Blow-up suppression of the Patlak-Keller-Segel-Navier-Stokes system via Taylor-Couette flow

TL;DR

This paper demonstrates that strong Taylor-Couette flow can prevent blow-up in the Patlak-Keller-Segel-Navier-Stokes system, ensuring global boundedness of solutions regardless of initial conditions.

Contribution

It provides a mathematical proof that sufficiently strong Taylor-Couette flow stabilizes the system and prevents blow-up, a novel insight into flow-induced stability mechanisms.

Findings

Strong flow prevents blow-up in the system.

Solutions are globally bounded with large flow strength.

No smallness restriction on initial conditions is needed.

Abstract

Motivated by the use of Taylor-Couette flow in extracorporeal circulation devices [K$\ddot{\rm o}$rfer et al., 2003, 26(4): 331-338], where it leads to an accumulation of platelets and plasma proteins in the vortex center and therefore to a decreased probability of contact between platelets and material surfaces and its protein adsorption per square unit is significantly lower than laminar flow. Increased platelet adhesion or protein adsorption on the device surface can induce platelet aggregation or thrombosis, which is analogous to the ``blow-up phenomenon" in mathematical modeling. Here we mathematically analyze this stability mechanism and demonstrate that sufficiently strong flow can prevent blow-up from occurring. In details, we investigate the two-dimensional Patlak-Keller-Segel-Navier-Stokes system in an annular domain around a Taylor-Couette flow $U(r,\theta)=A\big(r+\frac{1}{r} \big)(-\sin\theta, \cos\theta)^{T}$ with $(r,\theta)\in[1,R]\times\mathbb{S}^{1}$, and prove that the solutions are globally bounded without any smallness restriction on the initial cell mass or velocity when $A$ is large.