SISSA-International School for Advanced Studies, Sector of Functional Analysis and Applications, Via Bonomea 265, Trieste, Italy.
J Biol Dyn. 2012;6 Suppl 1:88-103. doi: 10.1080/17513758.2011.611260. Epub 2011 Aug 31.
We prove an existence, uniqueness, and regularity result for the motion of a self-propelled micro-swimmer in a particulate viscous medium, modelled as a Brinkman fluid. A suitable functional setting is introduced to solve the Brinkman system for the velocity field and the pressure of the fluid by variational techniques. The equations of motion are written by imposing a self-propulsion constraint, thus allowing the viscous forces and torques to be the only ones acting on the swimmer. From an infinite-dimensional control on the shape of the swimmer, a system of six ordinary differential equations for the spatial position and the orientation of the swimmer is obtained. This is dealt with standard techniques for ordinary differential equations, once the coefficients are proved to be measurable and bounded. The main result turns out to extend an analogous result previously obtained for the Stokes system.
我们证明了在颗粒粘性介质中，由 Brinkman 流体模拟的自主推进微游泳者的运动的存在性、唯一性和正则性结果。通过变分技术，引入了一个合适的函数设置来求解 Brinkman 系统的速度场和压力。通过施加自推进约束来写出运动方程，从而使粘性力和扭矩成为唯一作用于游泳者的力。从游泳者形状的无限维控制中，得到了一个关于游泳者空间位置和方向的六个常微分方程系统。这是通过对常微分方程的标准技术来处理的，一旦证明了系数是可测和有界的。主要结果扩展了先前在 Stokes 系统中获得的类似结果。
