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Institut für Theoretische Physik I

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- Force response function on a moving sphere in a viscoelastic medium
- supervised by Michael Schmiedeberg
- finished 2019-01
- [Full text as PDF]

The Intention of this thesis is to find an analytical solution or at least an approximation of the force response on a sphere moving in a viscoelastic medium. We use a two fluid model , c.f. [1], to describe a viscoelastic medium as the coupling of an elastic network to an incompressible Newtonian fluid. The coupling is done via a friction term, which is proportional to the relative motion between the elastic network and the fluid. Our study is limited to the regime of low Reynolds numbers which is justified for systems like biological tissues, polymere solutions or gels, where the moving object and the velocity are relatively small. A perturbation ansatz is used to decouple the differential equations of the two fluid model. Physically that means we consider only a small coupling of the Newtonian fluid and the elastic network. In zeroth order we are able to solve the stationary Navier Stokes equation identically to the derivation of the Stokes’ law in [2], whereas the displacement field is solved by a radial symmetric field for a resting sphere in the origin. The time dependence of the field will then be added by substituting r with r(t). The first order correction terms are calculated by the use of spherical harmonics and multipole expansion of the Green matrix of the equilibrium Navier Cauchy equation. We find that the resulting fields are not physically meaningful since they are divergent for large distances. In case of the first order correction term we find that this problem cannot be solved by adding homogeneous solutions of the differential equation due to properties of spherical harmonics. We argue that the divergent behaviour is due to a seeming incompatibility of the zeroth order velocity field to the zeroth order displacement field. This is caused by the long range of the r −1 decrease of the viscous velocity compared to the faster r−3 decrease of the velocity of the elastic network. Therefore we use the resulting fields only to get a rough estimation of the correction needed to the force response function. We discover that this correction seems physically plausible despite the underlying divergent fields. Surprisingly we do not find a R2 proportionality in the correction terms of the force response function. In zeroth order we find the linear dependency on R and the first order correction is already cubic in R.

[1] LEVINE A. J. and LUBENSKY T. C. Phys. Rev. Lett., 85 (2000) 1774.

[2] LANDAU L. D. and LIFSCHITZ E. M. Lehrbuch der Theoretischen Physik, Band VI: Hydrodynamik. Akademie Verlag, Berlin, 3. auflage edition, 1978.