Turbulent Dynamics of Polymer Solutions

We study properties of dilute polymer solutions. The probability density function (PDF) of polymer end-to-end extensions $R$ in turbulent flows is examined. We show 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 $R\gg R_0$. This confirms and extends the results of J. L. Lumley [Symp. Math. 9, 315 (1972)]. There is no essential influence of polymers on the flow in this regime. At $\lambda \tau >1\mathrm{}$ the majority of molecules is stretched to the linear size $R_{\mathrm{op}}\gg R_0$, which can be much smaller than the maximal length of the molecules due to their back reaction.