Moving boundary problems

Elliot C.M., Ockendon J.R. (1982) Weak and variational methods for moving boundary problems. Pitman, Research Notes Math. 59. 

These two processes provide examples of the moving boundary problem in diffusing systems in which a solid solution precedes the formation of a compound. The diickness of the separate phase of the product, carbide or... 

A. Boesch, H. Miiller-Krumbhaar, O. Shochet. Phase field models for moving boundary problems Controlling metastability and anisotropy. Z Physik B 97 161, 1995. 

The analysis of fluid-solid reactions is easier when the particle geometry is independent of the extent of reaction. Table 11.6 lists some situations where this assumption is reasonable. However, even when the reaction geometry is fixed, moving boundary problems and sharp reaction fronts are the general rule for fluid-solid reactions. The next few examples explore this point. 

The mathematical solution to moving boundary problem involves setting up a pseudo-steady-state model. The pseudo-steady-state assumption is valid as long as the boundary moves ponderously slowly compared with the time required to reach steady state. Thus, we are assuming that the boundary between the salt solution and the solid salt moves slowly in the tablet compared to the diffusion... 

Because it is more complicated to solve the moving boundary problem for the rotation of the screw, the barrel rotation models described above have been extensively adopted and investigated. In practice the screw is rotated and not the barrel. The barrel rotation theory has several limitations when describing the real extrusion process, so correct interpretation of the calculated results based on barrel rotation becomes necessary. Most screw design practitioners, with substantial previous design experience, make major adjustments in design specifications to obtain effective correiations. 

Farkas, B. E (1994). Modeling immersion frying as a moving boundary problem. Ph.D. dissertation, University of California, Davis. 

Other moving boundary problems such as crystal dissolution may be treated the same way. For example, for crystal dissolution, one way is to treat w as a negative parameter in the above equation. Alternatively, one may redefine u to... 

When we discussed moving boundary problems, we transformed the problem into boundary-fixed reference frame and converted the moving boundary to a... 

Mathematically, diffusive crystal dissolution is a moving boundary problem, or specifically a Stefan problem. It was treated briefly in Section 3.5.5.1. During crystal dissolution, the melt grows. Hence, there are melt growth distance and also crystal dissolution distance. The two distances differ because the density of the melt differs from that of the crystal. For example, if crystal density is 1.2 times melt density, dissolution of 1 fim of the crystal would lead to growth of 1.2 fim of the melt. Hence, AXc = (pmeit/pcryst) where Ax is the dissolution distance of the crystal and Ax is the growth distance of the melt. 

Free and Moving Boundary Problems. Oxford, UK Clarendon Press. 

To solve the preceding set of equations, Equation 5.62 is plugged into Equation 5.60. By separately determining the compaction properties of the fiber bed [32] an evolution equation for the pressure can be obtained. Because this is a moving boundary problem the derivative in the thickness direction can be rewritten [32] in terms of an instantaneous thickness. The pressure field can then be solved for by finite difference or finite element techniques. Once the pressure is obtained and the velocity computed, the energy and cured species conservation equations can be solved using the methodology outlined in Section 5.4.1. 

R.F. Sekerka, C.L. Jeanfils, and R.W. Heckel. The moving boundary problem. In H.I. Aaronson, editor, Lectures on the Theory of Phase Transformations, pages 117-169. AIME, New York, 1975. 

In the moving-boundary problems treated above, it was assumed that the interface retained its basic initial shape as it moved. It is important to realize that such problems are a subset of a much wider class of problems known as free-boundary problems, in which the boundary is allowed to change its shape as a function of time [2]. A mathematically correct solution for the motion of a boundary of a fixed ideal shape is no guarantee that it is physically realistic. 

Note that the above approximation is a first order approximation. If we were to use a central difference, we would increase the order, but contrary to what is expected, this choice will adversely affect the accuracy and stability of the solution due to the fact the information is forced to travel in a direction that is not supported by the physics of the problem. How convective problems are dealt will be discussed in more detail later in this chapter. The following sections will present steady state, transient and moving boundary problems with examples and applications. 

By solving the moving boundary problem associated with a swellable dispersed system, the following analytical solution results ... 

---