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

The non-inertia motive coordinate system is introduced so that the analysis of the flow ofa Newtonian fluid in annulus with the inner cylinder executing a planetary motion is greatlysimplified;the stream-function is introduced due to the continuity of the flow so that thetotal number of variables of the motion equations is reduced. According to thetransformation from the Descartes coordinate system to the bipolar coordinate system, theflow region is changed from the eccentric annular region to the rectangle region so thatthe boundary condition is greatly simplified. Therefore, the motion equations of the floware established in the non-inertia motive bipolar coordinate system.Based on Hanks' theory of the flow stability and the analysis of the motion equation,the mathematical expression of the stability parameter H′ is established in this paper.And by the using of the mathematical expression, the laminar-turbulent transition for theflow can be discriminated.By the using of the finite difference method, the distribution of the stabilityparameter of water in the wide and thin clearances in annulus is numerically calculatedand the relation between the stability parameter and the rotation and revolution of theinner cylinder, the eccentric distance and the pressure gradient is analyzed. Taking theflow of a Newtonian fluid in annulus with the inner cylinder executing a planetary motionas an example, the distribution of the stability parameter which are calculated by usingthe numerical method mentioned in this paper are compared with those which arecalculated by the using of the analytical method. The two results are consistent with eachother very well. It shows that the numerical method of the expression of stabilityparameter of the flow established in the paper is correct.