PhD thesis – Martin Reichelsdorfer

Abstract

Navier’s slip boundary condition, which assumes a linear velocity-dependent friction force between a flowing liquid and a solid, is almost 200 years old. Yet, the subject of external friction in hydrodynamics is far from settled, and many experimental realisations of increasingly slippery substrates as well as promising potential nano- and microfluidic applications have sparked a renewed interest. One of the strengths of the field – the large variety and complexity – is also one of the greatest challenges and motivates the search for universal and fundamental insights and theories. The present work contributes to this endeavour by the development and application of different mathematical descriptions of slip and external friction in fluids with a focus on small length scales ranging down to molecular dimensions.

A major part of the thesis is dedicated to an extension of (generalised) fluctuating hydrodynamics to liquids experiencing external friction – for instance at the interface to a solid. The theory includes thermal fluctuations in a natural way, and both a top-down as well as a microscopic bottom-up approach yield fluctuating external forces, which are related to a corresponding friction coefficient and the slip length by a fluctuation dissipation theorem. From a microscopic perspective, it turns out that some form of coarse graining like a wavelength cut-off in Fourier space is required for this kind of formulation. The resulting fluctuating slip boundary condition is subsequently applied to the subject of a current debate, leading to the conclusion that the autocorrelations of the tangential external force at a liquid-solid boundary are related to an effective friction coefficient of the entire system rather than that of the specific interface the forces are measured at. Moreover, a possible explanation for deviating results of other authors is found. A second application addresses the popular experimental system of a thin liquid film, for the dynamics of which the influence of thermal noise has already been proven. The fluctuating boundary condition now makes it possible to leave the no-slip regime and derive and discuss stochastic thin film equations for weak and strong slip as well.

Complementary to the above treatment, an alternative description for arbitrary small scales emphasises the microscopic mechanism of the generation of friction between a liquid and an external potential. Momentum is transferred by the static interaction of a flowing liquid’s deformed particle density with the potential, whereas energy is dissipated through additional internal viscous friction. The formalism is able to transfer notions of conventional macroscopic hydrodynamics to the microscale and to capture them in the equation of motion. In addition, an expression for the friction coefficient is obtained, which underlines the role of viscous dissipation. Eventually, an effective theory for the dynamics on larger scales, which integrates out the microscopic details, is devised. The theoretical considerations are visualised and tested by molecular dynamics simulations, which – besides – show an apparent complete breakdown of external friction above a critical flow velocity.