Analogy between Vortex Stretching by Drag-reducing Additives and Vortex Stretching by Fine Suspensions

Abstract

THE reduction of drag in water and other solvents by the addition of traces of certain long-chain molecules is a well known effect, which Gadd¹ explains by a vortex stretching mechanism. He discusses the transient diffusion of a two-dimensional vortex in a Newtonian liquid for which is given by the speed coefficient and r₀ the effective eddy radius Initial values are indicated by the subscript 1. Gadd considers that the contaminant molecules make the fluid viscoelastic; this means that the tangential stress is given by an effective viscosity It is therefore easy to see that the eddies can be expected to decay more rapidly because of the larger tangential stress. A faster vortex decay through diffusion causes a smaller energy dissipation, however, because it is—according to Von Karman²—proportional to (∂u_i/∂u_j)². In other words, a faster vortex diffusion of the single vortices—the turbulent elements of the flow—is equivalent to a reduction of the velocity gradient in the vortex, which is responsible for the energy dissipation. Gadd¹ has tested this model qualitatively by showing the decrease of turbulent mixing in a jet of water and “polyox”.