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