The motivation to study elastic structures such as filaments and shells stemmed from its application in the construction of tall buildings, bridges etc. Interest in this field has rekindled in the past decades due to growing interest in understanding biological materials and because of possible applications in nanoscience and medicine. This also poses new challenges as the biological materials show both solid and fluid-like behavior, and in addition, they are active. In this thesis, we study the mechanical properties of shells and filament and their interaction with fluid. The thesis is divided into two themes. First, how to model nano-vesicles and how are the mechanical properties of a spherical shell is affected if they are thermal and active? Second, can non-linear interaction between fluid and filament generate turbulence and hence mixing in the Stokes flow?

To model nano-vesicles, we develop an open-source software package – MeMC. The MeMC models nano-vesicles as an elastic objects. It interprets the force-distance data generated by indentation of biological nano-vesicles by atomic force microscopes and uses Monte-Carlo simulations to compute elastic coefficients of a nano-vesicle. Further, we use this code and break the detailed balance in Monte-Carlo simulation – thereby driving the shell active and out of thermal equilibrium – to study the effect of activity on mechanical properties of elastic shells, in particular, buckling. Such a shell typically has either higher (active) or lower (quiescent) fluctuations compared to one in thermal equilibrium depending on how the detailed balance is broken. We show that for the same set of elastic parameters, a shell that is not buckled in thermal
equilibrium can be buckled if turned active. Similarly, a shell that is buckled in thermal equilibrium can unbuckle if turned quiescent. Based on this result, we suggest that it is possible to experimentally design microscopic elastic shells whose buckling can be optically controlled.

In the next part of the thesis, we visit the problem of mixing in Stokes flow using elastic filament. We study the interaction of the filament with Stokes flow. As it is known, the flow of Newtonian fluid at low Reynolds number is, in general, regular, and time-reversible due to absence of nonlinear effects. For example, if the fluid is sheared by its boundary motion that is subsequently reversed, then all the fluid elements return to their initial positions. Consequently, mixing in microchannels happens solely due to molecular diffusion and is very slow. Here, we show, numerically, that the introduction of a single, freely floating, flexible filament in a time-periodic linear shear flow can break time reversibility and give rise to chaos due
to elastic nonlinearities, if the bending rigidity of the filament is within a carefully chosen range. Within this range, not only the shape of the filament is spatiotemporally chaotic, but also the flow is an efficient mixer. We model the filament using the bead-rod model.

We consider two different models for the viscous forces: (a) they are modelled by the Rotne-Prager tensor. This incorporates the hydrodynamic interaction between every pair of beads. (b) we consider only the diagonal term of the Rotne-Prager tensor which makes the viscous forces local. In both cases, we find the same qualitative result:
the shape of a stiff filament is time-invariant — either straight or buckled for large enough bending rigidity; it undergoes a period-n bifurcation (n = 2,3, 4, etc) as the filament is made softer; becomes spatiotemporally chaotic for even softer filaments. For case (a) but not for (b) we find that the chaos is suppressed if bending rigidity is decreased further. For (b), in the chaotic phase, we show that the iterative map for the angle, which the end–to–end vector of the filament makes with the tangent its one end, has period three solutions; hence it is chaotic.