Homogenization of the Equations Governing the Flow Between a Slider and a Rough Spinning Disk

The main challenge in the design of hard-disk drives is to increase the density of data storage. Traditional disk-drive designs involve a read/write head positioned on a slider bearing that flies over a smooth spinning magnetic disk. The demand for larger data storage has made it necessary to reduce the minimum gap height for various design features of the air-bearing surface to be in the range of tens of nanometers to just a few nanometers in present-day drives. A further reduction in the minimum gap height has become physically impractical so that in future designs other approaches may be used. One approach is to use “patterned” magnetic disks in which the disk is rough on the nanometer scale. A potential difficulty in the design process for such patterned disks is that the behavior of the flow in the gap may no longer be described with sufficient accuracy by the solution of the compressible Reynolds equation in lubrication theory. Since many flow solutions are needed in the inner loop of the design process, discarding the numerical solution of the Reynolds equation in favor of the numerical solution of the full Navier-Stokes equations implies a huge increase in computational cost. Thus, a significant problem of interest to Hitachi is whether modified versions of the Reynolds equation may be derived for patterned disks using some asymptotic homogenization approach, and whose solutions provide sufficient accuracy for the design process. Figure 1 shows a schematic of a patterned disk. The slider bearing is located at the end of a suspension structure that acts in a manner of a tone-arm support to control the radial position of the slider as it flies over the spinning disk. The patterned disk has regions of concentric grooves and discrete islands, the former representative of Discrete Track Media (DTM) while the later representative of Bit Patterned Media (BPM). For purposes of mathematical analysis we consider the two cases separately, and view DTM as longitudinal roughness running in the direction of the relative motion between the slider bearing and the disk, whereas we view BPM as transverse roughness running perpendicular to the relative motion, i.e. “speed bumps.” Since the length scale of the roughness is on the order of nanometers which is much smaller than the millimeter length scale of the slider bearing, a homogenization of the equations governing the flow is considered for both cases. In order to carry out a mathematical analysis of both cases, we need to consider the various parameters of the problem. These parameters are collected in Table 1. Here U is a representative value for the speed of the slider relative to the disk, h0 is the minimum gap height, L is the length scale for the slider, λ is a length scale for the roughness along the surface of the disk (see Figures 2 and 5), and ρa , Pa and μ are ambient values for the density, pressure and dynamic viscosity of the flow, respectively. From these, we can construct several relevant dimensionless parameters, namely,