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16100lecture2_cg

# 16100lecture2_cg - Kinematics of a Fluid Element Rotation...

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Unformatted text preview: Kinematics of a Fluid Element Rotation Convection Shear Strain Compression/Dilation (Normal strains) Convection: u i Rotation rate: Ω= 1 1∂ ∇×u = 2 2 ∂x u j k ∂ ∂y v ∂ ∂z w ω = vorticity = ∂v ∂u 1 ∂w ∂v ∂u ∂w − i + − j + − k 2 ∂y ∂z ∂z ∂x ∂x ∂y Normal strain rates: dLx ∂u ε xx = dt = ∂x Lx dL ∂v ε yy = y = ∂z dt dL ∂w ε ZZ = z = dt ∂z Ly Lx Shear strain rates: ε ij = 1 2 ∂u ∂u j i+ ∂x j ∂xi Strain rate tensor: ε xx ε yx ε zx ε xy ε xz ε yy ε yz ε zy ε zz 1 d A ngle betw een edge = = ε ji 2 dt along i and along j Kinematics of a Fluid Element Divergence ∇•u = ∂u ∂v ∂w d (Volume ) / Volume + + = dt ∂x ∂y ∂z Substantial or Total Derivative D ∂ ∂ ∂ ∂ = +u +v +w Dt ∂t ∂x ∂y ∂z u •∇ =rate of change (derivative) as element move through space Cylindrical Coordinates u = ux ex + ur er + uθ eθ ∂u ∂u ε xx = x ε rr = r ∂x ∂r 1 ∂ u 1 ∂ur ε rθ = r θ + 2 ∂r r r ∂θ 1 ∂u εθθ = 1 ∂uθ ur + r ∂θ r ∂u ε rx = r + x ∂r 2 ∂x 1 1 ∂u ∂u x + θ εθ x = ∂x 2 r ∂θ 1 ∂ur 1 ∂ 1 ∂ux ∂uθ ∂ur ∂ux ∇×u = ( ruθ ) − ex + r ∂θ − ∂x er + ∂x − ∂r eθ r ∂θ r ∂r ∂u 1 ∂ ( rur ) 1 ∂uθ ∇•u = x + + ∂x r ∂r r ∂θ 16.100 2002 2 ...
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16100lecture2_cg - Kinematics of a Fluid Element Rotation...

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