<p>The present dissertation focuses on two themes: (i) elastic instability of flow and (ii) elastic instability of microscopic filaments.</p> <p><br></p> <p>(i) The presence of macromolecules often leads to the viscoelastic nature of industrial and biological fluids. The flow of viscoelastic fluids in porous media is important in many industrial, geophysical, and biological applications such as enhanced oil recovery, groundwater remediation, biofilm formation, and drug delivery. The stretching of polymeric chains as the viscoelastic fluid passes through the microstructure of the porous media induces large elastic stresses, which leads to viscoelastic instability at the Weissenberg number greater than a critical value, where the Weissenberg number quantifies the ratio of elastic to viscous forces. Viscoelastic instability can lead to a time-dependent chaotic flow even at negligible inertia, which is sometimes also known as elastic turbulence due to its analogous features to traditional inertial turbulence. In the present thesis, we investigate the pore-scale viscoelastic instabilities and the flow states induced by the instabilities in symmetric and asymmetric geometries. We found that the topology of the polymeric stress field regulates the formation of different flow states during viscoelastic instabilities. Viscoelastic instability-induced flow states exhibit hysteresis due to the requirement of a finite time for the transformation of polymeric stress topology. Further, we study viscoelastic flows through ordered and disordered porous geometries and explore the effect of viscoelastic instability on sample-scale transport properties. Viscoelastic instability enhances transverse transport in ordered porous media and longitudinal transport in disordered porous media. We also derive a relationship between the polymeric stress field and the Lagrangian stretching field. The Lagrangian stretching field helps to predict the feature of flow states and transport in complex flows. The experimental measurement of the polymeric stress field is extremely challenging. The framework established here can be used to obtain the topology of the polymeric stress field directly from the easily measured velocity field.  </p> <p><br></p> <p><br></p> <p>(ii) The interaction between flow and elastic filaments plays an important role in sperm and bacterial motility and cell division. The sperm cells of many organisms use long elastic flagellum to propel themselves and also face complex flows and boundaries during their search for egg cells. Strong flows have the potential to mechanically inhibit flagellar motility through elastohydrodynamic interactions. We explore the effects of an extensional flow on the buckling dynamics of sperm flagella through detailed numerical simulations and microfluidic experiments. Compressional fluid forces lead to rich buckling dynamics of the sperm flagellum beyond a critical dimensionless sperm number, which represents the ratio of viscous force to elastic force. Shear flows navigate the sperm cells in complex geometries and flows. We have also studied the effect of flow strength and flagellar elastic deformation on the sperm trajectory in simple shear and Poiseuille flows.</p>