Mechanism of instability in non-uniform dusty channel flow

TL;DR

This paper investigates two novel low-Reynolds number instabilities in inhomogeneously loaded dusty channel flows, revealing their mechanisms, energy dynamics, and the stabilizing effect of particle volume fraction, with implications for flow transition.

Contribution

It introduces and analyzes two new instability modes caused by particle concentration variations, extending classical stability theory to dusty flows and deriving a minimal composite model.

Findings

Overlap modes have similar energy budgets despite different eigenstructures.

Particle volume fraction stabilizes the flow.

Critical layer variations drive particle-induced instabilities.

Abstract

Particles in pressure-driven channel flow are often inhomogeneously distributed. Two modes of low-Reynolds number instability, absent in Poiseuille flow of clean fluid, are created by inhomogeneous particle loading, and their mechanism is worked out here. Two distinct classes of behaviour are seen: when the critical layer of the dominant perturbation overlaps with variations in particle concentration, the new instabilities arise, which we term overlap modes. But when the layers are distinct, only the traditional Tollmien-Schlichting mode of instability occurs. We derive the dominant critical layer balance equations in this flow along the lines done classically for clean fluid. These reveal how concentration variations within the critical layer cause two the particle-driven instabilities. As a result of these variations, disturbance kinetic energy production is qualitatively and majorly altered. Surprisingly the two overlap modes, though completely different in the symmetry of the eigenstructure and regime of exponential growth, show practically identical energy budgets, highlighting the relevance of variations within the critical layer. The wall layer is shown to be unimportant. We derive a minimal composite theory comprising all terms in the complete equation which are dominant somewhere in the flow, and show that it contains the essential physics. When particles are infinitely dense relative to the fluid, the volume fraction is negligible. But for finite density ratios, the volume fraction of particles causes a profile of effective viscosity. This is shown to be uniformly stabilizing in the present flow. Gravity is neglected here, and will be important to study in future. So will transient growth of perturbations due to non-normality of the stability operator, in a quest for the mechanism of transition to turbulence.