粘弹性流体的分数元模型及圆管起动流

Fractional Element of Viscoelastic Fluids and Start-up Flow in a Pipe

分数元模型所描述的非牛顿流体属于复杂粘弹性流体,其应力与应变的分数阶时间导数成正比。本文提出一种用弹簧和油壶连接组成的分形网络结构来比拟分数元模型的应力-应变特性,利用Heaviside运算微积,证明了该分形网络结构对应的粘弹性流体为1/2阶导数的分数元。并证明了构成其他分数阶导数分数元模型需要引入弹簧和油壶的多重分形网络结构。本文还导出了分数元模型的圆管起动流的解析解,研究了分数元模型起动过程振荡特征与该模型导数阶β之间的关系;发现在β≠1的情况下,随时间的进程,圆管内分数元模型的运动最终均将趋于静止,只有β=1的情况是一个例外。

Fractional elements describe complex viscoelestic fluids, which stress is proportional to fractional derivtive of strain, namely σ = Gλ^βd^βε/dt^β,（0≤β≤1）. The stress-strain relation of fractional elements was realized physically through a kind of spring-dashpot fractance. Based on Heaviside operational calculus, the stress-strain relation of the fractance is the same as the fractional element with 1/2 order derivative. Furthermore the spring-dashpot fractance with stress-strain relation of general fractional elements was presented. The exact solution of start-up flow of fractional elements in a pipe was derived by using fractional calculus. The oscillation of flow was studied. It is found that in the case of β≠1,with the exception β= 1, the motion of fractional element in the pipe will be rest, if the time is long enough.