  We construct sense-preserving univalent harmonic mappings which map the unit
disk onto a domain which is convex in the horizontal direction, but with
varying dilatation. Also, we obtain minimal surfaces associated with such
harmonic mappings. This solves also a recent problem of Dorff and Muir (Abstr.
Appl. Anal. (2014)). In several of the cases, we illustrate mappings together
with their minimal surfaces pictorially with the help of \texttt{Mathematica}
software.
