A meaning to these mathematics

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In our most recent paper, we reported the thickness of the boundary layers that occur in laminar flows between a rotating and a stationary disc. While investigating this, we stumbled upon something remarkable…

To simulate the velocity profiles in the rotor-stator gap we adopted two different techniques for solving the equations governing the flow; a numerical solution of the differential equations, and an analytical series approximation of the form:

\frac{v_z}{\Omega h}=-2\sum\limits_{i=0}^{\infty}a_i\left(\frac{z}{h}\right)^i

\frac{v_\theta}{\Omega r} = \sum\limits_{i=0}^{\infty}b_i\left(\frac{z}{h}\right)^i

v_r = -\frac{r}{2h}\frac{\textrm{d}v_z}{\textrm{d}z}

where vz, vθ, and vr are the axial, azimuthal, and radial components of velocity, respectively, and Ω, h, r, and z are the rotor speed, rotor-stator gap, and the coordinates in the radial and axial direction. As is detailed in the paper, a recurrence relation is obtained for the coefficients ai, and bi, following the no-slip conditions on both the rotor and the stator, so that the flow field can be described with these analytic functions within a certain radius of convergence, Rc:

R_c^{-1} = \lims_{i\rightarrow\infty}^i\sqrt{||a_i||}

The as yet unexplained, but remarkable feature of this radius of convergence is that it coincides with the boundary layer thickness within 2%-4%. Mathematically this means that at least one of the derivatives of v with respect to z at the radius of convergence approaches (positive or negative) infinity. Obviously, this cannot happen within the region between the rotor-and stator for physically realistic flows and in the appendix to our paper we have shown that, in fact, the singularity lies outside of the rotor-stator domain, with the same distance from it as the boundary layer thickness.

Why does such an abstract mathematical concept as the radius of convergence so closely follow the theoretical prediction on the boundary layer thickness, which has a concrete physical meaning? Is there a more fundamental link between the two concepts? What does it mean that a mathematical singularity outside of the fluid domain influences the flow field inside of it?

The contents of this blog are based on the following publication:

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