Flow of a Rarefied Gas between Parallel and Almost Parallel Plates

Transcription

1 Flow of a Rarefied Gas between Parallel and Almost Parallel Plates Carlo Cercignani, Maria Lampis and Silvia Lorenzani Dipartimento di Matematica, Politecnico di Milano, Milano, Italy 033 Abstract. Rarefied gas flows in ultra-tin film slider bearings are studied in a wide range of Knudsen numbers. Te generalized Reynolds equation first derived by Fukui and Kaneko [], [] as been extended by allowing for bounding surfaces wit different pysical structures, as an issue of relevance for applications. Since te solution of tis equation requires tat te Poiseuille and Couette flow rates between two parallel plates ave to be accurately calculated in advance, we ave used our recent results on Poiseuille flow [3], [4] and new results on Couette flow to evaluate te lubrication caracteristics. FLOW OF A RAREFIED GAS BETWEEN TWO PARALLEL PLATES: THE POISEUILLE-COUETTE PROBLEM Let us consider two plates separated by a distance and a gas flowing parallel to tem, in te x direction, due to a pressure gradient. Te lower boundary (placed at z ) moves to te rigt wit velocity U, wile te upper boundary (placed at z ) is fixed. Bot boundaries are eld at a constant temperature T o. If te pressure gradient is taken to be small as well as te velocity U, it can be assumed tat te velocity distribution of te flow is nearly te same as tat occurring in an equilibrium state. Tis means tat te Boltzmann equation can be linearized about a Maxwellian f 0 by putting [5]: f f 0 () were f x z c is te distribution function for te molecular velocity c expressed in units of RT o (R being te gas constant), z is te coordinate normal to te plates and z c is te small perturbation upon te basic equilibrium state. If one assumes te linearized BGK model for te collision operator [6], te Boltzmann equation reads [7]: were by definition k c Z z z θ π Z z c z π e c z Z z c z dc z Z z c z () p k p x e c x c y c x z c dc x dc y ρ ρ x wit p and ρ being te gas pressure and density, respectively, and θ is te collision time. Consequently, te integral equation for te bulk velocity of te gas can be written as follows: q z From Eq. () we obtain in integral form [3], [8]: π e c z Z z c z dc z (3) 79

2 Report Documentation Page Form Approved OMB No Public reporting burden for te collection of information is estimated to average our per response, including te time for reviewing instructions, searcing existing data sources, gatering and maintaining te data needed, and completing and reviewing te collection of information. Send comments regarding tis burden estimate or any oter aspect of tis collection of information, including suggestions for reducing tis burden, to Wasington Headquarters Services, Directorate for Information Operations and Reports, 5 Jefferson Davis Higway, Suite 04, Arlington VA Respondents sould be aware tat notwitstanding any oter provision of law, no person sall be subject to a penalty for failing to comply wit a collection of information if it does not display a currently valid OMB control number.. REPORT DATE 3 JUL 005. REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE Flow of a Rarefied Gas between Parallel and Almost Parallel Plates 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Dipartimento di Matematica, Politecnico di Milano, Milano, Italy PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 0. SPONSOR/MONITOR S ACRONYM(S). DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited. SPONSOR/MONITOR S REPORT NUMBER(S) 3. SUPPLEMENTARY NOTES See also ADM0079, International Symposium on Rarefied Gas Dynamics (4t) Held in Monopoli (Bari), Italy on 0-6 July ABSTRACT 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT UU a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified 8. NUMBER OF PAGES 6 9a. NAME OF RESPONSIBLE PERSON Standard Form 98 (Rev. 8-98) Prescribed by ANSI Std Z39-8

3 Z z c z exp z sgnc z z sgnc z c z θ Z sgnc z c z z t exp c z θ q t kθ c z θ dt (4) wit te values at te boundary, Z sgnc z c z, depending on te model of boundary conditions cosen. In te following, we will consider te Maxwell boundary conditions and specialize te analysis to walls aving different pysical properties so tat two accommodation coefficients (α ) can be defined. In tis case, te boundary conditions can be written as [3], [8]: Z c z α Z c z (5) Z c z α U α Z c z (6) were U is expressed in units of RT o ; Z c z, Z c z are te distribution functions of te molecules impinging upon te walls and Z c z, Z c z te distribution functions of te molecules reemerging from tem. Once te function at te boundary, Z sgnc z c z, as been evaluated following te analytical procedure reported in [3], [4], te substitution of te integral formula (4) in te definition (3) of q z gives te integral equation for te bulk velocity of te gas: q z kθ ψ p u Uψ c u (7) Eq. (7) sows tat te gas velocity is induced by te superposition of two distinct effects. Te gas moves by an imposed pressure gradient (Poiseuille flow) and by te sear driven flow due to te motion of te bottom surface (Couette flow). Te non-dimensional functions ψ p u and ψ c u, giving te Poiseuille and Couette contributions, respectively, can be written as: ψ p u π dwψ p w α S δ u w α S δ u w α α S δ u w S δ u w T u w (8) α ψ c u T o π δ u α S o 3 δ u α α S o 5 δ u π dwψ c w α S δ u w α S δ u w α α S δ u w S δ u w T u w (9) were T n x is te Abramowitz function defined by T n x S n x is a generalized Abramowitz function defined by 0 t n exp t x t dt S n x δ α and te following non-dimensional variables ave been introduced: 0 t n exp t x t α α exp δ t dt δ θ w t θ u z θ 70

4 l z x γ 0 x=0 x=l U FIGURE. Geometry of a slider bearing. Using Eq. (7), te flow rate (per unit time troug unit tickness) defined by [7]: F ρ q z dz (0) can be expressed as te sum of te Poiseuille flow (F p ) and te Couette flow (F c ) as follows: were F F p F c p x Q p δ α ρu Q c δ α () Q p δ α δ δ ψ p u du Q c δ α δ ψ c u du are te non-dimensional volume flow rates. THE GENERALIZED REYNOLDS EQUATION Te analysis developed in te previous section can be applied to te slider bearing problem in lubrication teory. Te basic geometry of te two-dimensional gas film is outlined in Fig.. Unlike te configuration considered in te previous section, te upper plate is sligtly inclined at a small angle γ. Since te pitc angle γ is typically less tan o, te pressure p in te gas is taken to be constant in te vertical direction, wile at te left (x 0) and rigt (x L) boundaries it is fixed at ambient pressure p o. In standard ydrodynamic lubrication teory, te pressure distribution in te gas film can be computed from te classical Reynolds equation (an approximation of te full Navier-Stokes equation): d dx PH 3 dp ΛPH 0 () dx were te no-slip boundary conditions for te velocities at te bottom and top surfaces ave been used. Equation () is written in terms of te following dimensionless quantities and te bearing number is defined as X x l ; P p ; H (3) p o 6µUl Λ p o (4) o were µ is te dynamic viscosity of te gas. To extend te generalized Reynolds equation, te mass flow conservation equation across te film tickness must be applied [], [], wit te flow rate F given by Eq. (): 7

5 d dx dp dx Q p δ α ρu Q c δ α 0 (5) For te purpose of a direct comparison wit te classical Reynolds equation (), let us introduce te Poiseuille relative flow rate [], []: Q p δ α Q p δ α Q con (6) were Q con δ 6. Furtermore, te rarefaction parameter δ can be expressed as: δ δ o PH, were δ o is te caracteristic inverse Knudsen number defined by te minimum film tickness,, and te ambient pressure p o as: δ o p o µ RT o Finally, assuming tat te eat generation in te gas is very small, so tat an isotermal process can be considered, te non-dimensional generalized Reynolds equation reads: d dx Q p δ o PH α PH 3 dp Q c dx δ o PH α ΛPH 0 (7) If te two walls are identical (α α α), te Couette flow rate is independent of te Knudsen number regardless of te value of te accommodation coefficient α: Q c δ o PH α α δ Terefore, in tis case, Eq. (7) reduces to te generalized Reynolds equation introduced by Fukui and Kaneko [], [], wile in te continuum regime, Eq. (7) reduces to te classical Reynolds equation (). Writing te non-dimensional film tickness H in terms of te longitudinal coordinate X, suc tat Eq. (7) can be analytically integrated to give: l L H dp dx l L l L X (8) dp dh Q p δ o PH α PH 3 dp dh Q c δ o PH α ΛPH K (9) were K is a constant of integration. Te substitution of in Eq. (9) gives: PH ζ (0) dζ dh ζ H Q c δ o ζ α Λζ K l L Q p δ o ζ α Hζ Eq. () as been solved numerically using te relaxation metods. To apply tis numerical sceme, te differential equations ave to be replaced by finite-difference equations on a point mes. Te solution of te resulting set of equations is determined by starting wit a guess and improving it iteratively using Newton s metod. Te Poiseuille flow rate coefficient Q p δ α as been evaluated by means of te numerical metod described in [3] and a variational tecnique wic applies to te integrodifferential form of te Boltzmann equation based on te BGK model [4]. In order to compute te Couette flow rate Q c δ α we ave solved numerically Eq. (9), extending a finite difference tecnique first introduced by Cercignani and Daneri [7]. () 7

6 O O.6 O O.8 O O.3 O O.7 O.8 0. FIGURE. Pressure profile for 8 0 = 0.5. Te line styles indicate a = 0.8 (solid), a = 0.3 (dased), and a = 0. (dot dased). From left to rigt, te bearing number A is 0, 50, 00. O O.6 O O.8 O O.7 O FIGURE 3. Pressure profile for 8 0 = 0.5. Te line styles indicate a t = 0.5 a = 0.8 (solid), a t = 0.5 a = 0.3 (dased), and a l 0.5 a - 0. (dot dased). From left to rigt, te bearing number A is 0, 50, 00. NUMERICAL RESULTS OF LUBRICATION CHARACTERISTICS Once (//) as been numerically evaluated on a grid tat spans te domain of interest, Eqs. (8) and (0) give te pressure field in te gas film as a function of X. Furtermore, a prediction of te vertical force acting on te upper surface of te slider bearing, crucial for practical design, may be obtained from te load carrying capacity W, defined as W= - (P-l)dX () In order to investigate te effects of te rarefaction parameter 8 0 and te bearing number A on te basic lubrication caracteristics (pressure distribution and load carrying capacity), te parameters describing te gas film geometric configuration were fixed at te following values: / 0 =, L/ 0 = 00. Figure sows te pressure field as a function of te longitudinal coordinate X at tree different bearing numbers: A = 0, 50, 00. To assess te influence of te boundary conditions, te profiles corresponding to different accommodation coefficients (for bounding walls supposed pysically identical) are drawn in eac panel of Fig.. Looking at te pictures, one sees tat te pressure distribution in te gas film increases wit increasing A. Furtermore, at fixed bearing number, te pressure field reduces by increasing te fraction of gas molecules specularly reflected by te walls. Fig. 3 reports te pressure profiles for te same parameters as in Fig., except tat now te two bounding plates are allowed to re-emit te impinging gas molecules differently, so tat two accommodation coefficients can be defined. We keep te accommodation coefficient of te upper wall (a ) fixed and vary te oter one (a ). A comparison wit Fig. sows tat, for every A, te pressure distribution significantly depends on a and only weakly on a. Tis picture remains true at different Knudsen numbers 73

7 FIGURE 4. Pressure profile for A = 50. Te line styles indicate a = 0.8 (dot dased), a t = 0.5 a = 0.8 (long dased), a = 0.3 (solid), and a = 0.5 a = 0.3 (dased). Te inverse Knudsen number 8 0 is 0~ 3 (left) and 0 (rigt). FIGURE 5. Pressure profile versus X. Comparison between our results (solid line) and DSMC data [0] (open circles). From left to rigt, te parameters are: (a) 8 0 = 0.7, A = 6.6, a = ; (b) 8 0 = 0., A = 64, a = ; (c) 8 0 = 0.7, A = 6.6, a = 0.7. progressing from free molecular, troug transitional, to continuum regions (see Fig.4). It is wort noting tat, wen A increases, te Couette contribution to te lubrication flow rate becomes dominant compared wit te Poiseuille flow. Terefore, if te two walls are identical, te influence of te Knudsen number on te load carrying capacity decreases as A increases, since Q c is independent of 5 and a. On te contrary, if te two walls ave a different pysical structure te load carrying capacity sows a dependence on bot te Knudsen number and te accommodation coefficients a, a. For te validation of te code, our outputs ave been compared wit te results from DSMC (Direct Simulation Monte Carlo) simulations publised by Alexander et al. [0] in te case of Maxwell's boundary conditions on two pysically identical walls (see Fig.5). REFERENCES. Fukui, S., Kaneko, R., Journal oftribology 0, 53-6 (988).. Fukui, S., Kaneko, R., JSME International Journal 30, (987). 3. Cercignani, C., Lampis, M., Lorenzani, S., Transp. Teory Stat. Pys. (to be publised). 4. Cercignani, C., Lampis, M., Lorenzani, S., Pys. Fluids (004, in press). 5. Cercignani, C., Rarefied Gas Dynamics, Cambridge Texts in Applied Matematics, Cambridge University Press (000). 6. Batnagar, P. L., Gross, E. P., Krook, M., Pys. Rev. 94, 5-55 (954). 7. Cercignani, C., Daneri, A., Journal of Applied Pysics 34, (963). 8. Cercignani, C., Lampis, M., Lorenzani, S., Proceedings of 3 rd Int. Symp. on Rarefied Gas Dynamics 663, 4-48 (003). 9. Burgdorfer, A., /. Basic Eng. 8, (959). 0. Alexander, F. J., Garcia, A. L., Alder, B. J. Pys. Fluids 6, (994). 74