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273 K.
Ī²I + (Īø ā’ 1) Ī²i
G For the Stefan problem we assume that there is no expansion across
Ī²i+1 = (23.20)
Īø the front (i.e., [Ļ] = 0), reducing the Rankine-Hugoniot jump conditions
according to equation (23.16). for mass and momentum to [VN ] = 0 and [p] = 0, respectively. Then
In multiple spatial dimensions, the equations are discretized in a equation (23.23) reduces to
dimension-by-dimension manner using the one-dimensional discretization
ā’ĻS ([ho ] + [cp ] (TI ā’ To )) = k T Ā· N , (23.24)
outlined above independently on (Ī²ux )x , (Ī²uy )y , and (Ī²uz )z . Figure 23.1
shows a typical solution obtained in two spatial dimensions with a spatially
where Ļ = Ļu = Ļr . Finally, the standard interface boundary condition of
varying Ī².
TI = To reduces this last equation to
The same techniques can be used to discretize the spatial terms in equa-
tion (23.8) or (23.10) to obtain symmetric linear systems of equations for ā’ĻS [ho ] = k T Ā· N , (23.25)
the unknown temperatures Tin+1 . Again, the symmetry allows us to exploit
a number of fast solvers such as PCG. where [ho ] is calculated at the reaction temperature of TI = To .
The Stefan problem is generally solved in three steps. First, the interface
velocity is determined using equation (23.25). This is done by ļ¬rst com-
23.4 Stefan Problems puting TN = T Ā· N in a band about the interface, and then extrapolating
these values across the interface (see equation (8.1)) so that both (TN )u
Stefan problems model interfaces across which an unreacted incompressible and (TN )r are deļ¬ned at every grid point in a band about the interface,
material is converted into a reacted incompressible material. The interface allowing the reaction speed S to be computed in a node-by-node fashion.
256 23. Heat Flow 23.4. Stefan Problems 257

Next, the level set method is used to evolve the interface to its new location.
Finally, the temperature is calculated in each subdomain using a Dirich-
let boundary condition on the temperature at the interface. This Dirichlet
boundary condition decouples the problem into two disjoint subproblems
that can each be solved separately using the techniques described earlier
in this chapter for the heat equation. For more details, see [44].
Figure 23.2 shows a sample calculation of an outwardly growing inter-
face in three spatial dimensions. Figure 23.3 shows two-dimensional results
obtained using anisotropic surface tension. The interface condition is the
fourfold anisotropy boundary condition
8
sin4 (2(Īø ā’ Īø0 )) Īŗ
T = ā’0.001
3
with (left) Īø0 = 0 and (right) Īø0 = Ļ/4. The shape of the crystal in the right
ļ¬gure is that of the crystal in the left ļ¬gure rotated by Ļ/4, demonstrating
that the artiļ¬cial grid anisotropy is negligible.

Figure 23.2. Stefan problem in three spatial dimensions. A supercooled material
in the exterior region promotes unstable growth.
258 23. Heat Flow

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0.8 0.8

0.6 0.6

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0.2 0.2

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