G Solidification at a Planar Interface

For all of the flows that can be classified as unidirectional, the analysis of Section A shows that the governing equations reduce to solving a single heat equation for the magnitude of the scalar velocity component in the flow direction. For heat transfer applications, mathematically analogous problems involve heat transport by pure conduction. As noted earlier, there are excellent comprehensive books devoted exclusively to the solution of this class of problems.4 Here, we consider a related problem, which is chosen because it addresses the physically important coupling of heat transfer in the presence of phase change and also because it is another ID problem that exhibits a self-similar solution. 

We consider a vertically oriented cylindrical container, which may be treated as infinitely long, that is initially filled with a pure, single-component liquid at a temperature 0 that exceeds the freezing temperature of the liquid m. Suddenly, at a time that we may denote as t = 0, the base of the cylinder is lowered to a temperature 90 that is below the freezing point and a thin layer of the liquid instantaneously freezes. The problem that we wish to 

The governing equation in each of the two bulk-phase materials is the thermal energy equation, (2-110). In the solid phase, u = 0, and thus 

We have assumed that the fluid properties are independent of temperature, and hence no natural convection will occur. However, the kinematic condition, (2.114b), at an interface involving a phase transformation from liquid to solid requires that there be a relative velocity in the liquid relative to the velocity of the interface  

If pL = ps the only motion in the system will be the motion of the solid-liquid interface as more liquid is solidified, and the problem would consist of heat transfer by conduction in both phases. Otherwise, when pi / ps, we must retain the convection term in (3-170). 

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