High Temperature Superconducting Magnetic Levitation

7.5 Two-dimensional modelings and simulations Ë 227

the formulation may be yielded as

T

X

0

X

ΩHTSC

휌|∇×h|^2 =

T

X

0

X

ΩHTSC

휌J^2 c(|h+He|). (7.38)

Correlating the above four functions (7.34), (7.36), (7.37), and (7.38), we obtain

휇 0

T

X

0

X

R^3

휕(h+He)

휕t

⋅ (휑−h)= −

T

X

0

X

ΩHTSC

휌∇×h⋅ ∇×(휑−h)

⩾

T

X

0

X

ΩHTSC

휌(J^2 c(|h+He|)−|∇×휑||∇×h|)⩾ 0. (7.39)

This proves thathis a solution of the problem

Find functionh∈K(h)such thatœ휕(h+He)

휕t

,(휑−h)⩾ 0

for any휑∈K(h),h儨儨儨儨t= 0 =h 0 , (7.40)

where(u,w)=

T

X

0

X

R^3

u⋅wis the scalar product of two vector functions, withh 0 =

B 0 /휇 0 −He|t= 0 denoting the initial condition of the problem.
In the two-dimensional case, the magnetic field has only two components and it
can be expressed as

H=Hxx̂+Hyŷ. (7.41)

The current flowing in the HTSC has only thezcomponent, which is derived as

J=

( 0

0

8

x̂ ŷ ẑ

휕

휕x

휕

휕y

휕

휕z

Hx Hy Hz

) 1

1

9

=¤

휕Hy

휕x

−

휕Hx

휕y

¥ẑ. (7.42)

Since there is no imposed current in the HTSC, the net current induced by the variation
of the external magnetic field must satisfy

X

ΩHTSC

J=Itotal(t)= 0. (7.43)