This chapter includes contour solidification, a phenomenon impacting the solidification of castings with complex shapes like spheres, cylinders, and custom parts. These shapes introduce surface tension effects that influence the inward solidification process. The chapter explores the Gibbs-Thomson effect, a key concept in understanding how surface tension (represented by the coefficient \(\Gamma _{ls}\) ) affects solidification in alloys. Surface tension enhances the solidification front’s growth rate, which can be calculated using Stefan’s thermal energy balance based on heat transfer principles. The chapter presents analytical solutions for one-dimensional melting and solidification problems in cylindrical geometries (complementing the rectangular solutions in Chap. 5 ). Similar to rectangular coordinates, the L-S interface is denoted by \(r = s(t)\) . Solutions for these “Stefan problems” are based on temperature field equations involving the exponential integral function \(E_{i}\left ( -\mu \right )\) . For comparison, Chap. 5 introduced solutions in rectangular coordinates using the error function \(\operatorname {erf}\left ( n\right )\) and complementary error function \(\operatorname {erfc}\left ( n\right )\) for the temperature field in the liquid and solid phases, respectively. Both approaches utilize a similarity parameter ( \(n=x/4\alpha t\) ) that incorporates thermal diffusivity and solidification time. This chapter prepares readers with the analytical tools to understand solidification in cylindrical geometries and lays the groundwork for exploring solidification in more complex shapes.