1. For incompressible flows, their velocity field 2. In the case of axisymmetric 2D incompressible flows, where is Stokes' stream function, and u = VxS, uz = S(r, z, t) = where {r, y, z} are the cylindrical coordinates in which the flow is independent on the coordinate and hence 1 d4 r dr 1 dy r dz Show that in spherical coordinates {R, 0, 0} with the same z axis, this result reads Y(R, 0, t) R sin UR S(R, 0,t) = = Y(r, z, t) r Ug = and Ur = 1 ay R² sin de 1 ӘҺ R sin 0 OR ∙eq, 9

Elements Of Electromagnetics
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1. For incompressible flows, their velocity field
2. In the case of axisymmetric 2D incompressible flows,
where is Stokes' stream function, and
u = VXS,
S(r, z, t) =
Uz =
where {r, y, z} are the cylindrical coordinates in which the flow is independent on the coordinate and hence
1 Ꭷ
r dr
1 dy
r dz
Show that in spherical coordinates {R, 0, 0} with the same z axis, this result reads
Y(R, 0, t)
R sin 0
S(R, 0, t)
UR
uo
Y(r, z, t)
r
=
=
-eq,
and
Up = =
1
ay
R2 sin Ꮎ ᎧᎾ
1 ƏY
R sin Ꮎ ᎧR
-eq
2
(1)
(2)
(3)
Transcribed Image Text:1. For incompressible flows, their velocity field 2. In the case of axisymmetric 2D incompressible flows, where is Stokes' stream function, and u = VXS, S(r, z, t) = Uz = where {r, y, z} are the cylindrical coordinates in which the flow is independent on the coordinate and hence 1 Ꭷ r dr 1 dy r dz Show that in spherical coordinates {R, 0, 0} with the same z axis, this result reads Y(R, 0, t) R sin 0 S(R, 0, t) UR uo Y(r, z, t) r = = -eq, and Up = = 1 ay R2 sin Ꮎ ᎧᎾ 1 ƏY R sin Ꮎ ᎧR -eq 2 (1) (2) (3)
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