High Temperature Superconducting Magnetic Levitation

7.6 Three-dimensional modeling and simulations Ë 249

When we replace휎sby a tensor resistivity휌sto take the anisotropy of the HTSC into
account in Eq. (7.73) and consider that

휌s(∇×T)=( 0

8

휌ab 0 0

0 휌ab 0

0 0 훼휌ab

) 1

9

( 0

(^00)
(^00)
(^00)
(^00)
(^00)
8
¤휕Tz
휕y

−

휕Ty

휕z

¥x̂

œ휕Tx

휕z

−휕Tz

휕x

ŷ

¤

휕Ty

휕x

−

휕Tx

휕y

¥ẑ

) 1

(^11)
(^11)
(^11)
(^11)
(^11)
9
=휌ab ¤휕Tz
휕y

−

휕Ty

휕z

¥x̂+œ휕Tx

휕z

−휕Tz

휕x

ŷ+훼¤

휕Ty

휕x

−휕Tx

휕y

¥ẑ¡.

(7.74)

The following equality for the first term in the left side of Eq. (7.73) holds,

∇×휌s(∇×T)=휌ab ̈훼

휕^2 Ty

휕x휕y

−훼휕

(^2) Tx
휕y^2

−휕

(^2) Tx
휕z^2

+휕

(^2) Tz
휕x휕z
©x̂
+휌ab ̈휕
(^2) Tz
휕y휕z

−

휕^2 Ty

휕z^2

−훼

휕^2 Ty

휕x^2

+훼휕

(^2) Tx
휕y휕x
©ŷ
+휌ab ̈휕
(^2) Tx
휕z휕x

−휕

(^2) Tz
휕x^2

−휕

(^2) Tz
휕y^2

+

휕^2 Ty

휕z휕y

©ẑ. (7.75)

According to the Coulomb gauge, we have∇(∇ ⋅T)=0, i.e.

¬

휕^2 Tx

휕x^2 +

휕^2 Ty

휕y휕x+

휕^2 Tz

휕z휕x­

x̂+¬휕

(^2) Tx
휕x휕t+
휕^2 Ty
휕y^2 +
휕^2 Tz
휕z휕y­
ŷ

+¬

휕^2 Tx

휕x휕z+

휕^2 Ty

휕y휕z+

휕^2 Tz

휕z^2 ­

ẑ= 0. (7.76)

The following identities can be derived from Eq. (7.76),

휕^2 Tz

휕z휕x

= −휕

(^2) Tx
휕x^2

−

휕^2 Ty

휕y휕x

, 휕

(^2) Tz
휕z휕y

= −휕

(^2) Tx
휕x휕y

−

휕^2 Ty

휕y^2

, 휕

(^2) Tx
휕x휕z

+

휕^2 Ty

휕y휕z

= −휕

(^2) Tz
휕z^2

. (7.77)