Stream
Function/Potential Flow
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In a Nut Shell: For
steady, incompressible, two-dimensional fluid motion in the xy-plane conservation
of mass (also called the
continuity equation) reduces to ∂u/∂x + ∂v/∂y =
0 (1) It
proves useful to define a new function,
ψ(x,y)
, called the stream function
u =
∂ ψ /∂y and
v = - ∂ ψ /∂x that
automatically satisfies the conservation of mass (the continuity equation). 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 which
likewise automatically satisfies conservation of mass: ∂(rvr/∂r) +
∂vθ/∂θ = 0
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