Phase-Textured Complex Viscosity in Linear Viscous Flows: Non-Normality Without Advection, Corner Defects, and 3D Mode Coupling

TL;DR

This paper investigates how spatially varying complex viscosity phase textures in 3D incompressible flows induce non-normality and mode coupling, affecting flow stability and response without the need for advection or corner defects.

Contribution

It introduces a mathematical framework for analyzing phase textures in complex viscosity, revealing their role in non-normality and mode interactions in viscous flows.

Findings

Spatial variation of phase texture causes non-normality in viscous operators.

Spanwise dependence of viscosity leads to mode coupling and sidebands.

Existence and stability bounds are established for oscillatory flow operators.

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 $\mu_0(\omega)\|\nabla\varphi\|_{L^\infty}$.