On the Turbulent Dynamics of Polymer Solutions

We consider properties of dilute polymer solutions which are known to depend strongly on polymer elongation. The probability density function (PDF) of polymer end-to-end extensions $R$ in turbulent flows is examined. We demonstrate that if the value of the Lyapunov exponent $\lambda$ is smaller than the inverse molecular relaxation time $1/\tau$ then the PDF has a strong peak at the equilibrium size $R_0$ and a power tail at $R\gg R_0$. In the drag reduction regime $\lambda$ is greater than $1/\tau$ and the majority of molecules have linear size $R_{\rm back}$ which is determined by the back reaction of polymers on the flow. The size $R_{\rm back}$ is much larger than $R_0$ and is smaller than the maximal length of the molecules.