6 Lubrication Transformation and Nanoscale Thin Film Lubrication
6.4 Nano-Gas Film Lubrication
6.4.4 Calculation of Magnetic Head/Disk of Ultra Thin Gas Lubrication [21]
The dimensionless Reynolds equation for steady ultra-thin gas lubrication of the head/disk can be written as
Λx𝜕(PH)
𝜕X +AΛy𝜕(PH)
𝜕Y = 𝜕
𝜕X (
PH3Q𝜕p
𝜕X )
+A2 𝜕
𝜕Y (
PH3Q𝜕p
𝜕Y )
. (6.24)
In order to solve Equation 6.24, the following two problems must be solved first.
k k 6.4.4.1 Large Bearing Number Problem
In the ultra thin gas lubrication, the bearing numberΛis very large, usually more than 1 million.
This will cause the solution to be severely unstable. In order to solve the large bearing number problem in ultra-thin gas bearing lubrication, we analyzed Equation 6.24 as follows:
1. The bearing consists of two shear flow items,ΛxandΛy. Whenh0is small, they are far greater than the other items. Therefore, if the traditional lubrication calculation is still used, the large number will make the iterative process instable.
2. In the incompressible Reynolds equation, the shear flow does not contain the pressure so that the pressure must be solved from the left-hand items of Equation 6.24. However, due to gas compressibility, the right-hand items contain pressurepnow. This provides a new possibility for solving the equation.
Concerning the ultra-thin gas lubrication for the above two points, by using the upwind scheme, Equation 6.24 can be shown as
2Λx(Pi,jHi,j−Pi−1,jHi−1,j)∕ΔXi+2AΛy(Pi,jHi,j−Pi,j−1Hi,j−1)∕ΔYj
=0.5[(QH3)i+1∕2,j(P2i+1,j−P2i,j) − (QH3)i−1∕2,j(P2i,j−P2i−1,j)]∕ΔX2i
+0.5A2[(QH3)i+1∕2,j(P2i,j+1−P2i,j) − (QH3)i−1∕2,j(P2i,j−P2i,j−1)]∕ΔY2j. (6.25) After finishing, we have
P̂i,j=
[2P̂i−1,jΛxHi−1,j∕ΔXi+0.5((QĤ 3)i+1∕2,jP̃2i+1,j+ (QĤ 3)i−1∕2,jP̂2i-1,j)∕ΔX2i
+2P̂i,j−1AΛyHi,j−1∕ΔYj+0.5A2((QĤ 3)i,j+1∕2P̃2i,j+1+ (QĤ 3)i,j−1∕2P̂2i,j-1)∕ΔY2j ]
[2ΛxHi,j∕ΔXi+2AΛyHi,j∕ΔYj+0.5P̃i.j((QĤ 3)i+1∕2,j+ (QĤ 3)i−1∕2,j)∕ΔX2i +0.5A2P̃i,j((QĤ 3)i+1∕2,j+ (QĤ 3)i−1∕2,j)∕ΔY2j
]
(6.26) where the variables with ‘–’ above are the values before iteration, with ‘∧’ above are the values after iteration.
6.4.4.2 Sudden Step Change Problem
In Chapter 3, we introduced a method to deal with such cases, using the formula qi+1∕2,j= 2qi+1,jqi,j
qi+1,j+qi,j, (6.27)
whereq=PHQ3.
The flow coefficient in the y direction has a similar formula to Equation 6.27. It should be noted that if we did not use the above conversion formula, the results obtained may contain errors.
6.4.4.3 Solution of Ultra-Thin Gas Lubrication of Multi-Track Magnetic Heads
In the actual multi-track heads, the main sizes, conditions and structures are given in Table 6.3 and Figure 6.19.
k k Table 6.3 Size and condition parameters.
Parameters Value Unit
Head lengthl 1.235×10−3 m
Head widthb 0.7×10−3 m
Minimum film thicknesshmin 10×10−9 m
Disk linear velocityU 25 m/s
Gas viscosity𝜂 1.8060×10−5 Pa.s
External pressurep0 1.0135×105 Pa
Skew angle𝛼s 12.361 ∘
Rolling angle𝛼r 6.14653×10−6 rad
Pitch angle𝛼p 1.87×10−4 rad
Figure 6.19 Head structures and pressure distributions.
Here, the studied magnetic heads have two steps, 114 nm and 1.77μm with respect to the top surface of the head. Therefore, the film thickness is expressed as
h(x,y) =h0(x,y) +l(xN−x)sin(𝛼p+𝛼0) +b(yM−y)sin𝛼r
+ (x+x0−l)(x−x0)∕(7.8125×106) +hmin, (6.28) where h0(x,y) is the original head height (i.e. original film thickness); items 2 and 3 of the right-hand side of Equation 6.28 are for the tilt; item 4 is a coupled arc with the maximum height of 33 nm;hminis the minimum film thickness,hmin=10 nm;x0and𝛼0are the coordinate and
k k initial inclination angle of the point of the minimum thickness;xNandyMare the coordinates
of the head end.
In addition, the velocity components of the disk are calculated from Ux=Ucos𝛼s
Uy =Usin𝛼s. (6.29)
By using Equations 6.26 and 6.27 to solve the problems of the large bearing number and the step sudden change, and combined with Equations 6.28 and 6.29, we can finally obtain the pressure of Equation 6.24 and then the other lubrication performances. Figure 6.19 gives the pressure distribution of six different kinds of structures of the actual heads. Compared with the approximate analytical solutions, the two solutions are very close to each other. This indicates that the results are reasonable.
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