Blow-up suppression of the Patlak-Keller-Segel-Navier-Stokes system via Taylor-Couette flow
Shikun Cui, Lili Wang, Wendong Wang
TL;DR
This work demonstrates that a sufficiently strong Taylor–Couette flow in a 2D PKS-NS system on an annular domain suppresses blow-up by inducing enhanced dissipation. The authors develop a detailed mode-decomposed framework, establishing elliptic and energy estimates for chemoattractant concentration, stream function, and density/vorticity, and they implement a bootstrap argument around an energy functional that remains uniformly controlled for large A. A crucial outcome is that global boundedness of solutions is achieved without smallness assumptions on the initial data, highlighting a robust blow-up suppression mechanism via non-planar shear flow. The findings have potential implications for design and safety in biomedical devices employing TC-like flows to improve hemocompatibility and reduce thrombogenic risk.
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 $U(r,θ)=A\big(r+\frac{1}{r} \big)(-\sinθ, \cosθ)^{T}$ with $(r,θ)\in[1,R]\times\mathbb{S}^{1}$, and prove that the solutions are globally bounded without any smallness restriction on the initial cell mass or velocity when $A$ is large.