Area-preserving dynamics of a long slender finger by curvature: A test case for globally conserved phase ordering - art. no. 066101

A long and slender finger can serve as a simple "test bed" for different ph ase-ordering models. In this work, the globally conserved, interface-contro lled dynamics of a long finger is investigated, analytically and numericall y, in two dimensions. An important limit is considered when the finger dyna mics is reducible to area-preserving motion by curvature. A free boundary p roblem for the finger shape is formulated. An asymptotic perturbation theor y is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of the Mullins finger (a well -known analytic solution) whose width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area . The subleading terms of the asymptotic theory are also calculated. Finall y, the finger dynamics is investigated numerically, employing the Ginzburg- Landau equation with a global conservation law. The theory is in very good agreement with the numerical solution.