Simulation of Gas Film Lubrication in Magnetic Disc
Storage: DSMC Method Coupled with a Continuum
Solution
Denisikhin S.V., Memnonov V.P., and Zhuravleva S.E.
St.Petrsburg State University, St.Petersburg, Russia
Abstract. In the paper we present some results from Monte Carlo simulations of two-dimensional unsteady problem of
gas flow in an extremely narrow channel with an inclined upper wall and moving lower one. This is a model of gas film
lubrication which occurs in modern magnetic disk storage, now being under development. Due attention is given to nonuniform inner and outer structure of the boundary incoming and outgoing flows. This is accomplished by simultaneous
continuum finite element solution for the outer flow around the magnetic head. The coupling of DSMC simulation with
this continuum solution is realized as step by step approximation. It is found that the molecular density before and after
the channel differs markedly from the atmospheric one because of the slowing down of the flow by magnetic head. Thus
previously much used atmospheric pressure boundary conditions there are quite far from reality. Space and time
distributions of different parameters of the flow are given.
INTRODUCTION
Flows, which have for some of its linear dimension R the order of several nanometers, are now often found in
high technological processes and thus are of growing importance. As the mean free path (mfp) λ for NTP in air is
equal to 62 nm so the Knudsen number Kn=λ/R may be close to unity. They are frequently in transitional regime as
it is the case of the gap flow in the modern Winchester-type disk drives [1], [2]. Thus the current trend toward
nanoscale technology in the slider air bearing causes the previous design calculations based on continuum Reynolds
equation or on the simulation of the whole slider region with surrounding external flow by solving the Navier-Stokes
equations [3] to be changed by the solution of the Boltzmann equation at least in some part of the computational
region. In this occasion the direct simulation Monte Carlo (DSMC) method introduced by Bird [4] is particularly
suitable and was employed in [1], [2], [5] for a stationary flow problem by calculating some important parameters of
a slider air bearing.
In this method molecular collisions and their free motion are directly simulated, but comprehensive description
of microscopic dynamics, especially for molecular collisions, is approximately replaced by a stochastic process.
Under fulfillment of certain restrictions on the mean numbers of molecules in the computational cells and the value
of the time step this process can be approximated by a Poisson process [6]. But the same restrictions demand
changing Bird’s selection procedure for calculating of molecular collisions to a selection without replacement back
already collided in this time step molecules. Thus removing an odious phenomenon in the Bird’s algorithm namely
potential repeated collision of the pair just collided molecules which is possible because of too strong violation of
the microscopic dynamics produced by that selection with replacement.
Yet it should be mentioned that for performance evaluation of a Winchester-type disk drive under the transient
conditions one needs to know unsteady characteristics of the flow. In this paper we present some results from
Monte Carlo simulations of two-dimensional unsteady problem of gas flow in an extremely narrow channel with an
inclined upper wall and moving lower one. This is a model of gas film lubrication which occurs in modern magnetic
disk storage, now being under development. Due attention is given to non-uniform inner and outer structure of the
boundary incoming and outgoing flows. This is accomplished by simultaneous continuum finite element solution
for the outer flow around the magnetic head.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
STATEMENT OF THE PROBLEM
It is considered a two-dimensional simulation problem for flow around the magnetic head and inside the channel
which is formed by the latter with the hard disc, see
Fig.1. In this channel and in the additional regions
of 20λ long before and after the head numerical
solution was obtained with the help of DSMC
method which was coupled along the boundaries at
0C and CD on the left and GF, FE on the right with
a continuous finite element solution of NavierStokes equations. The latter produces boundary
conditions for DSMC simulations by forming
fluxes, which are calculated from mean velocities
and its derivatives on these boundaries. Molecular
interactions are taken to be the hard sphere one and
their interaction with the wall was diffuse reflection. The square cells in DSMS computational region had the linear
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size of approximately λ/3 and time step ∆t was less than one third of the molecular mean free time (mft) ∆t =4·10 .
The height of the inlet and outlet in the channel was equal to 2λ and λ accordingly. The initial rotation of the lower
wall which models the magnetic media of the disk drive for small time interval can be approximated as being set
moving instantly with linear speed Uw from the rest in a direction parallel to itself and thereafter maintained in
uniform motion. The speed Uw was taken to be 75m/s. The flow itself is formed because the air molecules after
diffuse reflection from this wall are carried along this motion.
The coupling procedure was realized as a step by step approximation. First in the whole computational region
Navier-Stokes equations were solved by finite element method with diminishing cell size in these additional regions
and the channel. As boundary conditions for this solution far from the head at x=Xb and x=-Xb asymptotic solution
of the Raleigh problem obtained by Cercignani and et al. [7] was used.
1
1

y 

U as = U w ⋅ 1 − erf (q ) − 1.24 ⋅ g 2 ⋅ exp − q 2 + 0.292 ⋅ g 2 ⋅ exp 7.86 ⋅  ,
λ 


(
)
(1)
where erf(q) is the probability integral [4], q = 0.77 y ⋅ (λ ⋅ t ⋅ VT ) 2 , g = λ (πtVT ) 2 , t is the time in
seconds.
Though employment of continuum equations near inlet, outlet and inside of the channel is inadequate one can
expect that on the boundaries OCDEFG which are 20λ apart from them the influence of these small bad parts will be
insignificant. So, at the next step within this part DSMC method was employed with incoming molecular fluxes
calculated from the stored mean velocities and their derivatives at these boundaries obtained at the previous step.
Details of this procedure are outlined in the next section.
−1
−1
SIMULATIONS OF THE INCOMING FLUXES
The linear size of the cells in finite element solution was four times and its time step 25 times larger than in
DSMC simulation, so that an interpolation of the stored macroscopic parameters was used. Molecular properties in a
Navier-Stokes solution are represented through the Chapmen-Enscog velocity distribution function f(u,v,w) which
for our case of negligible temperature changes and with decart velocity components u=(u’-uav)/VT, v=(v’-vav)/VT,
w=w’/VT being nondimensionalized relative to VT can be represented in the form:
(
~+
f (u , v, w) = f 0 (u, v, w) 1 − α 1u 2 − α 2 v 2 + α 3 w 2 − αuv
where
[(
f 0 (u, v, w) = π −3 2 exp − u 2 + v 2 + w 2
)]
)
And coefficients αi are given by the formulas:
∂v
1  ∂u
α 1 = c0 ⋅  2 av − av
3  ∂x
∂y

;

∂u
1  ∂v
α 2 = c0 ⋅  2 av − av
3  ∂y
∂x

;

∂v
1  ∂u
α 3 = c0 ⋅  av + av
3  ∂x
∂y

;

 ∂u
∂v 
α = c0 ⋅  av + av ;
∂x 
 ∂y
c0 ≡
5 λ π
⋅
4 VT
The value of the constant c0 is only 0.35·10-9s. The largest in our problem combination of derivatives in α is
equal to or less than 108s-1. As the other combinations are much less, so that the distribution f(u,v,w) can be
approximately written as follows
f
+
(u , v, w) = f 0 (u, v, w)(1 − α ⋅ u ⋅ v )
Simulation of the decart velocity component for the incoming molecules from this multidimensional distribution
can be proceeded as follows. As an example let us consider molecules incoming through the boundary CD. So that
potential values for v lie in the interval -∞<v<-vav/VT and for those values of u one has -∞<u<∞. First by introducing
conditional distribution P2(u|v) and by integrating out the independent w velocity component one has
f 2+ (u, v ) =
∞
∫
f
+
P2 (u | v1 ) =
(u, v, w)dw;
−∞
f 2+ (u , v1 )
P1 (v1 )
where v1 is a value of v component already simulated by rejection technique [4] from the distribution
f1 (v ) = (− v − vav / VT )⋅ P1 (v )
with the one dimensional density P1(v) for the incoming molecules being given by the formula
P1 (v ) =
∞
∫ f (u, v )du =
+
2
e −v
−∞
2
π
Now rejecting very rarely realized values v1 greater than 6 and taking into account that the maximum value of α is
only 0.035 one can with accuracy one part to 100 neglect in the below listed formula the small term (αv1/2)2
P2 (u | v1 ) ≅
e
 α 
−  u + v1 
2 

π
2

α 2 2 

⋅ 1 + O
⋅ v1  
4



And finally simulating from the pure exponent term we obtain [4]
u = − ln γ 1 ⋅ cos(2π ⋅ γ 2 ) −
α
v1
2
with γ1 and γ2 being two next random numbers.
SOME NUMERICAL RESULTS
Simulations for our DSMC parallel code were performed on the St. Petersburg university cluster consisting of 40
processors Intel Pentium ΙΙΙ 933 MHz. Parallelization of the code was proceeded through independent realizations
when each processor produced the whole solution and the results being gathered then on some leading one for
averaging and outputting. This scheme allowed us sometimes use an additional cluster located far away but
connected with one of the ours through the Internet channel into a metacomputer with very good efficiency. For
details see [8].
All previous researchers [1], [2], [5] had used the assumption that before the inlet and after outlet of the channel
unperturbed atmospheric pressure could be used as the boundary condition. Thus they entirely neglected the slowing
down of the flow by magnetic head. We are now in a position to check it. For this purpose we simulated particle
-9
density before the channel and represented it for the height y=λ/3 on the Fig.2 by the solid curve for time t1=5·10 s
-8
and by the dashed one for time t2 = 6·10 s. It is clearly seen that the slowing down of the flow is present by 15%
n av
1,05
1
t1
0,95
t12
0,9
1
4
7
10
13
16
FIGURE 2. Increase of the density n av on the distance x
(mfp) before the inlet for different times.
19 x
increase in density nav there. It should be mentioned that at the edges of the inlet and outlet the density changes are
even larger and they are shown in their dependence on the height y at Fig.3 and Fig.4 with the same times and
n av
n av
1,06
t1 0,93
t12
0,88
1,04
0,83
1,08
t1
t12
0,78
1,02
0
5
10
y
15
0
FIGURE 3. The density n av on the height y (mfp)
before the inlet for different times.
5
10
y
15
FIGURE 4. The density n av on the height y (mfp) just
after the outlet for different times.
curves. Streamwise velocity uav at these positions are shown on Fig.5 and Fig.6.
u av
u av
48
78
38
58
t1
t1
28
38
t12
18
t12
18
8
-2
0
5
10
15
-2
y
FIGURE 5. The streamwise velocity u av on the height y (mfp)
just before the inlet for different times.
0
5
10
15
FIGURE 6. The streamwise velocity u av on the height y
(mfp) just after the outlet.
Thus one may conclude that the pressure boundary conditions stated in [5] at least for Uw≥75m/s in reality are
not fulfilled. Both of the pressures are not atmospheric and are quite different near the inlet and outlet.
y
On the Fig.7 it is shown the pressure along the channel for different times with the curves: solid – for t1=5⋅10-9s,
pointed – for t2=3⋅10-8s and dashed – for t3=6⋅10-8s.
p
1,4
1,3
1,2
1,1
1,0
1
6
11
16
21
26
31 x
FIGURE 7. Pressure along the channel for different times.
It is interesting to note that the maximum of the pressure for t2 here shifts much more from the value xm=2a/3,
which follows from classical lubrication solution [1], than the density maximum on the Fig.8. On the Fig.9
streamwise velocity uav is presented for different height values. Near the end of the channel all of them are growing
thus forming the decrease rapidness of the latter maxima.
u av
n av
85
1,4
75
1,3
65
1,2
55
1,1
45
1
35
0
5
10
15
20
25
30
x
-8
FIGURE 8. Density n av along the channel for t=6*10 s.
0
5
10
15
20
And finally on the Fig.10 it is shown the increase of pressure on the magnetic head.
p
1,25
1,20
1,15
1,10
1,05
1,00
1
3
5
25
30
FIGURE 9. Velocity u av along the channel: dashed y=λ /3, solid - y=2λ /3, pointed - y=λ .
7
9
11
FIGURE 10. General pressure on the upper wall, time t in
-9
units of 5*10 s.
t
x
CONCLUSIONS
Based on the coupling of the DSMC method with a continuum solution at the outer regions it is found
considerable deviations from the atmospheric pressure before and after the channel. Thus quality of the pressure
boundary conditions much used in the previous papers on this topic seems to be doubtful. Space and time
distributions of different parameters of the flow are also presented.
ACKNOWLEDGMENTS
This work was partially supported by the RFBR grant N01-01-00315 and by the grant "Integration" B 0008.
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