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Kinematics of a fluid element

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Kinematics of a fluid element

  1. 1. Kinematics of a Fluid ElementConvection Rotation Compression/Dilation Shear Strain (Normal strains) Convection: u i j k 1 1 ∂ ∂ ∂ Ω= ∇×u = Rotation rate: 2 2 ∂x ∂y ∂z u v w ω = vorticity 1   ∂w ∂v   ∂u ∂w   ∂v ∂u   =  − i +  − j +  − k  2   ∂y ∂z   ∂z ∂x    ∂x ∂y   Normal strain rates: dLx ∂u ε xx = dt = Lx ∂x Ly dL ∂v ε yy = y = dt ∂z dL ∂w ε ZZ = z = Lx dt ∂z Shear strain rates: 1  ∂u ∂u j  1 d  A ngle betw een edge  ε ij =  i + =   = ε ji  ∂x j ∂xi  2   2 dt  along i and along j  Strain rate tensor: ε xx ε xy ε xz    ε yx ε yy ε yz   ε zx  ε zy ε zz  
  2. 2. Kinematics of a Fluid ElementDivergence ∂u ∂v ∂w d (Volume ) ∇•u = + + = / Volume ∂x ∂y ∂z dtSubstantial or Total Derivative D ∂ ∂ ∂ ∂ = +u +v +w Dt ∂t ∂x ∂y ∂z u •∇ =rate of change (derivative) as element move through spaceCylindrical Coordinates u = ux ex + ur er + uθ eθ ∂u ∂u 1 ∂uθ ur ε xx = x ε rr = r εθθ = + ∂x ∂r r ∂θ r 1  ∂  u  1 ∂ur  ε rθ =  r  θ  +  2  ∂r  r  r ∂θ  1  ∂u ∂u  ε rx =  r + x  2  ∂x ∂r  1  1 ∂u ∂u  εθ x =  x + θ 2  r ∂θ ∂x  1 ∂ 1 ∂ur   1 ∂ux ∂uθ   ∂ur ∂ux  ∇×u =  ( ruθ ) −  ex +  r ∂θ − ∂x  er +  ∂x − ∂r  eθ  r ∂r r ∂θ      ∂u 1 ∂ ( rur ) 1 ∂uθ ∇•u = x + + ∂x r ∂r r ∂θ16.100 2002 2

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