BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 15
chosen in a such way that the set { er, eθ
, e z
} to be a right-oriented system of
vectors. In this case, the relationships between the Cartesian coordinates
x , x , x and the cylindrical coordinates ( r, θ , z)
are
1 2 3
⎧x = rcos θ,
⎪
⎨
⎪
⎩
⎧ 2 2
r = x1 + x2,
⎪
⎪ x
cosθ
=
1
1 2 2
⎪ x1 + x2
x2
= rsin θ,
⇔ ⎨
x2
x3
= z; ⎪
⎪
sinθ
= ,
2 2
x1 + x2
⎪
⎪
⎩z
= x3.
,
(22)
These equations give the meaning of each cylindrical coordinate. By
using Achenbach (1973), the operators in the previous sections, in cylindrical
coordinates, have the expressions:
∂ 1 ∂ ∂
∇ = er
+ eθ
+ ez
,
(23)
∂r r ∂θ
∂z
f 1 f f
∇f
= grad f = ∂ er + ∂ e
∂
θ
+ e
z
,
∂r r ∂θ ∂z
(24)
1 ∂ur
1∂uθ
∂uz
∇ · u= divu= ur
+ + +
r ∂r r ∂θ ∂z ,
(25)
er reθ ez
1 ∂ ∂ ∂
∇ × u= curl u=
,
(26)
r ∂r ∂θ ∂z
u ru u
r θ z
where the vector u, written in cylindrical coordinates, has the analytical
expression
u= ure .
r
+ uθeθ + uze
z
The scalar Laplace operator of a vector u is more complicated, but we will
write its expression by using the identity
u ∇∇ ( · u) ∇ ∇ u grad (div u) curl curl u .
(27)
2
∇ = − × × = −
By employing the expressions for the gradient (22), the divergence (23), and the
curl (24), from (25) we obtain
2 ⎛ 2 2 ∂uθ
⎞ ⎛ 2 2 ∂uθ
⎞ 2
∇ u = ⎜∇0ur −
2 r
+ ∇
0uθ + ,
2
θ
+∇
z z
r ∂θ ⎟ ⎜
r ∂θ
⎟
⎝ ⎠ e ⎝ ⎠
e u e (28)