dorsal/arxiv
View SchemaPhase-Textured Complex Viscosity in Linear Viscous Flows: Non-Normality Without Advection, Corner Defects, and 3D Mode Coupling
| Authors | Lillian St. Kleess |
|---|---|
| Categories | |
| ArXiv ID | 2601.08231vv1 |
| URL | https://arxiv.org/abs/2601.08231 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
We consider time-harmonic incompressible flow with a spatially resolved complex viscosity field $\mu^*(\mathbf{x},\omega)$ and, at fixed forcing frequency $\omega>0$, its constitutive phase texture $\varphi(\mathbf{x})=\arg\mu^*(\mathbf{x},\omega)$. In three-dimensional domains periodic in a spanwise direction $z$, $z$-dependence of $\mu^*$ converts coefficient multiplication into convolution in spanwise Fourier index, yielding an operator-valued Toeplitz/Laurent coupling of modes. Consequently, even spanwise-uniform forcing generically produces $\kappa\neq 0$ sidebands in the harmonic response as a \emph{linear, constitutive} effect. We place $\mu^*$ at the closure level $\hat{\boldsymbol{\tau}}=2\,\mu^*(\mathbf{x},\omega)\mathbf{D}(\hat{\mathbf{v}})$, as the boundary value of the Laplace transform of a causal stress-memory kernel. Under the passivity condition $\Re\mu^*(\mathbf{x},\omega)\ge \mu_{\min}>0$, the oscillatory Stokes/Oseen operators are realized as m-sectorial operators associated with coercive sectorial forms on bounded Lipschitz (including cornered) domains, yielding existence, uniqueness, and frequency-dependent stability bounds. Spatial variation of $\varphi$ renders the viscous operator intrinsically non-normal even in the absence of advection, so amplification is governed by resolvent geometry (and associated pseudospectra), not by eigenvalues alone. In the pure-phase class $\mu^*(\mathbf{x},\omega)=\mu_0(\omega)e^{i\varphi(\mathbf{x})}$, the texture strength is quantified by $\mu_0(\omega)\|\nabla\varphi\|_{L^\infty}$.
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"abstract": "We consider time-harmonic incompressible flow with a spatially resolved complex viscosity field $\\mu^*(\\mathbf{x},\\omega)$ and, at fixed forcing frequency $\\omega\u003e0$, its constitutive phase texture $\\varphi(\\mathbf{x})=\\arg\\mu^*(\\mathbf{x},\\omega)$. In three-dimensional domains periodic in a spanwise direction $z$, $z$-dependence of $\\mu^*$ converts coefficient multiplication into convolution in spanwise Fourier index, yielding an operator-valued Toeplitz/Laurent coupling of modes. Consequently, even spanwise-uniform forcing generically produces $\\kappa\\neq 0$ sidebands in the harmonic response as a \\emph{linear, constitutive} effect.\n We place $\\mu^*$ at the closure level $\\hat{\\boldsymbol{\\tau}}=2\\,\\mu^*(\\mathbf{x},\\omega)\\mathbf{D}(\\hat{\\mathbf{v}})$, as the boundary value of the Laplace transform of a causal stress-memory kernel. Under the passivity condition $\\Re\\mu^*(\\mathbf{x},\\omega)\\ge \\mu_{\\min}\u003e0$, the oscillatory Stokes/Oseen operators are realized as m-sectorial operators associated with coercive sectorial forms on bounded Lipschitz (including cornered) domains, yielding existence, uniqueness, and frequency-dependent stability bounds.\n Spatial variation of $\\varphi$ renders the viscous operator intrinsically non-normal even in the absence of advection, so amplification is governed by resolvent geometry (and associated pseudospectra), not by eigenvalues alone. In the pure-phase class $\\mu^*(\\mathbf{x},\\omega)=\\mu_0(\\omega)e^{i\\varphi(\\mathbf{x})}$, the texture strength is quantified by $\\mu_0(\\omega)\\|\\nabla\\varphi\\|_{L^\\infty}$.",
"arxiv_id": "2601.08231",
"authors": [
"Lillian St. Kleess"
],
"categories": [
"math.AP",
"math-ph",
"math.MP",
"physics.flu-dyn"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Phase-Textured Complex Viscosity in Linear Viscous Flows: Non-Normality Without Advection, Corner Defects, and 3D Mode Coupling",
"url": "https://arxiv.org/abs/2601.08231",
"version": "v1"
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