Nonlinear Interaction between Motion of a Sphere and an Unsteady Fluid Flow in a Small-diameter Tube

近年来，为了设计和制造血管微型机械，人们进行了大量的研究和开发。非定常流体在小直径管内的流动是层流的，与重力和惯性相比，粘度的影响变得非常大，不容忽视。由于球体与管之间的间隙很小，流体的流动行为将是动态不稳定的，并且会出现多种流型。然而，由于难以在三维空间中进行分析，对这种现象的研究仍然很少。本文通过数值和实验研究了密度ρ_s、直径D_s的小球在平行于管轴的重力加速度G的恒定作用下，密度ρ ρ、运动粘度υ的流体在管内的运动和流场。采用标记单元法进行数值计算，假设流场为轴对称流场。该系统中有三个无量纲参数，即Galilei No。Ga=D_s^3G/υ^2，管径比d= d /D_s，密度差比α=(ρ_s - ρ)/ρ。在Ga不变的情况下，管径比d和密度差比α发生了变化。通过数值分析，主要得出以下结论：(1)在d值较小的情况下，球的无因次终端速度ν_<bterr>=V_<Bter>υ/(α d_s ^2G)不随α与Ga乘积的增大而变化。(2)在小直径管内，围绕运动球体的流线呈椭圆形。球后流线长度随d和αGa的增大而增大。理论预测的非定常层流绕球流动特性和球的运动特性在实验中得到了定性证实。

Recently, there are so many researches and developments in order to design and make the micro machine in blood vessels. The unsteady fluid flow in a small diameter tube is laminar and the effect of viscosity becomes so large that it can no more be disregarded as compared to gravity and inertia. The behavior of fluid flow will be dynamically unstable and many flow patterns will occur due to the small clearance between a sphere and the tube. However, there is still so few researches associated with such phenomena because it is difficult to analyze in a three dimensional space.This research investigates numerically and experimentally the motion of a sphere of density ρ_s and diameter D_s, and flow field in small-diameter tube filled with fluid of density ρ and kinematic viscosity υ, under the external constant force of gravity acceleration G parallel to the pipe axis. The marker and cell method is used to calculate numerically, and the flow field is assumed to be axisymmetric. There are three dimensionless parameters in this system, i.e., Galilei No.Ga=D_s^3G/υ^2, the tube-diameter ratio d=D/D_s, and the density difference ratio α=(ρ_s - ρ)/ρ. The tube-diameter ratio d and the density difference ratio α are changed in this research, though Ga is constant.As a main result, the following conclusions are obtained from the numerical analysis ;(1)The dimensionless terminal velocity of a sphere ν_<bterr>=V_<Bter>υ/(αD_s^2G) does not change with increasing the product of α and Ga for the case that d is small value.(2)The streamline around a moving sphere in a small-diameter tube is elliptical. The length of streamline behind a sphere increases with increasing d and αGa.The theoretically predicted characteristics of unsteady laminar fluid flow around the sphere and the motion of the sphere are confirmed qualitatively in experiments.