Scaling Relations in Elastic Turbulence

We report the scaling relations between the exponents of the power-law decays of kinetic and elastic energies, pressure, as well as torque fluctuations in elastic turbulence (ET). The relations are derived by estimating that the divergent part of the elastic stress is much larger than its vortical part, and its contribution into the full elastic stress is dominant in the range of the power spectrum amplitudes observed experimentally in ET. The estimate is in line with polymer stretching by flow: the polymers are stretched mostly by the divergent part associated with a strain rate, whereas a rotational, or vortical, flow plays a minor role in the polymer stretching. The scaling relations agree well with the exponent values obtained experimentally and numerically in the ET regime of a viscoelastic fluid in different flow geometries.