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as time goes by, the hot piece gets cooler and the cold one gets hotter. After a long
time, the temperature is the same in both pieces, i.e., the temperature is constant
throughout both metal pieces. The physical process where heat flows from hot to
cold regions is called
heat conduction
and is caused by the motion of the molecules
in the metal pieces.
If the two metal pieces are surrounded by air, there will also be heat exchange
with the air. Heat will flow into the air if the metal is hotter than the air. On the
contrary, the air will heat the metal if the metal is initially cooler than the air. Let
us imagine a physical experiment where we can neglect the heat exchange with the
surrounding air. Each metal piece is then a rod with some outer isolating material
such that hardly any heat can flow through the isolating material. At one end of the
rod, there is no isolation, and this end is brought in contact with the similar end
of the other rod. In this situation there will be a very small temperature difference
throughout the cross sections of the rods. We can therefore assume that the temper-
ature, called T , only varies along the rods, and in time. We then set T
D
T.x;t/,
where x is a coordinate along the center axis of the rods. If the lengths of the left
and right rods are L
L
and L
R
, respectively, we let x go from 0 to the total length
L
D
L
L
C
L
R
.TheT.x;t/ function is then defined from x
2
Œ0; L and t>0.
Initially, the metal pieces have different temperatures: T
D
T
L
in the left rod,
x
2
Œ0; L
L
,andT
D
T
R
in the right rod, x
2
.L
L
;L.SayT
L
>T
R
.Then
heat will flow from the left to the right piece. The initial jump in T is smoothed and
the T.x;t/function becomes a constant T
1
as t
!1
. Because no heat can escape
from this isolated system, T
1
D
.T
L
C
T
R
/=2. Later, we will compute the T.x;t/
function in detail and see how it evolves in space and time.
7.1.2
Diffusive Transport of a Substance
Another diffusion process occurs when you drop some dark blue ink into a glass
of water. At once, you can observe the ink in a small, localized region of space.
As time goes by, the ink diffuses into the water, i.e., the ink molecules mix with the
water molecules. The water gets bluer as the initial dark blue ink spreads throughout
the glass. After quite some time, all the water is light blue. The color is reflected by
the concentration of ink molecules. We can use a function c of space and time to
represent the concentration of ink molecules. When c.x; y;
z
;t/
D
1, we have only
ink molecules inside a small volume
1
surrounding the spatial point .x; y;
z
/ at time t .
Initially (i.e., when t
D
0), c
D
1 in a small region corresponding to the droplet
of ink. In the water there are no ink molecules and thus c
D
0. The significant
jump from c
D
0 in the water to c
D
1 in the ink is smoothed out in time. After
some time, c might end up at a value larger than 0, but significantly less than 1,
since there is much more water than ink in the glass. The final light blue color we
1
A small volume in the present context can be a cube with a width of, say, 0.1 mm.
