具有分数微分的广义Oldroyd-B流体螺旋流动

Some helical flows of a generalized Oldroyd-B fluid with fractional derivative

针对无限长两同心圆柱间的广义Oldroyd-B流体的螺旋流,圆柱管道绕对称轴方向振荡,在自身所在的平面上沿对称轴方向做加速运动拖动流体在环形管道内做螺旋流动.采用分数微分建立流动模型,应用分数微分的Laplace变换和Hankel变换方法研究非稳态Oldroyd-B流体螺旋流动在周向和轴向的速度场和剪切力.结果表明:在t=0.5时,分数阶导数α的值越小,流体的速度衰减的越慢,β有与α相反的结果;广义二阶流体,广义Maxwell流体,牛顿流体及单圆柱管道内广义Oldroyd-B流体的螺旋流的解是该模型解的特殊形式.

Considering some helical flows of a generalized Oldroyd-B fluid between two infinite concentric cylinders, which is due to the cylinders oscillate around their common axis and accelerating slide in the direction of the same axis with prescribed velocities, this paper obtained the exact solutions of some unsteady helical flows by using Laplace transform coupled with Hankel transform for fractional calculus. The results show that the smaller the values of a, the more slowly the velocity decays with the flow, and the material parameter fl has quite the opposite effect to that of a. The corresponding solutions for generalized second grade fluid, generalized Maxwell fluid, Newtonian fluid and generalized Oldroyd-B fluid through a circular cylinder were obtained as limiting cases of general solutions.