基于分形导数对非牛顿流体层流的数值研究

非牛顿流体具有复杂的流变特性,揭示该流变特性可以更加合理地指导非牛顿流体在工农业生产中的应用.经典的非牛顿流体本构模型往往形式复杂,仅能应用于某些特定的情况.分数阶导数模型具有参数少和形式简单的特点,己成功地应用于描述非牛顿流体的运动.Hausdorff分形导数作为一个备选的建模方法,相比分数阶导数具有更简单的形式以及更高的计算效率.本文基于Hausdorff分形导数改进现有牛顿黏性模型,提出分形黏壶模型.通过研究分形黏壶在常应变率下表观黏度的变化情况,以及在加、卸载条件下的蠕变及恢复特性,发现分形黏壶模型适合于描述具有黏弹性的非牛顿流体(本文称之为分形流体).结合连续性方程及运动微分方程,推导出分形流体在平行板间层流的基本方程.按是否拖动上板和是否存在水平的压力梯度分为3种工况,分别用数值方法计算这3种工况下流速在板间的分布及其随时间变化的情况.通过分析不同工况下的流速分布,发现水平的压力梯度会改变流速随时间变化的形状,且会推迟流速到达稳定的时间.在水平压力梯度不存在的情况下,不同阶数的分形流体具有相同的流速分布或是演变过程.另外,在水平压力梯度存在的情况下,上板速度不影响不同阶数分形流体间稳定速度的差值.

NUMERICAL STUDY FOR LAMINAR FLOW OF NON-NEWTONIAN FLUID BASED ON FRACTAL DERIVATIVE

Non-Newtonian fluid has complex rheological characteristics. It is very helpful to reveal these characteris-tics for the applications of non-Newtonian fluid in industry and agriculture. The classical rheological models of non-Newtonian fluid usually have sophisticated forms and the limitations of specific materials or rheological situations. Frac-tional models have been successfully applied to describe the motion of non-Newtonian fluid due to their simplicity and few parameters. As an alternative method, the Hausdorff fractal derivative possesses simpler form and higher computa-tional efficiency compared with the fractional derivative. This paper proposes a fractal dashpot model that improves the current Newton's Law by using the Hausdorff fractal derivative. By investigating the apparent viscosity, the creep and recovery characteristics of the fractal dashpot, it shows that the proposed fractal dashpot model is suitable to describe the non-Newtonian fluid with viscoelasticity (the so-called fractal fluid). Combined the fractal dashpot model with the continuity and motion equations, the basic equation for the fractal fluid for the laminar flow between two parallel plates is derived. Moreover, the velocity distributions between two plates are numerically calculated in three cases, which can be obtained through whether there is horizontal pressure gradient or the initial velocity of upper plate. It is found that the horizontal pressure gradient can change the shape of velocity over time and delay the arrival of stable velocity. The fractal fluid with different orders has the same velocity distribution and evolution when the horizontal pressure gradient doesn't exist. In addition, the velocity of upper plate doesn't influence the difference of stable velocity between different orders of fractal fluid when the horizontal pressure gradient exists.