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A Cascade Theory of Drag Reduction

We describe homogeneous, isotropic, three-dimensional turbulence in a dilute solution of neutral, flexible chains. Energy flows by the usual cascade down to a scale r* such that the shear rates U(r*)/r* (U being the velocity) become equal to the relaxation rate of one coil. At small scales r, the molecules follow affinely the deformation of a local volume element. At a certain smaller scale r** the elastic stresses in the coils become comparable to the Reynolds stresses. The polymer truncates the cascade when r** becomes larger than the usual Kolmogorov limit rk. This defines a critical concentration, which depends on both polymer and flow parameters.