Sebastian Zimmermann

• The functionality of superhydrophobic (SHS) and liquid-infused surfaces (LIS) rely on the confinement of air or other fluids within micro- and nanostructures. They have attracted considerable interest due to their ability to significantly influence properties on a macroscale, including self-cleaning, enhanced heat transfer and reduction of flow resistance in sub-millimetre fluidic systems. Therefore, SHS and LIS have the potential to significantly increase the overall efficiency of functional surfaces in various industrial applications. Despite these promising properties, the development of such tailored surfaces requires a fundamental understanding of their local behaviour and properties. However, the underlying fluid dynamics governing low-Reynolds-number flow along such surfaces remains complex and difficult to model, in particular due to the complicated boundary conditions and the need to consider both solid-fluid and fluid-fluid interactions. As a result, analytical solutions are only available for a limited number of (idealised) scenarios. This thesis is therefore generally devoted to the analytical modelling of such flow regimes, with the objective of providing accurate and accessible solutions that can support surface design and optimisation. Mathematically, one-dimensional velocity field expressions u=(0,0,w) governed by either the Poisson or Laplace equation and subject to mixed-type no-slip/shear boundary conditions are derived to describe pressure-driven or shear-driven laminar Stokes flow. Using powerful complex analysis techniques, in particular conformal mappings and prime functions, analytical models are obtained that capture the essential alternating features of such flow past SHS and LIS. From these, further expressions are derived and presented for key quantities such as volume flux and the effective slip length, which are fundamental to the evaluation of slip performance and overall efficiency. In general, the novel derivation methods based on special complex analysis tools, and the solutions they produce, are of general relevance to a wide range of mixed-type boundary value problems involving Poisson's or Laplace's equation, which arise in numerous other applications that exceed the scope of the presented thesis, such as planar elasticity problems and electrostatics.