DEAN VORTICES WITH WALL FLUX IN A CARVED CHANNEL MEMBRANE SYSTEM .2. THE VELOCITY-FIELD

The velocity and pressure fields and the effect of wall flux on these fields in a spiral channel are presented. As fluid flows inward throug h a spiral channel with constant gap and permeable walls, the streamwi se flux decreases while the curvature increases. Thus, by balancing th e stabilizing effect of wall suction with the destabilizing effect of increasing curvature, established vortices can be maintained along the spiral channel. This approach is used to prescribe spiral geometries with different wall fluxes. Using a weakly nonlinear stability analysi s, the influence of wall flux: on the characteristics of Dean vortices is obtained. The critical Dean number is reduced when suction is thro ugh the inner wall only, is slightly reduced when suction is equal thr ough both walls, and is increased when suction is through the outer wa ll only. The magnitude of change is proportional to a ratio of small n umbers that measures the importance of the effect of curvature. In mem brane filtration applications the wall flux is typically 2 to 5 orders of magnitude less than the streamwise flow. If the radius of curvatur e of the channel is of the order of 100 times the channel gap, the eff ect on the critical Dean number is within 2% of the no-wall flux case. If the radius of curvature is sufficiently large, however, it is poss ible to observe effects on the critical Dean number that approach O(1) in magnitude for certain parameter ranges.