On the effect of a large cloud of rigid particles on the motion of an incompressible non--Newtonian fluid
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929308720693248 |
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| author | Feireisl, Eduard Roy, Arnab Zarnescu, Arghir |
| author_facet | Feireisl, Eduard Roy, Arnab Zarnescu, Arghir |
| contents | We show that the collective effect of $N$ rigid bodies $(\mathcal{S}_{n,N})_{n=1}^N$ of diameters $(r_{n,N})_{n=1}^N$ immersed in an incompressible non--Newtonian fluid is negligible in the asymptotic limit $N \to \infty$ as long as their total packing volume $\sum_{n=1}^N r_{n,N}^d$, $d=2,3$ tends to zero exponentially -- $\sum_{n=1}^N r_{n,N}^d \approx A^{-N}$ -- for a certain constant $A > 1$. The result is rather surprising and in a sharp contrast with the associated homogenization problem, where the same number of obstacles can completely stop the fluid motion in the case of shear thickening viscosity. A large class of non--Newtonian fluids is included, for which the viscous stress is a subdifferential of a convex potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06782 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the effect of a large cloud of rigid particles on the motion of an incompressible non--Newtonian fluid Feireisl, Eduard Roy, Arnab Zarnescu, Arghir Analysis of PDEs We show that the collective effect of $N$ rigid bodies $(\mathcal{S}_{n,N})_{n=1}^N$ of diameters $(r_{n,N})_{n=1}^N$ immersed in an incompressible non--Newtonian fluid is negligible in the asymptotic limit $N \to \infty$ as long as their total packing volume $\sum_{n=1}^N r_{n,N}^d$, $d=2,3$ tends to zero exponentially -- $\sum_{n=1}^N r_{n,N}^d \approx A^{-N}$ -- for a certain constant $A > 1$. The result is rather surprising and in a sharp contrast with the associated homogenization problem, where the same number of obstacles can completely stop the fluid motion in the case of shear thickening viscosity. A large class of non--Newtonian fluids is included, for which the viscous stress is a subdifferential of a convex potential. |
| title | On the effect of a large cloud of rigid particles on the motion of an incompressible non--Newtonian fluid |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.06782 |