Inertial Effects in the Response of Viscous and Viscoelastic Fluids

We consider the effect of inertia on the high frequency response of a general linear viscoelastic material to local deformations. We calculate the displacement response and correlation functions for points separated by a distance $r$. The effects of inertia and incompressibility lead to anticorrelations in the correlation or response functions, which become more pronounced for more elastic materials. Furthermore, the stress propagation in viscoelastic media is no longer diffusive, as for simple liquids.

Figure 1

The velocity response of a Newtonian fluid for parallel and perpendicular motion. The solid lines show the in-phase (real) velocity response, which decays on the scale of the penetration depth. The dashed lines show the out-of-phase velocity response (specifically, $\omega {\alpha }^{\prime }$). In the noninertial limit of small $r$, the Oseen tensor is recovered, for which the (velocity) response is real. The decay of the various components of the response illustrates the finite penetration depth for the response. The strong dip in the perpendicular response is a manifestation of the vortexlike flow at short times.

Figure 2

The parallel response for viscoelastic media with $G(\omega )=\overline{g}(-i\omega {)}^z$, where $z=1/2$, $3/4$, 1. For reference, the response function for a Newtonian liquid ($z=1$) is shown in gray. In each case, $z=3/4$ is intermediate between $z=1$ and $z=1/2$. Both real (a) and imaginary (b) parts are shown.

Figure 3

The perpendicular response for viscoelastic media with $z=1/2$, $3/4$, 1. Again, the response for a Newtonian liquid is shown in gray, and the case of $z=3/4$ is intermediate between $z=1$ and $z=1/2$. Both real (a) and imaginary (b) parts are shown.

Figure 4

The displacement field displays a clear vortexlike structure. Here, a force in the $\widehat{x}$ direction is applied at the origin (as shown by the filled circle and arrow). Distances are shown in units of the penetration depth $\delta =\sqrt{|G|/(\rho {\omega }^2)}$. This example has been calculated for the Rouse model with $z=1/2$.