  We generalize the notion of dissipationless, topological Hall viscosity
tensor to optical phonons in thin film Weyl semimetals. By using the strained
Porphyrin thin film Weyl semimetal as a model example, we show how optical
phonons can couple to Weyl electrons as chiral pseudo gauge fields. These
chiral vector fields lead to a novel dissipationless two-rank viscosity tensor
in the effective dynamics of optical phonons whose origin is the chiral
anomaly. We also compute the contribution to this two rank Hall viscosity
tensor due to the presence of an external magnetic field, whose origin is the
conventional Hall response of Weyl electrons. Finally, the phonon dispersion
relations of the system at the long-wavelength limit with and without an
electromagnetic field are calculated showing a measurable shift in the Raman
response of the system. Our results can be investigated by Raman scattering or
infrared spectroscopy by attenuated total reflectance experiments.
