In this thesis, mathematical systems of coupled differential equations arising in fluid dynamics are studied; motivated by the insight they provide in understanding and prompting experimental results as well as the opportunity they present to develop tools to investigate their rich dynamical system.
To cover a wide range of physical arrangements and parabolic PDEs, three problems are posed; connected via the types of equations which govern their motion and the techniques used to study them.

The multiplicity of the equations is first generated by the interaction between a fluid and a flexible substrate, where the evolution of the fluid's free-boundaries are coupled through the fluid.
The resonant interaction is shown to cause linear instabilities at a range of Reynolds numbers, from the Stokes flow limit to very high values where a Chebyshev-Tau numerical approach was required; providing a foundation for future nonlinear studies.

Similarly, the evolution of the two free-boundaries in a three-layer channel flow, applicable in microfluidics, form a system of PDEs, where novel, asymmetric instabilities were found in the inertialess regime.
Numerical simulations demonstrated complex, nonlinear behaviour; supporting the transition of the parabolic PDEs' advection component from hyperbolic to elliptic.

Modern modelling techniques of falling films (namely WIBL) consist of multiple evolution equations for the properties of the fluid, and are applied here to study the stability and dynamics of rivulets supporting travelling lenses, as observed in previous experiments.
A quasi-static linear stability analysis about the rivulets provides the growth rate, spacing and small-amplitude surface profiles of the subsequent linear instability.
Further numerical simulations extended these into the nonlinear regime where three-dimensional travelling waves, qualitatively comparable to snapshots of those seen in experiment, were found for the Benney equation and similar form, large-time dewetting waves and time-periodicity for the higher-order models.

Supplementary code is provided in accompanying repositories.