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中的微型机械，人们进行了大量的研究和开发。小直径管内的非定常流体流动是层流的，粘性的影响变得非常大，与重力和惯性相比，粘性的影响是不容忽视的。由于球体和管子之间的小间隙，流体流动的行为将是动态不稳定的，并且会出现许多流型。本文对密度为ρ_S、直径为D_S的球体在重力加速度G平行于管轴的恒定外力作用下，在充入密度为ρ、运动粘度为υ的流体的小直径圆管内的运动和流场进行了数值模拟和实验研究。数值计算采用标记法和单元法，流场假设为轴对称。该体系有三个无量纲参数，即伽利莱数Ga=D_S、3G/υ^2、管径比d=D/D_S和密度差比α=(ρ_S-ρ)/ρ。通过数值分析得出以下主要结论：(1)在d较小时，球体的无量纲末端速度ν_&lt；bterr&gt；=V_&lt；Bter&gt；υ/(αD_S^2G)不随α和Ga的乘积的增大而变化。(2)小直径圆管中运动球体的流线为椭圆形。球体后流线长度随d和α的增大而增大，实验定性地证实了理论预测的球体周围非定常层流流动和球体运动的特征。

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.