When can viscosity be ignored? I am working on understanding the motion of a solid ball in an infinite elastic (or possibly viscoelastic) medium when subjected to a sinusoidal driving force. The frequency-dependent complex wave number of such a ball can be shown to be [1]
$$ k(\omega) = \sqrt{\frac{\rho \omega^2}{\mu-i \omega \eta}}, $$
where $\mu$, $\rho$, and $\eta$ are the shear modulus, density, and viscosity of the medium.
My question is, how do I know when I can ignore viscoelastic effects in the medium? In other words, what specific condition can I write to specify the case where 
$$ k(\omega) \approx \sqrt{\frac{\rho \omega^2}{\mu}}? $$
My first thought was to simply set $\mu \gg i \omega \eta$, but I am not sure what that means, given one of those is complex, and the other is real. Am I instead supposed to drop the $i$, and set $\mu \gg \omega \eta$? Please help me find a relationship between $\mu$ and $\eta$ (and possibly other things if necessary) to determine when it is ok to ignore $\eta$.

[1] Aglyamov, S. R., Karpiouk, A. B., Ilinskii, Y. A., Zabolotskaya, E. A. & Emelianov, S. Y. Motion of a solid sphere in a viscoelastic medium in response to applied acoustic radiation force: Theoretical analysis and experimental verification. J. Acoust. Soc. Am. 122, 1927–1936 (2007).
 A: Theoretically, a wavenumber can be split into real and imaginary parts in the following way: $k = k_{real} + ik_{imaginary}$, 
from $k_{real}(w)$ you can then extract phase velocity dispersion curves with no damping, whereas from $k_{imaginary}(w)$ - attenuation dispersion curves. Attenuation dispersion curves are related to the viscoelasticity of the underlying material. If you do not want to consider viscoelasticity, then consider only a real part of your wavenumber. I would compute a real part from you complex expression, instead of just dropping the term with $i \omega \eta$. 
To decide whether include viscoelasticity in your model or not would be up to you, it is an assumption you make based on available observations. To check if the material does not have significant viscosity, the most simple would be to perform mechanical tests - creep and/or relaxation at different rates of strain. Of course, if it is possible for your type of the material. If you want to quantify viscoelasticity in your material, depending on its type (but I assume you are considering something of polymers-soft tissues), then usually linear viscoelastic models can be fitted to that data. However, be careful here, cause probably the observation time of a solid ball in the medium is much shorter than slower mechanical tests for estimating viscoelastic properties obtained from creep-relaxation data. Then may be considering dynamic viscoelastic measurements would be better, allowing to know the dependency of viscoelasticity on frequency and estimating it within your used frequency range. But this step would only be necessary if you really really want to prove your assumption beyond the observations that you already have. 
