Hüdro- ja aeromehaanika
u v w
v u
w =w= -
x y
Subtituting to (1.27) we get the vorticity transport equation on the plane
w w w 2w 2w
+u +v = v 2 + 2
t x y x y
1.3 Vorticity transport equation for axisymmetric case
Navier-Stokes equation:
u 1
+ u ( u ) = - p + g + v 2 u
t
u = (v,0, u ) (e)
( e ) = (i , j , k )
=0
We derive the equation for the axisymmetic case (in the absence of gravity):
z component:
u u u 1 p 2 u 2 u 1 u
+u +v =- + v 2 + 2 +
t z r z z r r r
(1.31)
r component:
v v v 1 p 2 v 2 v 1 v v
+u +v =- + v 2 + 2 + -
t z r r z r r r r 2
(1.32)
Differentiating 1.31 in respect to r and 1.32 in respect to z, so we get:
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