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Direct simulations of two-dimensional plane channel flow of a viscoelastic fluid at Reynolds number
reveal the existence of a family of attractors whose structure closely resembles the linear Tollmien–Schlichting (TS) mode, and in particular exhibits strongly localized stress fluctuations at the critical layer position of the TS mode. At the parameter values chosen, this solution branch is not connected to the nonlinear TS solution branch found for Newtonian flow, and thus represents a solution family that is nonlinearly self-sustained by viscoelasticity. The ratio between stress and velocity fluctuations is in quantitative agreement for the attractor and the linear TS mode, and increases strongly with Weissenberg number,
. For the latter, there is a transition in the scaling of this ratio as
increases, and the
at which the nonlinear solution family comes into existence is just above this transition. Finally, evidence indicates that this branch is connected through an unstable solution branch to two-dimensional elastoinertial turbulence (EIT). These results suggest that, in the parameter range considered here, the bypass transition leading to EIT is mediated by nonlinear amplification and self-sustenance of perturbations that excite the TS mode.
The science of studying diamond inclusions for understanding Earth history has developed significantly over the past decades, with new instrumentation and techniques applied to diamond sample archives revealing the stories contained within diamond inclusions. This chapter reviews what diamonds can tell us about the deep carbon cycle over the course of Earth’s history. It reviews how the geochemistry of diamonds and their inclusions inform us about the deep carbon cycle, the origin of the diamonds in Earth’s mantle, and the evolution of diamonds through time.
This is an introduction to the dynamics of fluids at small scales, the physical and mathematical underpinnings of Brownian motion, and the application of these subjects to the dynamics and flow of complex fluids such as colloidal suspensions and polymer solutions. It brings together continuum mechanics, statistical mechanics, polymer and colloid science, and various branches of applied mathematics, in a self-contained and integrated treatment that provides a foundation for understanding complex fluids, with a strong emphasis on fluid dynamics. Students and researchers will find that this book is extensively cross-referenced to illustrate connections between different aspects of the field. Its focus on fundamental principles and theoretical approaches provides the necessary groundwork for research in the dynamics of flowing complex fluids.
Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear travelling wave solutions to the Navier–Stokes equations in the channel flow geometry – exact coherent states (ECS) – that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number
. At the bifurcation, the solution has a highly symmetric quasi-streamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented ‘head’ near the channel centreplane where the mean shear vanishes connected to counter-rotating quasi-streamwise ‘legs’ that extend toward the channel wall. At
, the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.