Computational Project # 7
Due April 29
Consider the unsteady heat conduction problem in a square region with variable heat
conductivity. The temperature at the top boundary is
0
T
, the left and right boundaries are
adiabatic. The thermal conductivity in region I is
1
k
and region II is
2
k
. The initial
temperature in the entire region is
0
T T
=
. At time
0
t
=
the constant heat flux is supplied
through the bottom boundary.
The equation describing the time evolution
of the temperature in the region is given
T T T
c k k
t x x y y
ρ
∂ ∂ ∂ ∂ ∂
= +
∂ ∂ ∂ ∂ ∂
.
The initial conditions are
(
)
0
, ,0
T x y T
=
,
While the boundary conditions are
( ) ( )
( ) ( )
0 1
0, , , , 0,
, , , ,0, .
w
T T
y t L y t
x x
T
T x L t T k x t q
x
∂ ∂
= =
∂ ∂
∂
= =
∂
This problem can be rewritten in non-dimensional form by using the following
transformation:
0 1 2
1 2
2
0 1 1 0
, , , , 1, , w
w
T T x y tk k q L
X Y q
T L L cL k k T
θ τ α α
ρ
−
= = = = = = =
ɶ.
With this transformation the problem becomes
,
X X Y Y
θ θ θ
α α
τ
∂ ∂ ∂ ∂ ∂
= +
∂ ∂ ∂ ∂ ∂
where
2
1 2 1 2
,
3 3 3 3
1 elsewhere
X Y
α
α
≤ ≤ ≤ ≤
=
The initial conditions are given by
(
)
, ,0 0
x y
θ
=
while the boundary conditions are
( ) ( )
( ) ( )
0, , 1, , 0,
,1, 0, , 0, .
w
Y Y
X X
X X q
x
θ θ
τ τ
θ
θ τ τ
∂ ∂
= =
∂ ∂
∂
= =
∂
ɶ
0
T
x
=
∂
∂
0
T
x
=
∂
∂
0
T
T=
w
q
L
L
Region I
Region
II
