A new approach to studying active suspensions is presented. They exhibit a specific behavior pattern, sometimes referred to as active turbulence. Starting from first principles, we establish a description for an active suspension, consisting of a Newtonian fluid and active Janus particles. The fluid phase is described by Navier–Stokes equations and the particles by Newton–Euler equations. A level set approach is used to separate the two phases, well-known from the representation of sharp interfaces in various numerical schemes. By introducing the multi-point probability density function (PDF)-approach known from hydrodynamic turbulence, we obtain a hierarchical ordered infinite set of linear statistical equations. However, the equations for the K-point PDF depend on the K + 1 and K + 2-point PDF, exposing the closure problem of active turbulence. As all statistical moments can be formed from the PDF, the latter set of equations already includes every statistical model for an active suspensions. To illustrate this, we derive the Eulerian spatial averaging theory from the hierarchy of multi-point PDF-equations.

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主动悬架的概率论

提出了一种研究主动悬架的新方法。它们表现出特定的行为模式，有时称为主动湍流。从第一性原理出发，我们建立了一个由牛顿流体和活性 Janus 粒子组成的活性悬浮液的描述。流体相由 Navier-Stokes 方程描述，粒子由 Newton-Euler 方程描述。水平集方法用于分离两个阶段，这在各种数值方案中的锐界面表示中是众所周知的。通过引入从流体动力学湍流已知的多点概率密度函数 (PDF) 方法，我们获得了线性统计方程的分层有序无限集。然而，K点 PDF的方程取决于K  + 1 和K  +  2 点 PDF，暴露了主动湍流的闭合问题。由于所有统计矩都可以从 PDF 中形成，后一组方程已经包含了主动悬架的每个统计模型。为了说明这一点，我们从多点 PDF 方程的层次结构中推导出欧拉空间平均理论。