7.1. ASTROPHYSICAL FLUID DYNAMICS 405
2) By (7.1.6) and (7.1.45),(u·∇)ukcan be written asukDkuθ=ur∂uθ
∂r+uθ∂uθ
∂ θ+uφ∂uθ
∂ φ+
2
r(7.1.50) uθur−sinθcosθu^2 φ
ukDkuφ=ur∂uφ
∂r+uθ∂uφ
∂ θ+uφ∂uφ
∂ φ+
2cosθ
sinθuθuφ+2
r(7.1.51) uφur,
ukDkur=ur∂ur
∂r+uθ∂ur
∂ θ+uφ∂ur
∂ φ−
r
α(u^2 θ+sin^2 θu^2 φ−α′
2 r(7.1.52) u^2 r).
3) The gradient operator:(7.1.53) ∇p=
(
1
α∂p
∂r,
1
r^2∂p
∂ θ,
1
r^2 sin^2 θ∂p
∂ φ)
.
4) By (7.1.8) and√
g=r^2 sinθ√
α, the divergent operator divuis(7.1.54) divu=
1
sinθ∂
∂ θ(sinθuθ)+∂uφ
∂ φ+
1
r^2√
α∂
∂r(r^2√
αur).Remark 7.4.The expressions (7.1.47)-(7.1.54) are the differential operators appearing in the
fluid dynamic equations describing the stellar fluids. However, we need to note that the two
componentsuθanduφare the angular velocities ofθandφ, i.e.
uθ=dθ
dt, uφ=dφ
dt.
In classical fluid dynamics, the velocity fieldv= (vθ,vφ,vr)is the line velocity. The relation
ofuandvis given by
(7.1.55) uθ=
1
rvθ, uφ=1
rsinθvφ, ur=vr.Hence, inserting (7.1.55) into (7.1.1) with the expressions (7.1.47)-(7.1.54), we derive the
Navier-Stokes equations in the usual spherical coordinateform as follows
(7.1.56)
∂vθ
∂t+ (u·∇)vθ=ν∆vθ−1
ρr∂p
∂ θ+fθ,∂vφ
∂t
+ (u·∇)vφ=ν∆vφ−1
ρrsinθ∂p
∂ φ
+fφ,∂vr
∂t+ (u·∇)vr=ν∆vr−1
ρ α∂p
∂r+fr,divv= 0 ,where
(7.1.57)
∆vθ= ̃∆vθ+2
r^2∂vr
∂ θ−
2cosθ
r^2 sin^2 θ∂vφ
∂ φ−
vθ
r^2 sin^2 θ−
1
2 α^2 rdα
dr∂
∂r(rvθ),∆vφ= ̃∆vφ+2
r^2 sinθ∂vr
∂ φ+
2cosθ
r^2 sin^2 θ∂vθ
∂ φ−
vφ
r^2 sin^2 θ−
1
2 α^2 rdα
dr∂
∂r(rvφ),∆vr= ̃∆vr−2
αr^2(
vr+
∂vθ
∂ θ+
cosθ
sinθvθ+1
sinθ∂vφ
∂ φ)
+
1
2 α∂
∂r(
1
αdα
drvr