From a raindrop rolling off a lotus leaf to a wave flowing over a sandy beach, flows over rough surfaces are a common sight in nature. In microscopic flows over rough surfaces confined by boundaries, either in thin films or in microchannels, surface textures may significantly influence the flow physics, especially as their size becomes comparable to the scale of the flow domain. Flow interaction with textured walls creates recirculation and transverse flows at the microscale, inducing deflection of the velocity at the wall relative to the free stream. Numerous macroscale effects arise from these microscale flow deflections, such as drag reduction or liquid-repellency. Unlike turbulent flows over rough surfaces in unbounded domains, which are well-studied, the effect of textures on laminar flows of viscous liquids under confinement is less well-understood, even for relatively common processes such as in microfluidics or coating. This thesis presents investigations of how textures modify viscous flows under confinement, and we study how the flow is modified depending on the shape and characteristic scale of the surface textures.

We begin in a system without flow by examining how capillary forces deform the shape of a textured solid surface in a quiescent outer fluid. A microfabrication technique is developed to produce textures with varied periodicity in extremely compliant gels. Capillary stress rounds and flattens the textures, and we find that elastic deformations increase for textures that are close enough to interact via their elastocapillary menisci. Next, we proceed to introduce a flow in an investigation of a microfluidic mixer, where mixing is promoted by a textured microchannel wall having either transverse grooves or micropillars of varied shapes. Via numerical simulations we demonstrate that mixing is amplified by texture anisotropy, but has a nonmonotonic dependence on texture periodicity, resulting in optimal mixing for intermediate periodicities. A homogenization framework is used to model the micromixer via a slip boundary condition at a flat plane, called the equivalent surface, located at the tip of the microtextures. The homogenized model forms the basis of an efficient adjoint-based design of microtextures for optimal mixing.

After analyzing a flow confined by solid walls, confinement by a liquid-air interface is studied in two viscous coating flows, dip coating and spin coating on textured surfaces, which we investigate experimentally and using the homogenized model. Though thinning in these two systems is driven by different forces, either capillary suction or centrifugal pressure, in both cases increasing the texture periodicity leads to amplified thinning, until the complete depletion of the free film atop the textures for large coating speeds. Following the studies of coating flows, the final study presented concerns the dynamics and resonance behavior of a drop bouncing on a vibrating wettable solid at varied vibration frequency. Two distinct bouncing states emerge, with a transition determined by variations in the drop's surface deformation. To conclude the thesis, we elucidate general principles determining how texture periodicity, shape, and size affect viscous flow over textures, which lays the foundation for further studies of flows over complex, natural rough surfaces.

Key words: coating flow, thin films, roughness, asymptotic homogenization, lubrication theory