MODELING OF CYLINDRICAL COUETTE FLOW OF RAREFIED GAS. THE CASE OF ROTATING INNER CYLINDER *

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1 th National Congess on Theoetical and Applied Mechanics, -5 Sept. 9, Boovets, Bulgaia MODELING OF CYLINDRICAL COUETTE FLOW OF RAREFIED GAS. THE CASE OF ROTATING INNER CYLINDER * PETER GOSPODINO Institute of mechanics, BAS, Sofia, Acad. G. Bonchev st. bl. png@imbm.bas.bg DOBRI DANKO Institute of mechanics, BAS, Sofia, Acad. G. Bonchev st. bl. dankov@imbm.bas.bg LADIMIR ROUSSINO Institute of mechanics, BAS, Sofia, Acad. G. Bonchev st. bl. vladimi@imbm.bas.bg STEFAN STEFANO Institute of mechanics, BAS, Sofia, Acad. G. Bonchev st. bl. stefanov@imbm.bas.bg ABSTRACT. The cylindical Couette flow of a aefied gas is studied in the case when the inne cylinde is otating while the oute cylinde is at est. elocity, density and tempeatue pofiles ae investigated by a Diect Monte Calo Simulation method and a numeical solution of the Navie-Stokes equations fo compessible flow is found. The esults obtained by both methods ae: in an excellent ageement at a small Knudsen numbe Kn=.; in a satisfactoy ageement at Kn=. and they vastly diffes each othe at a modeate Kn=.5. The compaison shows that the continuum appoach can be used successfully fo calculations of nonisothemal aefied gas flows at small Knudsen numbes Kn<.. These esults ae impotant fo applications in non-plana micofluidic poblems. KEY WORDS: Kinetic theoy, Raefied gas, Micofluidics,. Intoduction The Couette cylindical flow is a fundamental poblem in the aefied gas dynamics [,9,,]. As such, its modeling and numeical solving is of a geat impotance fo the micofluidics, which is the theoetical backgound fo analysis of new emeging Mico Electo Mechanical Systems MEMS [,,,]. The design of adequate mathematical models of gaseous flows in mico devices is one of the most impotant tasks of the studies. We conside both molecula * The eseach leading to these esults has eceived funding fom the Euopean Community's Seventh Famewok Pogamme FP7/7- unde gant ageement ITN GASMEMS n 55 and by the F of Bulgaia unde Gant No DO-5/8.

2 P. Gospodinov, D. Dankov,. Roussinov, S. Stefanov and continuum models teating the gaseous flow by using diffeent level of mathematical desciption. Both models take into account the specific micofluidic effects of gas aefaction and slip-velocity egime at the solid boundaies [6,7,8]. In the pesent pape we compae esults obtained by using the molecula Diect Simulation Monte Calo () method with those calculated by a numeical solution of the continuum Navie-Stokes equations fo compessible flow (). The aim of the pape is delineate the ange of validity of the with espect to the solution, which convegences to the Boltzmann equation when the numbe of modeling paticles is lage enough to be consideed as tended towads infinity.. Fomulation of the poblem and methods of solution We study a aefied gas between two coaxial cylindes (one dimensional, axissymmetical poblem) with equal tempeatues T =T. The inne cylinde has adius R and the oute R. The inne cylinde otates with a constant velocity and the oute one is static Figue. ϕ O R R z Fig.. Flow geomety.. Continuous Model and Numeical Simulation The continuous model is based on the Navie-Stokes equations fo compessible Newtonian fluid, completed with the equations of continuity and enegy tanspot. The govening equations ae witten in dimensionless fom as follows: ρ (.) + ( ρu) = t u u v P τ τ z (.) u ( ) ϕϕ ρ + = τ + t z v v uv τ ϕz (.) ρ + u + = ( τ ϕ ) + ρg ϕ t z DT (.) ρ c P = div( λgadt ) pdiv + μφ Dt

3 Modeling of cylindical couette flow of aefied gas (.5) P = ρrt Whee is the velocity vecto, u and v ae the velocity components along axis and ϕ. A athe standad notation is used in Eqs. (.)-(.5): ρ is density and T is the tempeatue. ρ, P, T, u, v = f (, t). The thems τ i, j ae the stess tenso components and Φ is the dissipation function [5]. The equations ae nomalized by using the following scales: fo density, ρ = mn, fo velocity, = RT, R is the gas constant, fo length - the distance between the cylindes L = R R, fo time t = L, fo tempeatue T = Tw - the wall tempeatue of both cylindes. Fo a pefect monatomic gas, the viscosity and the coefficient heat tansfe ead as []: 5 (.6) μ = μ( T ) = Cμρ l T, Cμ = π 6 5 (.7) λ = λ( T ) = Cλρ l T, Cλ = π The Knudsen numbe in (.)-(.5) is Kn = l L, whee the mean fee path is l o and γ = c c P = 5. Fo the poblem (.)-(.5) and t >, fist-ode slip bounday conditions ae imposed at both walls, which can be witten as follows [,]: u u (.8) u m.66kn = ui (.9) v = T (.) T ±.9Kn = Ti, at =R i,: i=,. In Eqs. (.8)-(.) ui = uw, i and T i = TW / To = ae the dimensionless wall velocity and tempeatue fo both cylindes. The equations of tansfe (.)-(.), togethe with the bounday conditions (.8)-(.) and zeo initial distibutions fo u, v and T, fomulate the initial nonsteady bounday-value poblem. A second ode of appoximation, implicit diffeence scheme to solve numeically the fomulated poblem is used. Stating fom the inne cylinde wall M gid knots ae intoduced along the coodinate. Thus, the diffeence value poblem, fo a given time t, is educed to the solution of lineaized systems of M algebaic equations.. Diect Simulation Monte Calo () Method The gas consideed is simulated as a stochastic system of N paticles [,5]. All quantities used ae non-dimensional, so that the mean fee path at equilibium is equal to. The basic steps of simulation ae as follows:

4 P. Gospodinov, D. Dankov,. Roussinov, S. Stefanov.5..5 ρ Kn= (A) ρ...9 Kn= (B).5 ρ..95 Kn= Kn=....8 (D) Kn=....8 (E) (C) (F) Fig.. Density (A, B, C) and ϕ velocity (D, E, F) pofile fo inne cylinde otating =.; =.5; =.7; =. A. The time inteval [ ;tˆ ] ove which the solution is found, is subdivided into subintevals with step Δ t. B. The space domain is subdivided into cells with sides Δ z, Δ. С. Gas molecules ae simulated in gap G using a stochastic system of N.6 points (paticles) having position zi ( t), i ( t) and velocities ξ i ( t).. Kn=.5

5 Modeling of cylindical couette flow of aefied gas D. N m paticles ae located in the m-th cell at any given time. This numbe vaies duing the compute simulation by the following two stages: Stage. Binay collisions in each cell ae calculated, wheeas paticles do not move. Collision modeling is ealized using Bid s scheme no time counte. Stage. Paticles move with new initial velocities acquied afte collisions, and no extenal foces act on paticles. No collisions ae accounted fo at this stage. E. Stage and Stage ae epeated until t = tˆ. F. Flow maco-chaacteistics (density, velocity, tempeatue) ae calculated as time-aveaged when steady egime is attained. G. Bounday conditions ae diffusive ove the cylindes and peiodical along axis Oy. The modeling paticles numbe fo method is..5. T Kn=..6. T.8 Kn= (A)..5 T (B) Kn= (C) Fig.. Tempeatue pofile fo inne cylinde otating velocity =.; =.5; =.7; =.

6 P. Gospodinov, D. Dankov,. Roussinov, S. Stefanov. Numeical esults We study fou cases at each of the Knudsen numbe.,. and.5, whee the adius of the oute cylinde (R =) is twice lage than that of the inne one (R =), and diffeent otation velocities of the inne cylinde ae consideed: =.;.5;.7;.. The oute cylinde is stationay. All tests ae implemented on the fou Gid clustes in the Institute of Mechanics and the Institute fo Paallel Pocessing, Bulgaian Academy of Sciences, which totally have moe than CPUs. These clustes ae included in the Pan- Euopean (EGEE) gid infastuctue. (see: The esults obtained by both methods ae: in an excellent ageement at a small Knudsen numbe Kn=.; in a satisfactoy ageement at Kn=. and they vastly diffes each othe at a modeate Kn=.5. The compaison shows that the continuum appoach can be used successfully fo calculations of non-plana isothemal aefied gas flows at small Knudsen numbes Kn<.. The using of simplified bounday conditions o bounday conditions, which content second deivative along, not leads to the esults impoving. R E F E R E N C E S [] CERCIGNANI C., F. SERNAGIOTTO, Physics of Fluids, (967) [] TIBBS K. W., F. BARAS, A. L. GARCIA, Phys, Rev. E 56, 8-8 (997) [] AOKI K., H. YOSHIDA, T. NAKANISHI AND A. GARCIA, Phys. Rev. E 68, 6 () [] BIRD G. A., Molecula Gas Dynamics and the Diect Simulation of Gas Flows, Oxfod Univesity Pess, Oxfod, 99. [5] CERCIGNANI C., Raefied Gas Dynamics, Fom Basic Concepts to Actual Calculations, Cambidge Univesity Pess, Cambidge,. [6] STEFANO S. K.,. M. ROUSSINO AND C. CERCIGNANI, Physics of Fluids, () [7] STEFANO S. K.,. M. ROUSSINO AND C. CERCIGNANI, Physics of Fluids, 7-88 () [8] DANKO D. Y.,. M. ROUSSINO Modeling and applications of the cylindical Couette flow of a aefied gas AIP Conf. Poc. 67, 9, (8) [9] NEITZEL G. P. Jounal of Fluid Mechanics,, p. (98) [] TANG G. H., X. J. GU, R. W. BARBER, D. R. EMERSON, Y. H. ZHANG Phys. Rev. E 78, 676 (8) [] MALIK M., J. DEY, MEHEBOOB ALAM Phys. Rev. E 77, 6 (8) [] HIROAKI Y. AND KAZUO A. Phys. Rev. E 7, (6) [] STEFANO S., P.GOSPODINO AND C.CERCIGNANI, Physics of Fluids, ol., Issue, pp. 89-, (998) [] MANELA A., I. FRANKEL, J. Fluid Mech., vol. 588, pp (7) [5] BIRD, R. B., STEWART, W. E., and LIGHTFOOT, E. N., "Tanspot Phenomena", nd edition, John Wiley, New Yok ()