Stability Parameter H' Of The Flow Of A Power-law Fluid In Annulus With The Inner Cylinder Executing A Planetary Motion

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 ofthe 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.