幂律流体在内管做行星运动的环空中流动的稳定性参数H’
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摘要
所谓内管做行星运动,即构成环空的内管在围绕自身轴线旋转(自转)的同时,也围绕外管的轴线旋转(公转)。内外管之间的环空中非Newton流体由于内管的公转和自转以及外加压力梯度引起的流动即为非Newton流体在内管做行星运动的环空中的流动。研究非Newton流体在内管做行星运动的环空中流动的稳定性参数具有学术理论和工程实际意义。
由非惯性运动坐标系的引入,使得流体在内管做行星运动的环空中流动的分析变得大大简化;由流动的连续性引进流函数,使流体动力学基本方程未知量的个数减少一个;由Descartes直角坐标系向双极坐标系的变换,使流体动力学基本方程的求解域由偏心环域变为矩形域,使对边界条件的处理大大简化,从而在非惯性运动双极坐标系中建立了幂律流体在内管做行星运动的环空中流动的动力学基本方程组。从运动双极坐标系下幂律流体在内管做行星运动的环空中流动的运动方程出发,引入Hanks流动稳定性理论,建立了判断幂律流体在内管做行星运动的环空中由层流向紊流转变的稳定性参数H′的数学表达式。
利用上述幂律流体在内管做行星运动的环空中流动的基本方程和稳定性参数H′的数学表达式,采用有限差分方法,对可视为幂律流体的CMC水溶液在内管做行星运动的环空中流动时在环空宽间隙和窄间隙处稳定性参数H′的分布进行了数值计算,并分析了稳定性参数H′与环空内管公转和自转速度、环空偏心距及压力梯度之间的关系。
以Newton流体在内管做行星运动的环空中的流动为例,把利用本文数值计算方法算得的稳定性参数H′的分布与利用解析解算得的稳定性参数H′的分布做了对比,结果吻合较好。说明了本文建立的幂律流体在内管做行星运动的环空中流动的稳定性参数H′的数学表达式及数值计算方法的正确性。
以CMC水溶液在内管自转和公转的环空中流动的实验数据为例,用本文建立的幂律流体在内管做行星运动的环空中流动的稳定性参数H′的数学表达式和数值计算方法算得了CMC水溶液稳定性参数H′的临界最大值H′_(max,c)。结果表明,本文建立的幂律流体在内管做行星运动的环空中流动的稳定性参数H′的临界最大值H′_(max,c),可以取为404。
The inner cylinder executing a planetary motion means the inner cylinder not onlyrotates around its own axis (the Rotation), but revolves around the axis of the outercylinder as well (the Revolution). The flow of non-Newtonian fluid in the annulus betweenthe inner cylinder and the outer cylinder, caused by the rotation and the revolution of theinner cylinder with the addition of the pressure gradient, is namely the flow ofnon-Newtonian fluid in annulus with the inner cylinder executing a planetary motion.Research on the stability parameter of the flow of the non-Newtonian fluid in annulus withthe inner cylinder executing a planetary motion is of certain academic value and practicalengineering significance.
The introduction of the non-inertia motive coordinate system makes the analysis ofthe flow of a power-law fluid in annulus with the inner cylinder executing a planetarymotion greatly simplified;the introduction of the stream function owing to the flowcontinuity makes the total number of the variables of the basic equations of fluid dynamicsless by one;with the transformation from the Descartes coordinate system to the bipolarcoordinate system, the changing of solving domain from the eccentric annular one to therectangle one makes the boundary conditions greatly simplified;therefore, the basicequations of fluid dynamics of the flow of a power-law fluid in annulus with the innercylinder executing a planetary motion are established under the non-inertia motive bipolarcoordinate system. Based on the equation of motion of the flow of a power-law fluid inannulus with the inner cylinder executing a planetary motion under the motive bipolarcoordinate system, the Hanks?ˉs theory of the flow stability is introduced, and themathematical expression of the stability parameter H?? is established, by which thetransition from the laminar flow to the turbulent flow of a power-law fluid in annulus withthe inner cylinder executing a planetary motion can be discriminated.
Based on the basic equations and the mathematical expression of the stabilityparameter H?? of the flow of a power-law fluid in annulus with the inner cylinder executinga planetary motion mentioned above, using the finite difference method, the distribution of
the stability parameter H?? of the carboxymethyl cellulose (CMC) aqueous solution,which can be regarded as a power-law fluid, in the wide clearance and the thin clearance ofannulus is numerically calculated and the relations among the stability parameter H?? andthe revolution and rotation velocity of the inner cylinder, the eccentric distance and thepressure gradient are analyzed.Taking the flow of a Newtonian fluid in annulus with the inner cylinder executing aplanetary motion for an example, the distribution of the stability parameter H?? which isobtained by using the method of numerical calculation mentioned in this paper is comparedwith that which is obtained by using the method of analytical solution, and both of the twodistributions coincide with each other very well. It shows that the mathematical expressionof the stability parameter H?? of the flow of a power-law fluid in annulus with the innercylinder executing a planetary motion established in this paper, as well as the method ofnumerical calculation, is correct.Taking the experiment data of the flow of the CMC aqueous solution in annulus withthe inner cylinder executing the motion of rotation and revolution for examples, using themathematical expression of stability parameter H?? of the flow of a power-law fluid inannulus with the inner cylinder executing a planetary motion and the method of numericalcalculation, which is established in this paper, the critical maximum value Hm?? ax,c of thestability parameter H?? of the CMC aqueous solution is obtained. The results show thatthe critical maximum value Hm?? ax,c of the stability parameter H?? of the flow of apower-law fluid in annulus with the inner cylinder executing a planetary motion, which isestablished in this paper, can be regarded as 404.
由非惯性运动坐标系的引入,使得流体在内管做行星运动的环空中流动的分析变得大大简化;由流动的连续性引进流函数,使流体动力学基本方程未知量的个数减少一个;由Descartes直角坐标系向双极坐标系的变换,使流体动力学基本方程的求解域由偏心环域变为矩形域,使对边界条件的处理大大简化,从而在非惯性运动双极坐标系中建立了幂律流体在内管做行星运动的环空中流动的动力学基本方程组。从运动双极坐标系下幂律流体在内管做行星运动的环空中流动的运动方程出发,引入Hanks流动稳定性理论,建立了判断幂律流体在内管做行星运动的环空中由层流向紊流转变的稳定性参数H′的数学表达式。
利用上述幂律流体在内管做行星运动的环空中流动的基本方程和稳定性参数H′的数学表达式,采用有限差分方法,对可视为幂律流体的CMC水溶液在内管做行星运动的环空中流动时在环空宽间隙和窄间隙处稳定性参数H′的分布进行了数值计算,并分析了稳定性参数H′与环空内管公转和自转速度、环空偏心距及压力梯度之间的关系。
以Newton流体在内管做行星运动的环空中的流动为例,把利用本文数值计算方法算得的稳定性参数H′的分布与利用解析解算得的稳定性参数H′的分布做了对比,结果吻合较好。说明了本文建立的幂律流体在内管做行星运动的环空中流动的稳定性参数H′的数学表达式及数值计算方法的正确性。
以CMC水溶液在内管自转和公转的环空中流动的实验数据为例,用本文建立的幂律流体在内管做行星运动的环空中流动的稳定性参数H′的数学表达式和数值计算方法算得了CMC水溶液稳定性参数H′的临界最大值H′_(max,c)。结果表明,本文建立的幂律流体在内管做行星运动的环空中流动的稳定性参数H′的临界最大值H′_(max,c),可以取为404。
The inner cylinder executing a planetary motion means the inner cylinder not onlyrotates around its own axis (the Rotation), but revolves around the axis of the outercylinder as well (the Revolution). The flow of non-Newtonian fluid in the annulus betweenthe inner cylinder and the outer cylinder, caused by the rotation and the revolution of theinner cylinder with the addition of the pressure gradient, is namely the flow ofnon-Newtonian fluid in annulus with the inner cylinder executing a planetary motion.Research on the stability parameter of the flow of the non-Newtonian fluid in annulus withthe inner cylinder executing a planetary motion is of certain academic value and practicalengineering significance.
The introduction of the non-inertia motive coordinate system makes the analysis ofthe flow of a power-law fluid in annulus with the inner cylinder executing a planetarymotion greatly simplified;the introduction of the stream function owing to the flowcontinuity makes the total number of the variables of the basic equations of fluid dynamicsless by one;with the transformation from the Descartes coordinate system to the bipolarcoordinate system, the changing of solving domain from the eccentric annular one to therectangle one makes the boundary conditions greatly simplified;therefore, the basicequations of fluid dynamics of the flow of a power-law fluid in annulus with the innercylinder executing a planetary motion are established under the non-inertia motive bipolarcoordinate system. Based on the equation of motion of the flow of a power-law fluid inannulus with the inner cylinder executing a planetary motion under the motive bipolarcoordinate system, the Hanks?ˉs theory of the flow stability is introduced, and themathematical expression of the stability parameter H?? is established, by which thetransition from the laminar flow to the turbulent flow of a power-law fluid in annulus withthe inner cylinder executing a planetary motion can be discriminated.
Based on the basic equations and the mathematical expression of the stabilityparameter H?? of the flow of a power-law fluid in annulus with the inner cylinder executinga planetary motion mentioned above, using the finite difference method, the distribution of
the stability parameter H?? of the carboxymethyl cellulose (CMC) aqueous solution,which can be regarded as a power-law fluid, in the wide clearance and the thin clearance ofannulus is numerically calculated and the relations among the stability parameter H?? andthe revolution and rotation velocity of the inner cylinder, the eccentric distance and thepressure gradient are analyzed.Taking the flow of a Newtonian fluid in annulus with the inner cylinder executing aplanetary motion for an example, the distribution of the stability parameter H?? which isobtained by using the method of numerical calculation mentioned in this paper is comparedwith that which is obtained by using the method of analytical solution, and both of the twodistributions coincide with each other very well. It shows that the mathematical expressionof the stability parameter H?? of the flow of a power-law fluid in annulus with the innercylinder executing a planetary motion established in this paper, as well as the method ofnumerical calculation, is correct.Taking the experiment data of the flow of the CMC aqueous solution in annulus withthe inner cylinder executing the motion of rotation and revolution for examples, using themathematical expression of stability parameter H?? of the flow of a power-law fluid inannulus with the inner cylinder executing a planetary motion and the method of numericalcalculation, which is established in this paper, the critical maximum value Hm?? ax,c of thestability parameter H?? of the CMC aqueous solution is obtained. The results show thatthe critical maximum value Hm?? ax,c of the stability parameter H?? of the flow of apower-law fluid in annulus with the inner cylinder executing a planetary motion, which isestablished in this paper, can be regarded as 404.
引文
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