  A long and slender finger can serve as a simple ``test bed'' for different
phase ordering models. In this work, the globally-conserved,
interface-controlled dynamics of a long finger is investigated, analytically
and numerically, in two dimensions. An important limit is considered when the
finger dynamics are reducible to the area-preserving motion by curvature. A
free boundary problem for the finger shape is formulated. An asymptotic
perturbation theory 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) which 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.
Finally, the finger dynamics is investigated numerically, employing the
Ginzburg-Landau equation with a global conservation law. The theory is in a
very good agreement with the numerical solution.
