Axial flow in an eroding Euler hairpin in ℝ³: does it prevent blow-up?

In our earlier paper [Childress, S. and Gilbert, A.D., Eroding dipoles and vorticity growth for Euler flows in (Formula presented.) : the hairpin geometry as a model for finite-time blowup. Fluid Dyn. Res. 2018, 50, 011418.], we derived equations of motion for a vortex dipole in the shape of a hairpin, and subject to erosion of vorticity, that is the shedding of vorticity at the rear stagnation point of the dipole and consequent loss of circulation. We applied these to the question of Euler blow-up of vorticity in (Formula presented.). In this paper, we shall calculate the axial flow within the vortex tubes of the hairpin and evaluate the resulting vorticity structure of their cores. The model is unusual in that it is not evolved from simple initial conditions. Rather, the hairpin structure is constructed at a time prior to possible blowup. It consists of a “nose”, where blow-up would occur, from which there extend two symmetric, quasi-two-dimensional “tails” of infinite length and infinitely large spatial scale. The quasi-self-similarity of the structure determines blow-up at the point of joining of the tails. During this growth, the dipole maintains a quasi-two-dimensional geometry. The analysis is believed to be the first study of blow-up incorporating both the deformation of the cores of the constituent vortex tubes and the axial flow within the tubes. The analysis raises problems which we will not be able to resolve fully here. Our results suggest that axial flow, coupled with erosion, may provide a mechanism preventing blow-up in finite time. The essential difficulty is that axial flow changes the local dipole structure and hence the dipole propagation speed. This expels the possibility of complete self-similarity. Possible ways to deal with this obstacle are discussed.