The presence of polymers in a viscous fluid confers elasticity to it, resulting in unexpected flow phenomena that appear in technologies both old and new, from conventional polymer processing to flows in microfluidic devices, and flow regimes both turbulent and laminar. For instance, turbulence in viscous liquid flows causes additional energy losses because of increased friction relative to laminar flow. Minute quantities of polymer molecules can dramatically reduce the turbulence and result in friction drag reduction, and consequently energy savings, by upto 60 %. The effect of polymers on flows of viscous liquid filaments and sheets (curtains) can be equally dramatic. Increased elasticity due to polymers can force a slightly sagged liquid filament fixed at its ends to defy gravity and rise against its own weight. Viscoelastic filaments are ubiquitous in polymer processing and electrospinning, and have recently been exploited to form biphasic ``Janus'' nanoparticles [K. Roh, D. Martin & J. Lahann, Nature Materials, 4, 759, (2005)]. A clear understanding of the underlying physics of these phenomena is essential for the development of novel technologies at all length scales.

To organize the phenomenology of drag reduction, we have developed a simple eddy-based model that predicts the mean velocity profile in wall-bounded parallel flows, and elucidates the role of the Trouton ratio as a dimensionless number controlling the extent of drag reduction. Furthermore, we have delved into the physics of self-sustenance of large scale coherent structures that appear in parallel shear flows and are the most significant contributors to turbulent drag. A great deal about turbulent coherent structures can be understood by focusing on the so-called exact coherent structures that appear as pre-cursors to full turbulence at Reynolds number much smaller than required for sustaining turbulence. We present low-dimensional models for the sustenance of exact coherent structures in shear flows of viscoelastic liquids aimed at helping interpret experiments and direct numerical simulations of turbulent drag reduction by polymers. These models are developed by systematically investigating the effect of incremental amounts of elasticity on the self-sustaining process maintaining exact coherent states in shear flows. The recently proposed self-sustaining process for shear flows [F. Waleffe, Phys. Fluids, 9, 83 (1997)] consists of streamwise rolls leading to redistribution of the mean shear into spanwise streaks. A Kelvin-Helmholtz instability of the spanwise streaky flow then results in the regeneration of the streamwise rolls via nonlinear interactions. Our low-dimensional models enable the identification of the part of the cycle that is interrupted or enhanced by the presence of elasticity. Additionally, we explore the effect of fluid rheology on the flow kinematics, particularly the role played by the first and second normal stress differences, as well as shear- thinning.

In the course of my research, I have also discovered the following curious phenomenon, which falls in the general class of instability of complex fluids. When a viscoelastic fluid blob is stretched out into a thin horizontal filament, it sags gradually under its own weight, forming a catenary-like structure that evolves dynamically. If the ends are brought together rapidly after stretching, the sagging filament tends to straighten by hoisting itself. These two effects have characteristic signatures of the elastic nature of the fluid which set it apart from the behavior of a purely viscous filament analyzed previously [J. Teichman & L. Mahadevan, J. Fluid Mech. 478, 71 (2003)]. Starting from the bulk equations for the motion of a viscoelastic fluid, we use perturbation theory to derive a simplified equation for the dynamics of a viscoelastic filament and analyze these equations in some simple settings to explain our observations.