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

Fractional elements describe complex viscoelestic fluids,which stress is proportional to fractional derivtive of strain,namely σ=Gλβdβedtβ,(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.