Abstract
This paper presents an approximate analytic solution for the (well-posed) one phase Stefan problem for the sphere using homotopy analysis. Unlike prior studies, the analytic approximation presented here uses homotopy analysis to deal with the non-linearity associated with the moving boundary; at no point in the analysis are any changes made to either the governing heat equation or the boundary and initial conditions. Explicit analytic expressions are developed in the form of separate Taylor series for the moving boundary location and the temperature profile within the sphere up to and including the fourth term in each series. The approximations are found to be in good agreement with the results from a number of prior studies.
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Notes
- 1.
The velocity of the moving boundary is unbounded at t = 0 but is finite for all t up to (but not including) the point when the moving boundary reaches the centre of the sphere (which point in time is denoted by te). Practically speaking, the moving boundary problem exists for all times between t = 0 and t = te. Beyond this point, the solution for the temperature profile is still well defined and, in this sense, the problem is “well behaved” for all values of t > te.
- 2.
For a detailed discussion of “homotopy analysis” including worked examples, see [2].
- 3.
The short time solutions are shown in [3] as Eqs. (4.3) and (4.4) and the leading term of Eq. (7.3).
- 4.
Refer to [2] for a detailed discussion of the role of the convergence control parameter in the homotopy analysis method.
- 5.
- 6.
The conditions set out in Eq. (16.34) will, of course, be automatically satisfied by the choice of w(z, t; 0).
- 7.
- 8.
- 9.
An example is given in Sect. 5 of [13] of a nonlinear PDE boundary value problem with a non-analytic, global solution (obtained using homotopy analysis) where the corresponding Taylor series solution can only be defined locally.
- 10.
The problem in [12] appears to be well behaved (and, by implication, the power series solutions for the temperature profile and the moving boundary location presented in [12] will be correspondingly well behaved) due to: (i) the condition that the crystal sphere stops growing when the maximum size of the sphere is reached; (ii) the constraint imposed on β0 that β0 is ≪ 1; and (iii) the spherical symmetry of the heat transfer problem. These features allow for only one (bounded) solution for each of the location of the moving boundary and the temperature profile at any given point in time up to and including the point in time where the spherical crystal reaches its maximum size.
- 11.
For example, the corresponding classical two phase Stefan problem for the sphere can be analysed using this approach (see [5] for details of the underlying problem). A suitable set of short time solutions for the classical two phase Stefan problem for the sphere are identified in [5] on pages 2068 and 2069 (see Eqs. 5.5 and 5.6 in [5] for the details).
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Shorten, R.B. (2019). Approximate Analytic Solution of the One Phase Stefan Problem for the Sphere. In: Gutschmidt, S., Hewett, J., Sellier, M. (eds) IUTAM Symposium on Recent Advances in Moving Boundary Problems in Mechanics. IUTAM Bookseries, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-030-13720-5_16
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DOI: https://doi.org/10.1007/978-3-030-13720-5_16
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