©Freund Publishing House Ltd. International Journal of Nonlinear Sciences and Numerical Simulation 4, 313-314, 2003

Variational Principle for Nano Thin Film Lubrication

Ji-Huan He College of Science, Donghua University, 1882 Yan 'an Xilu Road, Shanghai 200051, China, Email: jhhe@dhu. edu. cn

Abstract: A variational principle for nano thin film lubrication is established by the semi-inverse method, the stationary conditions of the obtained functional satisfy all field equations.

1. Introduction

Recent years see a rapid growth of interest in micro- and nano-scale film flows. Kucaba-Pietal and Migoun[l] examined squeeze film behavior, Ji-Huan He[2] established a variational model for micropolar fluids in lubrication journal bearing, and Huang Ping[3] proposed a physical-mathematical model for nano thin film lubrication. For detailed information on micropolar fluids, we refer to Lukaszewicz's monograph[4]. In this brief note, we will apply the semi-inverse method[5] to establish a variational model for the discussed problem.

2. Basic Equations

The basic equations for the discussed problem, by taking into account the rotation terms, read [3]:

V·Μ = 0, (1)

^ ( 2 μ + χ ) ν 2 ΰ + χ ν χ ώ - ν ρ = 0, (2)

νν2ώ + χνχΰ-2χώ = 0, (3)

where ü is the velocity vector, ώ microrotation vector, χ,ν characterize the properties of nano thin film flows.

2. Variational Model

According to the semi-inverse method[5], we begin with the following trial-functional:

J(w,<y, p) =

jjj{i(2/i + χ)ΰ • V2w + χΰ·νχώ-ΰ·νρ + F^iV

(4)

where F is an unknown function of ώ and its derivatives. There exist many alternative approaches to the construction of the trial-functional, details can be found in Ref.[5], The advantage of the above trial-functional lies on the fact that the stationary conditions with respect to ü and ρ result in two of the field equations, Eq.(l) and Eq.(2).

Calculating variation of ( 4) with respect to ώ , we have the following trial-Euler equation:

AF - χν xü + = 0, (5) δω

where SF / dw is called variational derivative with respect to w, defined by

öF _ dF d ^ dF ^ d (dF ) 8 ( dF ) <5W dw dx dwx dy dwy dz dwz

dw

Eq.(5) should be equivalent to the field equation (3), accordingly we set

SF 7 — = xS7xu = -\*7zcö + 2xco, (6) δω

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from which the unknown F can be identified as

(7) 1 2

F = ~—νώ-Vώ + χώώ.

We finally obtain the following generalized variational principle:

J(u,Ö),p) =

= |}}|i(2// + χ)ΰ·ν2ΰ + χΰ·ν*ώ-ΰ· Vp^dV

+ j j j j - i νώ • V2<ä + χώ · ö> JdV . (8)

For thin film lubrications, the field equations (1)~(3) can be approximately reduce to the forms[3]:

, 1 .d2u 9ω dp ... 2 fa1 dz dx

. 1 Λ 5 2 ν dcox dp . .... 2 ö z

2 dz dy

dp dz

= 0, (11)

d2<ox

Ί ζ 2

dv

d2ax

-2χωχ-χ — = 0, (12) dz

du ν γ-~2χω +χ— = 0, (13)

dz2 9z

ö2Ö>2 f £ öz2

- 2 ^ = 0 . (14)

Here we write ü = (u,v,w), ώ-(ωΧ,ωγ,ωζ).

The variational principle for the system (9)~( 14) can also obtained by the same way as illustrated above, which reads

J(u,V,W,0)x,C0y,6)z,p) =

< Τ Γ > 2 + Φ 2 dz dz

ill(Ζ"

ί ί ΐ ΐ '

dco

dz y dco

-XV dp

dV

dp dp, —— + U — + V— + w—)dV dz dx dy dz

dco, dC0ν -> ÖÖ)V ? öftj -5z

2 , „ 2

dz öz dV

- ίίίΐ(ω2+ω2 + ω2)dV

(15)

From the obtained functionals(4) and (15), by suitable approximation, Reynolds-type equation for the nano thin film lubrication can be obtained, we will discussed in later.

References

[1] Anna Kucaba-Pietal, Nikolai P. Migoun, Effects of non-zero values of microrotation vector on the walls on squeeze film behavior of micropolar fluid, International J. of Nonlinear Sciences and Numerical Simulation, 2(2), 2001: 127-138

[2] Ji-Huan He, A Variational Model for Micropolar Fluids in Lubrication Journal Bearing , International J. of Nonlinear Sciences and Numerical Simulation, 1(2), 2000:139-142

[3] Huang Ping, Physical-mathematical model and numerical analysis of nano thin film lubrication, Tribology, 23(1), 2003: 60-64(in Chinese)

[4] Grzegorz Lukaszewicz, Micropolar Fluids: Theory and Applications, Birkhauser, Boston, 1999

[5] Ji-Huan He, Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int. J. Turbo & Jet-Engines, 14(1), 1997: 23-28

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