Pulsatile Flows for Simplified Smart Fluids with Variable Power-Law: Analysis and Numerics
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| Format: | Preprint |
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2025
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| author | Berselli, Luigi C. Kaltenbach, Alex |
| author_facet | Berselli, Luigi C. Kaltenbach, Alex |
| contents | We study the fully-developed, time-periodic motion of a shear-dependent non-Newtonian fluid with variable exponent rheology through an infinite pipe $Ω:= \mathbb{R}\times Σ\subseteq \mathbb{R}^d$, $d\in \{2,3\}$, of arbitrary cross-section $Σ\subseteq \mathbb{R}^{d-1}$. The focus is on a generalized $p(\cdot)$-fluid model, where the power-law index is position-dependent (with respect to $Σ$), $\textit{i.e.}$, a function $p\colon Σ\to (1,+\infty)$. We prove the existence of time-periodic solutions with either assigned time-periodic flow-rate or pressure-drop, generalizing known results for the Navier-Stokes and for $p$-fluid equations.
In addition, we identify explicit solutions, relevant as benchmark cases, especially for electro-rheological fluids or, more generally, $\textit{`smart fluids'}$. To support practical applications, we present a fully-constructive existence proof for variational solutions by means of a fully-discrete finite-differences/-elements discretization, consistent with our numerical experiments. Our approach, which unifies the treatment of all values of $p(\overline{x})\in (1,+\infty)$, $\overline{x}\in Σ$, without requiring an auxiliary Newtonian term, provides new insights even in the constant exponent case. The theoretical findings are reviewed by means of numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pulsatile Flows for Simplified Smart Fluids with Variable Power-Law: Analysis and Numerics Berselli, Luigi C. Kaltenbach, Alex Numerical Analysis Analysis of PDEs 35Q35, 76A05, 76D05, 65M60, 35D30, 35K55, 35B10, 76M10, 35Q30 We study the fully-developed, time-periodic motion of a shear-dependent non-Newtonian fluid with variable exponent rheology through an infinite pipe $Ω:= \mathbb{R}\times Σ\subseteq \mathbb{R}^d$, $d\in \{2,3\}$, of arbitrary cross-section $Σ\subseteq \mathbb{R}^{d-1}$. The focus is on a generalized $p(\cdot)$-fluid model, where the power-law index is position-dependent (with respect to $Σ$), $\textit{i.e.}$, a function $p\colon Σ\to (1,+\infty)$. We prove the existence of time-periodic solutions with either assigned time-periodic flow-rate or pressure-drop, generalizing known results for the Navier-Stokes and for $p$-fluid equations. In addition, we identify explicit solutions, relevant as benchmark cases, especially for electro-rheological fluids or, more generally, $\textit{`smart fluids'}$. To support practical applications, we present a fully-constructive existence proof for variational solutions by means of a fully-discrete finite-differences/-elements discretization, consistent with our numerical experiments. Our approach, which unifies the treatment of all values of $p(\overline{x})\in (1,+\infty)$, $\overline{x}\in Σ$, without requiring an auxiliary Newtonian term, provides new insights even in the constant exponent case. The theoretical findings are reviewed by means of numerical experiments. |
| title | Pulsatile Flows for Simplified Smart Fluids with Variable Power-Law: Analysis and Numerics |
| topic | Numerical Analysis Analysis of PDEs 35Q35, 76A05, 76D05, 65M60, 35D30, 35K55, 35B10, 76M10, 35Q30 |
| url | https://arxiv.org/abs/2507.22449 |