Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion
| Type of publication: | Article |
| Citation: | |
| Publication status: | Published |
| Journal: | Advances in Nonlinear Analysis |
| Volume: | 10 |
| Number: | 1 |
| Year: | 2021 |
| Pages: | 501--521 |
| DOI: | 10.1515/anona-2020-0144 |
| Abstract: | We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity $\ve$, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor $\B$. By a proper choice of the constitutive relations for the Helmholtz free energy %Cauchy stress tensor $\T$ (which, however, is non-standard in the current literature despite the fact that this choice is well motivated from the point of view o physics) and for the energy dissipation, we are able to prove that $\B$ enjoys the same regularity as $\ve$ in the classical three-dimensional Navier-Stokes equations. This enables us to handle any kind of objective derivative of $\B$, thus obtaining existence results for the class of diffusive Johnson-Segalman models as well. Moreover, using a suitable approximation scheme, we are able to show that $\B$ remains positive definite if the initial datum was a positive definite matrix (in a pointwise sense). We also show how the model we are considering can be derived from basic balance equations and thermodynamical principles in a natural way. |
| Preprint project: | NCMM |
| Preprint year: | 2020 |
| Preprint number: | 01 |
| Preprint ID: | NCMM/2020/01 |
| Keywords: | |
| Authors | |
| Added by: | [MB] |
| Total mark: | 0 |
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