<p>From the sedimentation of fabric fibres in rivers, the dispersion of graphene particles in specialised inks, or the locomotion of flagella or slender cells in biological fluids, predicting the motion of slender bodies in viscous flows is vital for fields such as colloidal science and microbiology. In this talk, we present three distinct examples of how slender bodies can be modelled within viscous flows. First, we revisit the classic case of a settling rod, showing how environmental patchiness—specifically, heterogeneous viscosity fields—influences the rod's settling behaviour. To determine the leading effects of a viscosity gradient, we derive a new version of Resistive Force Theory (RFT) tailored for slender bodies in a spatial viscosity gradient field. This new RFT enables us to predict the orientation and drift of the settling rod under varying viscosity conditions. In our second example, we use the RFT to theoretically examine how viscosity gradients impact the locomotion of an actively waving filament. Crucially, we find that the viscosity gradient introduces an additional time-averaged angular rotation. Over time, this angular rotation enables actively waving filaments to control their orientation and direction, providing a physical mechanism for effective navigation through viscous media. Finally, in our third example, we show how slender body theory can be adapted to model two-dimensional ultra-thin sheets (2D SBT). We show how 2D SBT can offer valuable insights into the complex interactions of sheet-like 2D materials, such as graphene, in viscous shear flows.</p>
