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 boundaryLemma 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 bound￾ary 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