Kinematics of a Fluid Element

Convection: u

Rotation rate: 1 12 2

i j k

ux y zu v w

vorticityω

∂ ∂ ∂Ω = ∇× =

∂ ∂ ∂

=

12

w v u w v ui j ky z z x x y

∂ ∂ ∂ ∂ ∂ ∂ = − + − + − ∂ ∂ ∂ ∂ ∂ ∂

Normal strain rates:

x

xxx

dLudt

L xε ∂

= =∂

yyy

dL vdt z

ε ∂= =

∂

zZZ

dL wdt z

ε ∂= =

∂

Shear strain rates:

Angle between edge1 1along and along 2 2

jiij ji

j i

uu di jx x dt

ε ε ∂ ∂

= + = = ∂ ∂

Strain rate tensor:

xx xy xz

yx yy yz

zx zy zz

ε ε εε ε εε ε ε

Convection Rotation Compression/Dilation(Normal strains)

Shear Strain

Lx

Ly

Kinematics of a Fluid Element

16.100 2002 2

Divergence

( ) /d Volumeu v wu Volume

x y z dt∂ ∂ ∂

∇• = + + =∂ ∂ ∂

Substantial or Total Derivative

u

D u v wDt t x y z

•∇

∂ ∂ ∂ ∂= + + +∂ ∂ ∂ ∂

=rate of change (derivative) as element move through space

Cylindrical Coordinates x x r ru u e u e u eθ θ= + +

1x r rxx rr

u u u ux r r r

θθθε ε ε

θ=

∂ ∂ ∂= = +∂ ∂ ∂

1 12

rr

u urr r r

θθε θ

∂ ∂ = + ∂ ∂

12

r xrx

u ux r

ε ∂ ∂ = + ∂ ∂

1 12

xx

u ur x

θθε θ

∂ ∂ = + ∂ ∂

( )1 1 rx

uu ru er r rθ θ∂ ∂ ∇× = − ∂ ∂

1 xr

u u er x

θ

θ∂ ∂ + − ∂ ∂

r xu u ex r θ

∂ ∂ + − ∂ ∂

( )1 1rx ruu uux r r r

θ

θ∂∂ ∂

∇• = + +∂ ∂ ∂