Vorticity stretching

The authors postulate that a 1-vortex consists of viscous sublayer material and that it formed from a sheet of such material which rolls up at the edges into rods. This model is close to the wall combined with the model of a viscous tornado. Using such a A-vortex model GYR SCHMID (1984) show that the onset of the drag reducing effect can be explained by events just able to stretch the molecules. The local rheology in these events is changed by the stretched molecules and it is therefore of interest in which way the internal flow in these events is altered. Of main interest is the interaction of the vorticity stretching and the diffusion under these new material conditions. 

Cherdhirankorn T et al. Dynamics of vorticity stretching and breakup of isolated viscoelastic droplets in an immiscible viscoelastic matrix. Rheol Acta 2004 43(3) 246-256. 

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

For the two-dimensional problem the body force must be purely in the two-dimensional plane. Therefore Vxf must be purely orthogonal to the plane for example, in the r-6 problem, it must point in the z plane. It can be shown that the vortex-stretching term vanishes under these conditions. As a result the vorticity-transport equation is a relatively straightforward scalar parabolic partial differential equation,... 

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