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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
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