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