Kinematics of a fluid element

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 


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

Kinematics of a fluid element

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