  Let $M(2,\textbf{\underline{w}},\chi)$ be the moduli space of rank $2$
torsion-free sheaves over a reducible nodal curve with each component having
utmost two nodal singularities. We show that in each component of
$M(2,\textbf{\underline{w}},\chi)$, the closure of rank $2$ vector bundles with
fixed determinant is rational.
