Phase-Textured Complex Viscosity in Linear Viscous Flows: Non-Normality Without Advection, Corner Defects, and 3D Mode Coupling
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917206062792704 |
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| author | Kleess, Lillian St. |
| author_facet | Kleess, Lillian St. |
| contents | We consider time-harmonic incompressible flow with a spatially resolved complex viscosity field $μ^*(\mathbf{x},ω)$ and, at fixed forcing frequency $ω>0$, its constitutive phase texture $φ(\mathbf{x})=\argμ^*(\mathbf{x},ω)$. In three-dimensional domains periodic in a spanwise direction $z$, $z$-dependence of $μ^*$ 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 $κ\neq 0$ sidebands in the harmonic response as a \emph{linear, constitutive} effect.
We place $μ^*$ at the closure level $\hat{\boldsymbolτ}=2\,μ^*(\mathbf{x},ω)\mathbf{D}(\hat{\mathbf{v}})$, as the boundary value of the Laplace transform of a causal stress-memory kernel. Under the passivity condition $\Reμ^*(\mathbf{x},ω)\ge μ_{\min}>0$, 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 $φ$ 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 $μ^*(\mathbf{x},ω)=μ_0(ω)e^{iφ(\mathbf{x})}$, the texture strength is quantified by $μ_0(ω)\|\nablaφ\|_{L^\infty}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08231 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Phase-Textured Complex Viscosity in Linear Viscous Flows: Non-Normality Without Advection, Corner Defects, and 3D Mode Coupling Kleess, Lillian St. Analysis of PDEs Mathematical Physics Fluid Dynamics We consider time-harmonic incompressible flow with a spatially resolved complex viscosity field $μ^*(\mathbf{x},ω)$ and, at fixed forcing frequency $ω>0$, its constitutive phase texture $φ(\mathbf{x})=\argμ^*(\mathbf{x},ω)$. In three-dimensional domains periodic in a spanwise direction $z$, $z$-dependence of $μ^*$ 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 $κ\neq 0$ sidebands in the harmonic response as a \emph{linear, constitutive} effect. We place $μ^*$ at the closure level $\hat{\boldsymbolτ}=2\,μ^*(\mathbf{x},ω)\mathbf{D}(\hat{\mathbf{v}})$, as the boundary value of the Laplace transform of a causal stress-memory kernel. Under the passivity condition $\Reμ^*(\mathbf{x},ω)\ge μ_{\min}>0$, 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 $φ$ 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 $μ^*(\mathbf{x},ω)=μ_0(ω)e^{iφ(\mathbf{x})}$, the texture strength is quantified by $μ_0(ω)\|\nablaφ\|_{L^\infty}$. |
| title | Phase-Textured Complex Viscosity in Linear Viscous Flows: Non-Normality Without Advection, Corner Defects, and 3D Mode Coupling |
| topic | Analysis of PDEs Mathematical Physics Fluid Dynamics |
| url | https://arxiv.org/abs/2601.08231 |