On the regularity and uniqueness of conically self-similar free-vortex solutions to the Navier-Stokes equations

Abstract

In this paper the real analyticity of all conically self-similar free-vortex solutions to the Navier-Stokes equations is proven. Furthermore, it is mathematically established that such solutions are uniquely determined by the values of three derivatives on the symmetry axis, and hence a numerical method, invented and successfully used by Shtern & Hussain (1993,1996, is justified mathematically. In addition, it is proven that these results imply that for any conically self-similar free-vortex solution to the Navier-Stokes equations there exists a second order nonswirling correction term. For this term it is also shown that the second order contribution to the total axial flow force vanishes in the cases of the entire space and a half-space, but that it need not vanish for general conical domains. In doing so an old claim by Burggraf & Foster (1977) is established mathematically, however not for Long's problem but for Shtern & Hussain's (1996) extension of this problem to the full Navier-Stokes equations and the entire space. Mathematics Subject Classification (2000). 34A12, 34B15, 76D25, 76U05.

Quantities which define conically self-similar free-vortex solutions to the Navier–Stokes equations uniquely

“…The major technical tool required to prove theorem 1 is the following result proved in Stein (2001) Theorem 3. (Real analyticity) For any solution to (2.2a), (2.2c) and (4.1) with…”

“…In spite of their relative simplicity many fundamental properties of the conically self-similar solutions have remained poorly understood. Recently, however, results on the existence (Stein 2000) and regularity (Stein 2001) of these solutions have been established, and, in this paper, our primary aim is to find sets of physically relevant parameters, which determine a conically self-similar free-vortex solution uniquely. In doing so, we will also see that for the parameters used by Yih et al (1982) the uniqueness/non-uniqueness properties are rather intriguing.…”

“…To the author's knowledge only one uniqueness result has been proved for conically self-similar free-vortex solutions, and it is a result in Stein (2001) which tells us that not only is such a solution real analytic outside the origin, but it is also uniquely determined by the radial velocity, the polar derivative of the circulation and the radial friction at the symmetry axis. This result is required to justify mathematically the numerical method used in Shtern & Hussain (1993, 1996, but it is of rather limited physical interest, since the quantities involved are not directly related to any of the forces driving the flow.…”