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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).

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  • $\begingroup$ The point of the complex wave number is that it combines both frequency (in an underdamped situation) and damping information in one parameter. If the viscous forces are small compared with inertial forces in the system you can make the approximation that frequency part is the same as for an inviscid fluid, but IMO it doesn't make much sense to introduce viscosity into the physics and then ignore its main effect, which is energy dissipation or "damping". $\endgroup$ – alephzero Jan 17 at 23:11
  • $\begingroup$ there are dimensionless constants known as "similitude parameters" or "scaling parameters" that are used to guess when things like viscosity, surface tension, gravity, etc. become unimportant or important depending on things like flow velocity or scale length. Two examples are the reynolds number and the mach number and I'll bet there's one for your situation here which might be helpful. $\endgroup$ – niels nielsen Jan 18 at 0:37
  • $\begingroup$ @alephzero I guess another way to state my question is like this: Experimentally, I have observed that good results can be obtained by neglecting the viscosity. Loosely speaking, this material acts more like an elastic medium than a viscoelastic medium -- I don't observe damping. I want to show that this is to be expected mathematically. Are there specific properties of the material I can measure and compare in order to demonstrate that the viscous properties are negligible? $\endgroup$ – Bunji Jan 18 at 1:54
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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.

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