1. Drag Reduction by Polymers in Wall-bounded Turbulence Itamar Procaccia The Weizmann Institute of Science Work with: V.S. L’vov, A. Pomyalov and V. Tiberkevich (Weizmann Institute of Science) R. Benzi, E. De Angelis, C. Casciola (Universita di Roma, I, II

3. Von-Karman’s “logarithmic law of the wall” where For one observes a viscous sub layer

4. In the presence of long chain polymers the mean velocity profile changes dramatically. For sufficiently large concentration of polymers For smaller concentration of polymers there are non-universal crossovers. The relative increase of the mean velocity (for a fixed p’) due to the existence of the new law of the wall IS the phenomenon of drag reduction

5. Derivation of von-Karman’s log-law of the wall The momentum balance equation In the viscous sub-layer Outside the viscous layer W(y)=const, but we need to know more to find the velocity profile.

6. The energy balance equation The energy is created by the large scale motions at a rate of W(y) S(y) It is cascaded down the scales by a flux of energy, and is finally dissipated at a rate Experimentally, outside the viscous sub-layer

7. The visco-elastic case The effect of the polymer enters in the form of a conformation tensor

8. The derivation of the MDR (for large concentration) For the derivation of the MDR we consider the limit of large Deborah number

9. Consequence

11. The prediction is that the effective viscosity is linear in the distance from the wall! Is this blue-sky dreaming, or is a linear viscosity profile Sufficient for drag reduction?

13. The ab-initio calculation of the MDR