Viscosity variation in fluid flows across scales
Rev. Mod. Phys. 98, 045002 – Published 26 October, 2026
DOI: https://doi.org/10.1103/dj2s-sj4t
Abstract
A plethora of natural and engineering flows exhibit spatial and temporal variation in viscosity. The variations occur in flows across a colossal range of length scales and timescales, from microbial motion to Earth-scale mantle convection, and give rise to new physical mechanisms that are absent in constant-viscosity fluid flows. This review surveys such phenomena across scales and examines these mechanisms. In Stokes (zero-Reynolds-number) flows, viscosity gradients lead to translation-rotation coupling and enable novel particle dynamics such as rotation in response to uniform forcing—a feature that may be exploited by microorganisms. The dynamics of laminar shear flows across scales are transformed by viscosity variation, modifications to the base flow profile, and symmetry breaking. In high-Reynolds-number shear flow, viscosity stratification fundamentally alters the mathematical structure of the singular-perturbation problem that describes the production of disturbance kinetic energy. Even a minor stratification within the layer of kinetic energy production can dramatically enhance or suppress standard instabilities and create new instabilities. Shear flows are prone to transition to turbulence via linear mechanisms: here the linear stability operator is non-normal, resulting in algebraic perturbation growth. Viscosity variations introduce new non-normal and nonlinear pathways for perturbation energy growth, apart from altering the structure and dynamics of the perturbations via the broken symmetries. While laminar and fully developed turbulent flows have received extensive attention in recent decades, the process by which laminar flows transition to turbulence is not well understood, except in a few canonical constant-viscosity flows, and there only partially. There are myriad routes to fully developed turbulence, and viscosity variations will likely occupy center stage as more is learned about them. In fully developed turbulence, viscosity variations influence the structure of wall-bounded flows, jets, and mixing layers. At even larger scales, accounting correctly for the stratification of eddy viscosity in the ocean in global circulation models can lead to better weather predictions. Moving to the planetary scale, viscosity variations of several orders of magnitude occur in Earth’s mantle; these play a central role in geological evolution and mantle convection. Flows laden with solid particles and larger objects, whether in dilute suspension or in mushy regimes, are ubiquitous across scales. Effective viscosity variations from inhomogeneous particle loading can produce effects similar to actual viscosity variations. Throughout, this review emphasizes scenarios where viscosity variation qualitatively—and not merely quantitatively—alters flow physics. Although some aspects of viscosity-varying flows have been studied for decades, an understanding of the physics of many other aspects is just beginning, with a number of interesting questions remaining. The review has graduate students in mind, and every discussion attempts to identify well-posed, tractable research questions. Furthermore, the review highlights cross-scale, and therefore cross-disciplinary, connections, which are rarely made but will likely be insightful.
