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

TL;DR

The paper develops a frequency-domain framework for oscillatory incompressible flows with spatially resolved complex viscosity μ^*(x,ω). It shows that spatial phase textures φ(x,ω) introduce intrinsic non-normality in the viscous core, making harmonic amplification governed by resolvent geometry and pseudospectra rather than eigenvalues. In 3D z-periodic domains, z-dependent textures generate operator-valued Toeplitz/Laurent coupling of spanwise modes, enabling linear sideband generation even from spanwise-uniform forcing. The work introduces phase-only textures with a single texture-strength axis ∥∇φ∥_{L^∞}, provides phase-compensation tools, and furnishes physically meaningful observables (resolvent gains, impedance, vorticity localization) that connect operator theory to measurable quantities. Together, these results reveal a new linear mechanism for spanwise patterning and phase-sensitive amplification in complex- viscosity flows, with potential implications for designing metamaterials, wall layers, and structured fluids where phase lag plays a central role.

Abstract

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 $\varphi(\mathbf{x})=\argμ^*(\mathbf{x},ω)$. 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 $\varphi$ 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 $μ^*(\mathbf{x},ω)=μ_0(ω)e^{i\varphi(\mathbf{x})}$, the texture strength is quantified by $μ_0(ω)\|\nabla\varphi\|_{L^\infty}$.

Figures (1)

• Figure 1: Schematic of a constitutive phase texture. At a fixed forcing frequency $\omega$, the complex viscosity field is written in polar form $\mu^*(\mathbf{x},\omega)=|\mu^*(\mathbf{x},\omega)|e^{i\varphi(\mathbf{x},\omega)}$. The colormap encodes the spatially varying constitutive phase $\varphi(\mathbf{x},\omega)=\arg\mu^*(\mathbf{x},\omega)$ (a local stress--strain-rate lag), while the arrow glyphs indicate the associated phase-gradient direction (and relative magnitude), emphasizing the texture-gradient structure that enters the divergence-form viscous operator through $\nabla\mu^*= i\,\mu^*\,\nabla\varphi$ in the phase-only class. The inset provides the local phasor convention: the in-phase component $\mu'=\Re\mu^*$ (dissipative) and the quadrature component $\mu"=-\Im\mu^*$ (reactive) under the $e^{i\omega t}$ time dependence. The figure is qualitative and intended to visualize how spatial variation of $\varphi$ generates first-order, gradient-controlled couplings even when $|\mu^*|$ is uniform.

Theorems & Definitions (120)

• Remark 3.1: Sign conventions
• Proposition 3.1: Stieltjes representation and nonnegative dissipation (all $\omega\neq 0$)
• proof
• Remark 3.2: From $\Re\mu^*\ge 0$ to the uniform bound $\Re\mu^*\ge \mu_{\min}>0$
• Proposition 3.2: Harmonic power identity (no-slip or lifted boundary data)
• proof
• Remark 3.3: Traction/impedance boundary data and boundary power terms
• Remark 3.4: Spatially varying phase and local dissipation/reactive partition
• Proposition 3.3: Harmonic power identity
• proof