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 fowstwo 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 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 = 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 com￾ponents 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