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We compare an approximation of the singularity-free Wahlquist exact solutionwith a stationary and axisymmetric metric for a rigidly rotating perfect fluidwith the equation of state $\mu + 3p= \mu_0$, a sub-case of a globalapproximate metric obtained recently by some of us. We see that to have a fluidwith vanishing twist vector everywhere in Wahlquist's metric the only option isto let its parameter $r_0\rightarrow0$ and using this in the comparison allowsus in particular to determine the approximate relation between the angularvelocity of the fluid in a set of harmonic coordinates and $r_0$. Through somecoordinate changes we manage to make every component of both approximatemetrics equal. In this situation, the free constants of our metric take valuesthat happen to be those needed for it to be of Petrov type D, the lastcondition that this fluid must verify to give rise to the Wahlquist solution.

Wahlquist's metric versus an approximate solution with the same equation of state