Motion by curvature in generalized Cahn-Allen models

Motion by curvature and impurity drag: resolution of a mobility paradox

“…} We note that the key to the driving force paradox can already be found in the study undertaken by Fife and Lacey [14] of the AC equation alone where a kinetic coecient that depended on order parameter was assumed; however, in order to understand the singular behavior of the mobility in the vicinity of the tricritical point and reconcile the results presented in previous theories, it is necessary to consider the coupled Allen±Cahn/Cahn± Hilliard system, as we do in this paper.…”

“…this is a gradient system with respect to p that is based on an L 2 inner product with respect to the non-conserved order parameter v and an H À1 inner product with respect to the conserved order parameter u, which have been weighted to allow for kinetic coecients Q i , i = 1, 2 that are functions of u and v [14,18,19].…”

“…Existence has recently been proven in one dimension with a non-constant (degenerate) mobility and with a free energy de®ned as in (7) by Dal Passo et al [25]. Fife and Lacey [14] allowed Q 2 to depend on v and examined the limiting sharp interface motions predicted by the AC equation alone without compositional variation.…”

“…Let us assume for the moment that the composition is constant, or has negligible variation, as was assumed by Allen and Cahn [1,2], and Fife and Lacey [14]. Then equation (23) simpli®es to…”

Semi‐analytical analysis of Allen‐Cahn model with a new fractional derivative

“…There are many techniques in literature to handle with these kinds of PDEs such as perturbation iteration techniques, 3–10 optimal homotopy asymptotic method, 11–14 Chebyshev spectral collocation method, 15 homotopy decomposition method, 16 optimal q‐homotopy analysis method, 17 trial equation method, 18–20 (G ′ /G)‐expansion method, 21 homotopy analysis method, 22,23 the sine‐cosine method, 24,25 and cubic B‐spline collocation algorithm 26 . At present, the application of Equation () can be seen in many moving boundary problems in fluid dynamics and image processing through a phase‐field approach, 27,28 and it has been also used to model the impact of a droplet on a solid surface and image segmentation 29,30 . The fundamental concept of the AC model is to replace the singular macroscopic treatment of an interface (a discontinuous surface) with a smooth surface.…”