p p p p p V V dV V dr V V V u v V dx dy i dx dy j dx dy x y y x u v x u y v JG

# P p p p p v v dv v dr v v v u v v dx dy i dx dy j dx

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= p p p p p V V dV V dr V V V u v V dx dy i dx dy j dx dy x y y x u v x u y v = + = + + + = + + + + + JG JJG JG JJG G JG JG JG JJG G G Consider, add&subst. split in half 1 1 1 1 2 2 2 2 u u u v v dy dy dy dy dy y y y x x = + + ±²²³²²´ ±²²²³²²²´ p V dV + JJG G p V JJG dr G
Rearrange, 1 1 2 2 u u v u v dy dy dy y y x y x = + + Similarly, 1 1 2 2 v v u v u dx dx dx x x y x y = + + N N P P angu rotation linear def. trans. trans. linear def. rotation angular def. 1 1 1 1 = 2 2 2 2 p p u u v u v v v u v u V i u dx dy dy j v dy dx x y x y x y x y x y + + + + + + + + + µ²¶²· JG G G ±²³²´ ±²²³²²´ lar def. dx µ²²¶²²· or in terms of tensors, N N vorticity rate of tensor strain tensor 1 1 0 2 2 = 1 1 0 2 2 = . . p p ij u v u v u x x y x y V V dr dr u v v u v y x y y x V dr E dr + + + + + + JG JJG G G JJG G JJG G JG
Normally, write tensors (2-D) accounts for distorsion accounts for rotation , xx xy xx xy ij ij yx yy yx yy E = = ±²²³²²´ ±²²²³²²²´ Ex: Given a shear flow, y V U i h = JG G , determine components of deformations & rotation U h y x 0 xx u x ε = = 1 1 0 2 2 2 0 1 2 2 xy yy yx v u U U x y h h v y u v U y x h ∈ = + = + = = = ∈ = + = 0 always 1 1 0 2 2 2 0 1 2 2 xx xy yy yx v u U U x y h h u v U y x h = = = = − = = =
x y x y rotation rate of strain ( ) yx + ( ) xy yx xy Relation Between Stresses & Rate of Strains ij ij σ Strain rates : symmetric second-order tensor xx xy xz ij yx yy yz zx zy zz ∈ = , , xx yy zz u v w x y z ∈ = = ∈ = ij ji ∈ =∈ 1 2 xy yx v u x y ∈ =∈ = + 1 2 yz zy w v y z ∈ =∈ = + 1 2 zx xz u w z x ∈ =∈ = +
Remember: Transport properties of fluid; μ, k , α Viscosity:

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