  Acoustic torsion recently introduced in the literature (Garcia de
Andrade,PRD(2004),7,64004) is extended to rotational incompressible viscous
fluids represented by the generalised Navier-Stokes equation. The fluid
background is compared with the Riemann-Cartan massless scalar wave equation,
allowing for the generalization of Unruh acoustic metric in the form of
acoustic torsion, expressed in terms of viscosity, velocity and vorticity of
the fluid. In this work the background vorticity is nonvanishing but the
perturbation of the flow is also rotational which avoids the problem of
contamination of the irrotational perturbation by the background vorticity. The
acoustic Lorentz invariance is shown to be broken due to the presence of
acoustic torsion in strong analogy with the Riemann-Cartan gravitational case
presented recently by Kostelecky (PRD 69,2004,105009). An example of analog
gravity describing acoustic metric is given based on the teleparallel loop
where the acoustic torsion is given by the Lense-Thirring rotation and the
acoustic line element corresponds to the Lense-Thirring metric.
