Fluid Kinematics (continued)
In a Nut Shell: General motion
of a fluid element generally
includes translation, rotation,
linear deformation, and angular
deformation. The rotation vector, ω,
in general, has three components. ω =
ωx i +
ωy j +
ωz k where ωz = ˝
( ∂v/∂x - ∂u/∂y) , ωx = ˝
( ∂w/∂y - ∂v/∂z) , ωy = ˝
( ∂u/∂z - ∂w/∂x) The
rotation vector is important in that it leads to the definition of the
vorticity vector, ζ . ζ =
2 ω = Del
x V where
Del is the “del operator” Note: A special situation arises for two-dimensional
fluid flow in the x-y plane when ∂v/∂x =
∂u/∂y. In that
case rotation around the z-axis is
zero giving rise to the condition of
irrotational flow which simplifies
the analysis of complex flow fields. Special Case of Fluid
motion in the xy-plane For
steady, incompressible, two-dimensional fluid motion in the xy-plane
conservation of mass reduces
to ∂u/∂x
+ ∂v/∂y = 0
(1) This
equation is also called the continuity
equation. It
proves useful to define a new function,
ψ(x,y) , called the stream function such that
u =
∂ ψ /∂y
and v = - ∂
ψ
/∂x , then conservation
of mass (the continuity equation) is automatically satisfied. To see this result
substitute u and v into eq. (1). The
fluid velocity components in polar coordinates are: (Note here ψ
= ψ(r,θ) vr =
(1/r)∂ψ
/∂θ and vθ = -
∂ ψ
/∂r Click
here to skip to a discussion of stream function. Click
here to skip to a discussion of velocity potential. Click
here for examples of kinematics. |
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