Discussion: “Slip-Flow in Microchannels of Non-Circular Cross Sections”

Abstract

In a recent paper, Tamayol and Hooman [1] studied slip flow in regular polygonal, rectangular, and rhombic ducts by eigenfunction expansion and boundary collocation. The insight provided by the authors is appreciated. The boundary collocation approach [2,3] is general, applicable to many irregular shapes although the procedures are relatively involved. Tamayol and Hooman [1] presented the valuable comparison between their data and the model proposed by Duan and Muzychka [4]. It is invaluable to validate the proposed general models with extensive numerical, analytical, and experimental data over a wide range of geometries. In Ref. [1], the authors state that “from Table 4 it can be seen that the model of Duan and Muzychka is not that accurate for rhombus cross sections and the maximum relative difference between the approximate model and the present analytical solution is higher than 20%.” In this note, we would like to point out that the model may not be completely reflected in Ref. [1] and the results presented by Tamayol and Hooman [1] are supplemented. Tamayol and Hooman [1] appropriately employed the model; however, they used an earlier definition of the aspect ratio e1⁄4 b/a in the calculation, where b/a is the aspect ratio of the diagonals as shown in Fig. 6 of Ref. [1]. Therefore, in the paper of Tamayol and Hooman, a relative difference of 20% was reported. For the sake of brevity, the slip flow model [4] is examined directly to the point. Details will not be repeated here because it is quite space consuming and can be found in Ref. [4]. The dimensionless mean wall shear stress reduction depends on the geometry of the crosssection and rarefaction. The slip model is defined as the ratio of the dimensionless mean wall shear stress with slip to the dimensionless mean wall shear stress with no slip, which gives the following relationship [4,5]: PoL PoL ð Þns 1⁄4 f ReL f ReL ð Þns 1⁄4 1 1 þ a e ð ÞKn (1)