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Kinematics of a fluid element
1.
Kinematics of a
Fluid Element Convection 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.
Kinematics of a
Fluid Element Divergence ∂u ∂v ∂w d (Volume ) ∇•u = + + = / Volume ∂x ∂y ∂z dt 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 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 ∂θ 16.100 2002 2
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