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)·NOn 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 sur￾face 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