An asymmetrical periodic vortical structures and appearance of the self- induced pressure gradient in the modified Taylor flow

An incompressible liquid flow in the gap between two coaxial cylinders, such that the inner rotating (wavy) cylinder has a periodically varying radius along the axial direction while the outer stationary cylinder has a constant radius, is studied experimentally and theoretically. The basic attention is focused on the symmetry-breaking phenomenon of the vortex flow arising from the rotation of the inner wavy cylinder. It is found that the symmetry-breaking phenomenon of the vortical flow structures in this geometry is accompanied by occurrence of a self-induced axial pressure gradient. A generalized formulation of the problem of periodic vortical flow prevailing in the gap between two coaxial wavy cylinders having large axial length is presented. The comparison between the computed and the experimental results is presented and the underlying phenomena are discussed. INTRODUCTION An incompressible liquid flow between two coaxial cylinders, arising from the rotation of the inner cylinder, is well known in hydrodynamics as Taylor-Couette flow (see [1]). The modified Taylor flow is realized in geometry, where one or both rigid surfaces have axisymmetric wavy shape due to periodically varying radius along the axis of rotation. The existence of such flows, caused by the wavy surface of the rotor was investigated in the works [2] (1999) and [3] (2001). The modified Taylor flow, with both inner and outer cylinders having a wavy shape, has been analyzed numerically in [4] (2002). In the present work the experimental and numerical investigations of the case, where the fixed external cylinder has a constant radius, while the radius of the inner rotating cylinder varies along the axial direction following a cosine law, are presented. The attention is focused on the symmetry-breaking phenomenon of the vortical flow, which is accompanied by occurrence of a self-induced axial pressure gradient. The symmetrybreaking phenomenon of the flow in the region having geometrical symmetry has been scrutinized in a number of studies. The existence of periodic flow in the non-periodic pressure field with the given gradient is well known too. But the possibility of occurrence of a self-induced pressure gradient in the periodic flow field of an incompressible liquid in a closed region having large axial lengths has never been studied. In the present work this phenomenon is investigated theoretically and a generalized formulation of the problem of periodic vortical flow calculation in the region with the large axial lengths is presented. The computed results are compared with experimental observations and measurements. 1. Experimental apparatus, the methods and means of experimental investigation. We use the same experimental apparatus as in the work [2]. It consists of the transparent outer cylinder with the radius R=64.11 mm and the length H=263 mm. Inside the outer cylinder there is a coaxial rotating cylinder representing one of 6 bodies of rotation. The geometrical parameters of the variety of inner cylinders (rotor) used in this study are given in Table-1. Experiments are performed using one cylinder having a constant radius (CR1) and five cylinders (WR2,...,WR6) with periodically varying radius along the axial direction, as described by the formula: R1(z)=a0+a1cos(2πz/λ) (1) Table-1 (Made dimensionless by R) rotor a0 a1 λ CR1 0.8002 0 WR2 0.8002 0.1221 1.1192 WR3 0.8002 0.1221 0.5597 WR4 0.8002 0.0610 1.1192 WR5 0.8002 0.0610 0.5597 WR6 0.8002 0.0306 1.1192 Hereinafter the internal radius (R) of the cylindrical container is used as characteristic dimension.