation for Incompressible Flows Ap=pI(V (V⑧v)+KdlV Proof: Taking denoted as V the divergence operator on both sides of( 8), one has )=V(7+yvw+ym)+y ⅴvW)=(Vv)(V)+(四.)=(v)V,)+v(V)+v (VVS)(VsV)+KG(8 9Ls)VVi=(V8V): (V8V)+KGlV (V'VsVi) (VsV)=VSV(VsV)+R'tsViV+R'tlsvsVt VV(VsV)+KG(8s 9ts-98sVtV+Kg(oigts-985VsVi=Vs(V'vsVi ⅴV(VV)+V=V[Ke(663-929)V=V(KcV)=vVKG Then the identity is proved
3.2 Particular Theories For Incompressible Flows As generally, the incompressibility for two dimensional flows on fixed smooth surfaces is still defined as ˙θ = 0. Chomaz & Cathalau (1990) revealed some velocity domains of soap films in which they can be considered as two dimensional incompressible flows. Based on the continuity equation (7), one has θ = ∇sV s = ∂V s ∂xs (x, t) + Γs slV l = 1 √g ∂ ∂xs ( √ gV s )(x, t) = 0 then the stream function can be introduced through V s = ϵ st3 ∂ψ ∂xt (x, t). Subsequently, the stream function & vorticity algorithm can be derived. Lemma 6 (Stream Function & Vorticity Algorithm for Incompressible Flows) △ψ , g ij [ ∂ 2ψ ∂xi∂xj (x, t) − Γ k ij ∂ψ ∂xk (x, t) ] = −ω 3 ω˙ 3 = ∂ω3 ∂t (x, t) + V s ∂ω3 ∂xs (x, t) = µ ρ [ ∇s∇sω 3 + 2ϵ kl3∇k(KGVl) ] + 1 ρ ϵ kl3∇kfsur,l Proof: The Stream function Possion equation is just from the definition ω 3 = ϵ3ts∇tV s . On vorticity equation, accompanying (13) with (8) and taking account of θ = 0, one has ω˙ 3 = (∇ × a) · nΣ = ϵ kl3∇kal = − 1 ρ ϵ kl3 ∂ 2p ∂xkx l (x, t) + µ ρ [ ϵ kl3∇k(∇s∇sVl) + ϵ kl3∇k(KGVl) ] + 1 ρ ϵ kl3∇kfsur,l Thanks to the relation (see Xie et al., 2013) ϵ kl3∇k(∇s∇sVl) = ∇s∇s(ϵ kl3∇kVl) + ϵ kl3∇k(KGVl) = ∇s∇sω 3 + ϵ kl3∇k(KGVl) the proof is completed. Finally, the pressure Possion equation can be derived Lemma 7 (Pressure Possion Equation for Incompressible Flows) −∆p = ρ[(V ⊗ ∇) : (∇ ⊗ V ) + KG|V | 2 ] − 2µV · (∇KG) − ∇ · fΣ, |V | 2 := V sVs Proof: Taking denoted as ∇· the divergence operator on both sides of (8), one has ρ∇l ( ∂Vl ∂t (x, t) + V s∇sVl ) = −∇l (∇lp) + µ [ ∇l∇s∇sVl + ∇l (KGVl) ] + ∇l fsur,l where ∇l (V s∇sVl) = (∇lV s )(∇sVl) + V s (∇l∇sVl) = (∇lV s )(∇sVl) + V s [ ∇s(∇lVl) + R ·tl· l··sVt ] = (∇lV s )(∇sVl) + KG(δ l l δ t s − g tlgls)V sVl = (V ⊗ ∇) : (V ⊗ ∇) + KG|V | 2 ∇l (∇s∇sVl) = ∇l∇s (∇sVl) = ∇s∇l (∇sVl) + R ·tls s ∇tVl + R ·tls l ∇sVt = ∇s∇l (∇sVl) + KG(δ l s g ts − g tlδ s s )∇tVl + KG(δ l l g ts − g tlδ s l )∇sVt = ∇s (∇l∇sVl) = ∇s [∇s(∇lVl) + R ·tl· l··sVt ] = ∇s [KG(δ l l δ t s − g tlgls)Vt ] = ∇s (KGVs) = Vs∇sKG Then the identity is proved. 9
Fixed solid bondan o a b nx=g D Figure 2: Sketch of local parametrization of a patch of surface, Dl,2, D=l=2 and Drri denote parametric domains with respect to coordinates x, 22),a, 22) and (z, r) respectively 4 Conclusions Basically, some primary relations in general theory of vorticity dynamics have been extended to surface tensor fields based on Levi-Civita connection operator. Subsequently, the governing equation of vorticity, Lagrange theorem on vorticity, Caswell formula on strain tensor are at tained. Furthermore the stream function vorticity algorithm with pressure Possion equation has been set up for incompressible Hows. It has been revealed that Gaussian and mean cur vatures are taking part directly in some governing equations but all of the effects due to the geometry of the surface will disappear as flows on general fixed surfaces degenerates to ones or the plane. All of the theoretical results presented in this paper are exactly deduced without any approximation and constitute a theoretical framework for two dimensional flows on general fixed smooth surfaces Acknowledgements This work is supported by National Nature Science Foundation of China(Grant No. 11172069 and some key projects of education reforms issued by the Shanghai Municipal Education Com- mission(2011) Appendix 1. Construction of local curvilinear coordinates 3,J As sketched in Figure 2, one can suppose without lost of generality that the preimage of a segment of boundary on the surface in the parametric space [ x, 2) can be represented locally (a,b) v(x2) Then the following coordinates can be constructed locall (x2,x2)= The segment of the boundary is corresponding to il E(a, b)and 12=0. The local co-variant basis with respect to [_ is denoted by 9:12_. Thirdly, one introduces another coordinates
Figure 2: Sketch of local parametrization of a patch of surface, Dx1x2 , Dx˜ 1x˜ 2 and Dxqx⊥ denote parametric domains with respect to coordinates {x 1 , x2}, {x˜ 1 , x˜ 2} and {x q , x⊥} respectively. 4 Conclusions Basically, some primary relations in general theory of vorticity dynamics have been extended to surface tensor fields based on Levi-Civita connection operator. Subsequently, the governing equation of vorticity, Lagrange theorem on vorticity, Caswell formula on strain tensor are attained. Furthermore, the stream function & vorticity algorithm with pressure Possion equation has been set up for incompressible flows. It has been revealed that Gaussian and mean curvatures are taking part directly in some governing equations but all of the effects due to the geometry of the surface will disappear as flows on general fixed surfaces degenerates to ones on the plane. All of the theoretical results presented in this paper are exactly deduced without any approximation and constitute a theoretical framework for two dimensional flows on general fixed smooth surfaces. Acknowledgements This work is supported by National Nature Science Foundation of China (Grant No.11172069) and some key projects of education reforms issued by the Shanghai Municipal Education Commission (2011). Appendix 1. Construction of local curvilinear coordinates {x q , x⊥} As sketched in Figure 2, one can suppose without lost of generality that the preimage of a segment of boundary on the surface in the parametric space {x 1 , x2} can be represented locally as (a, b) ∋ x 1 7→ [ x 1 x 2 ] (x 1 ) = [ x 1 ψ(x 1 ) ] Then the following coordinates can be constructed locally [ x 1 x 2 ] 7→ [ x˜ 1 x˜ 2 ] (x 1 , x2 ) = [ x 1 x 2 − ψ(x 1 ) ] The segment of the boundary is corresponding to ˜x 1 ∈ (a, b) and ˜x 2 = 0. The local co-variant basis with respect to {x˜ i} 2 i=1 is denoted by {g˜i} 2 i=1. Thirdly, one introduces another coordinates 10
