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 proved3.2 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