The present work seeks to address three different problems that have a multiscale nature, we apply different techniques from multiscale analysis to treat these problems.

We introduce the field of multiscale analysis and motivate the need for techniques to bridge between scales, presenting the history of some common methods, and an overview of the current state of the field.

The remainder of the work deals with the treatment of these problems, one motivated by reaction rate theory, and two from multiphase flow. These superficially have little relation with each other, but the approaches taken share similarities and the results are the same - an average picture of the microscopic description informs the macroscale.

In Chapter 2 we address an asymmetric potential with a microscale, showing that the interaction between this microscale and the noise causes a first-order phase transition. This induces a metastable state which we observe and characterise: showing that the stability of this state depends on the strength of the tilt, and that the phase transition is inherently different to the symmetric case.

In Chapter 3 we investigate the nucleation and coarsening process of a two-phase flow in a corrugated channel using a Cahn--Hilliard Navier--Stokes model. We show that several flow morphologies can be present depending on the channel geometry and the initial random condition. We rationalise this with a static energy model, predicting the preferential formation of one morphology over another and the existence of a first-order phase-transition from smooth slug flow to discontinuous motion when the channel is strongly corrugated.

In Chapter 4 we address a model for interfacial flows in porous geometries, formulating an finite-element model for the equations. Within this framework we solve two equations in the microscale to obtain effective coefficients decoupling the two scales from each other. Finite-difference simulations of the macroscopic flow recover results from literature, supporting robustness of the method.