MODELING OF CYLINDRICAL COUETTE FLOW OF RAREFIED GAS BETWEEN ROTATING CYLINDERS * Peter Gospodinov, Dobri Dankov, Vladimir Roussinov, Stefan Stefanov

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1 МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2010 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2010 Poceedings of the Thity Ninth Sping Confeence of the Union of Bulgaian Mathematicians Albena, Apil 6 10, 2010 MODELING OF CYLINDRICAL COUETTE FLOW OF RAREFIED GAS BETWEEN ROTATING CYLINDERS * Pete Gospodinov, Dobi Dankov, Vladimi Roussinov, Stefan Stefanov The cylindical Couette flow of a aefied gas is studied in the case of two cylindes otating with diffeent velocities. Velocity, 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 = The calculations show that the gas lags o outstips in compaison with the walls o it has the elastic igid body behavio at the vaiety velocity bounday conditions. These esults ae impotant fo applications in non-plana micofluidic poblems. 1. Intoduction. Fluid tanspot in mico and maco channels yields the necessity to study flow in a cylindical coodinate system. Note that Couette cylindical flow is a fundamental poblem in the aefied gas dynamics [1, 6, 9 11]. 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 [2, 3, 12]. 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 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. In the pesent pape we compae esults obtained by using the molecula Diect Simulation Monte Calo (DSMC) method with those calculated by a numeical solution of the continuum Navie-Stokes equations fo compessible flow (NS) [7, 8]. Both methods ae used to model the cylindical Couette flow fo Knudsen numbe 0.02 and fo diffeent otating velocities of the both cylindes. The aim of the pesent pape is to study the gas flow in the gap depending on the vaiety velocity bounday conditions. The esults show that the gas lags o outstips in compaison with the velocity walls o it behaves as an elastic igid body. * 2000 Mathematics Subject Classification: 65C20. Key wods: Fluid Dynamics, Kinetic theoy, Raefied gas, DSMC. The eseach leading to these esults has eceived funding fom the Bulgaian National Science Foundation, unde Gant DID 02/ and the Euopean Community s Seventh Famewok Pogamme FP7/ unde gant ageement ITN GASMEMS GA

2 Fig. 1. Flow geomety 2. Fomulation of the poblem and methods of solving. We study a aefied gas flow between two coaxial cylindes (one dimensional, axially-symmetical poblem) with equal tempeatues T 1 = T 2. The inne cylinde has adius R 1 and the oute R 2. The oute cylinde otates with a constant velocity V 2 and the inne one with constant velocity V 1. Continuous Model and Numeical Simulation. The continuous model is based on the Navie-Stokes equations fo compessible fluid, completed with the equations of continuity and enegy tanspot. The govening equations ae witten in dimensionless fom as follows: (1) ( u ρ t + u u v2 ρ t + 1 ) = 1 ρt 2 (ρu) = 0, [ ] 1 (u) (2) C µkn T + 2 ( 3 C µkn 2 u u ) T, (3) ( v ρ t + u v + uv ) = C µ Kn T [ ] 1 (v) + C µ Kn ( v ( ) T ρ t + u T = C k Knγ 1 ( T T ) (γ 1) ρt (u)+ (4) 4 3 2C µkn (γ 1) [ ( u ) ] 2 T u u ( u ) C µ Kn (γ - 1) [ T (5) P = ρrt, ) T, ( u ) ] 2, whee u is the velocity along axis and v is the velocity along axis ϕ. A athe standad notation is used in Eqs (1) (5): ρ is density and T is the tempeatue. ρ, P, T, u, v = f (, t). The equations ae nomalized by using the following scales: fo density, ρ 0 = mn 0, fo velocity, V 0 = 2RT 0, R is the gas constant, fo length the distance between the cylindes L = R 2 R 1, fo time t 0 = L/V 0, fo tempeatue T 0 = T w - the wall tempeatue of both cylindes. In (2) the pessue is eliminated by using the pefect gas law (5) Fo a pefect monatomic gas, the viscosity and the heat tansfe coefficient ead 185

3 as [13]: (6) µ = µ (T) = C µ ρ 0 l 0 V 0 T, Cµ = 5 2π, 16 (7) λ = λ(t) = C λ ρ 0 l 0 V 0 T, Cλ = 15 2π. 32 The Knudsen numbe in (2) (5) is Kn = l 0 /L, whee the mean fee path is l o and γ = c P /c V = 5/3. Fo the poblem (1)-(5) and t > 0, fist-ode slip bounday conditions ae imposed at both walls, which can be witten as follows [13 14]: ( u (8) u Kn u ) = ū i, (9) v = 0, (10) T ± Kn T = T i, at = R i, i = 1, 2. In Eqs. (8) (10) ū i = u w,i /V 0 and T i = T W /T o = 1 ae the dimensionless wall velocity and tempeatue fo both cylindes. The equations of tansfe (1) (4), togethe with the bounday conditions (8) (10) and zeo initial distibutions fo u, v and T, fomulate the initial nonstationay boundayvalue 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 nodes ae intoduced along the coodinate. Thus, the diffeence value poblem, fo a given time t, is educed to the solution of 4 lineaized systems of M algebaic equations. Evey algebaic system has a diagonal and weakly filled matix. Due to poblem nonlineaity, an intenal iteation pocess is used. Afte attaining a peviously pescibed accuacy fo the cuent time, the next time step is consideed. Diect Simulation Monte Calo (DSMC) Method. The gas consideed is simulated as a stochastic system of N paticles [4, 5]. All quantities used ae non-dimensional, so that the mean fee path at equilibium is equal to 1. The basic steps of simulation ae as follows: A. The time inteval [ 0; ˆt ] ove which the solution is found, is subdivided into subintevals with step t. B. The space domain is subdivided into cells with sides z,. C. Gas molecules ae simulated in gap G using a stochastic system of N points (paticles) having position z i (t), i (t) and velocities ξ i (t) 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 1. Binay collisions in each cell ae calculated, wheeas paticles do not move. Collision modeling is ealized using Bid s scheme no time counte. Stage 2. 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 1 and Stage 2 ae epeated until t = ˆt. 186

4 Fig. 2. Density pofile in the cases 1 4 Fig. 3. ϕ velocity pofile in the cases 1 4 Fig. 4. Tempeatue pofile in the cases

5 F. Flow maco-chaacteistics (density, velocity, tempeatue) ae calculated as timeaveaged when steady egime is attained. G. Bounday conditions ae diffusive ove the cylindes and peiodical along axis Oy. All magnitudes used ae non-dimensional so that the mean fee path in equilibium state is equal to 1. The modeling paticles numbe fo DSMC method is Numeical esults. Let us note = V 1 R 2 R 1 V 2. We study the fou typical cases of aefied gas between otating cylindes: Case 1: V 1 =1, V 2 =1, T 1 = T 2 =1, =1.5; Case 2: V 1 =1, V 2 =0.5, T 1 = T 2 =1, =1.0; Case 3: V 1 =1, V 2 =0.3, T 1 = T 2 =1, =1.7; Case 4: V 1 =0.5, V 2 =1, T 1 = T 2 =1, =0.0. The numeical esults fo the ϕ velocity show that the gas velocity on the walls lags fom the inne cylinde velocity and it outstips fom the oute cylinde velocity in the case 1 and 2 (Figue 3). It is obseved the evese effect at lowe velocity of the oute cylinde the gas velocity lags fom the oute cylinde velocity and it outstips fom the inne cylinde velocity case 3 (Figue 3). It is inteest the ϕ velocity pofile in the case 4. The gas moves as an elastic igid body togethe with the inne and oute wall. The velocity pofile is linea (Figue 3) and the tempeatue is constant (Figue 4). The density pofile (Figue 2) is non-linea. It is invesely popotional to the gas pessue because in the pesence of centipetal acceleation which is highe fom the convection one, the gas pessue is popotional to 2 and, theefoe, the density is popotional to 1/ 2. Hee is only elastic defomation of the gas medium at =0. It is necessay to denote that the numeical esults show the same conduct of the gas medium between otating cylindes at Kn=0.2, 0.3, 0.5 and the keeping of the velocity and tempeatue bounday conditions as in case 4. REFERENCES [1] C. Cecignani, F. Senagiotto. Cylindical Couette Flow of a Raefied Gas. Physics of Fluids, 10 (1967), [2] K. W. Tibbs, F. Baas, A. L. Gacia. Anomalous flow pofile due to the cuvatue effect on slip length. Phys, Rev. E, 56 (1997), [3] K. Aoki, H. Yoshida, T. Nakanishi, A. Gacia. Inveted velocity pofile in the cylindical Couette flow of a aefied gas. Phys. Rev. E, 68 (2003), [4] G. A. Bid. Molecula Gas Dynamics and the Diect Simulation of Gas Flows, Oxfod Univesity Pess, Oxfod, [5] C. Cecignani. Raefied Gas Dynamics, Fom Basic Concepts to Actual Calculations, Cambidge Univesity Pess, Cambidge, [6] S. K. Stefanov, V. M. Roussinov, C. Cecignani. Rayleigh Bénad flow of a aefied gas and its attactos. I. Convection egime. Physics of Fluids, 14 (2002), [7] S. K. Stefanov, V. M. Roussinov, C. Cecignani. Rayleigh Bénad flow of a aefied gas and its attactos. II. Chaotic and peiodic convective egimes. Physics of Fluids, 14 (2002),

6 [8] D. Y. Dankov, V. M. Roussinov. Modelling and applications of the cylindical Couette flow of a aefied gas AIP Conf. Poc., 1067 (2008), 239. [9] G. P. Neitzel. Stability of cicula Couette flow with vaiable inne cylinde speed. Jounal of Fluid Mechanics, 123 (1982), [10] G. H. Tang, X. J. Gu, R. W. Babe, D. R. Emeson, Y. H. Zhang. Lattice Boltzmann simulation of nonequilibium effects in oscillatoy gas flow. Phys. Rev. E 78 (2008), [11] M. Malik, J. Dey, Meheboob Alam. Linea stability, tansient enegy gowth, and the ole of viscosity statification in compessible plane Couette flow. Phys. Rev. E, 77 (2008), [12] Y. Hioaki, A. Kazuo. Linea stability of the cylindical Couette flow of a aefied gas. Phys. Rev. E, 73 (2006), [13] S. Stefanov, P. Gospodinov, C. Cecignani. Monte Calo simulation and Navie Stokes finite diffeence calculation of unsteady-state aefied gas flows. Physics of Fluids, 10 (1998), No 1, [14] A. Manela, I. Fankel. On the compessible Taylo-Couette poblem. J. Fluid Mech., 588 (2007), Pete Gospodinov Dobi Dankov Vladimi Roussinov Stefan Stefanov Institute of Mechanics Bulgaian Academy of Sciences Acad. G. Bonchev St., Bl Sofia, Bulgaia png@imbm.bas.bg dankov@imbm.bas.bg vladimi@imbm.bas.bg stefanov@imbm.bas.bg МОДЕЛИРАНЕ НА ЦИЛИНДРИЧНО ТЕЧЕНИЕ НА КУЕТ ЗА РАЗРЕДЕН ГАЗ МЕЖДУ ВЪРТЯЩИ СЕ ЦИЛИНДРИ Петър Господинов, Добри Данков, Владимир Русинов, Стефан Стефанов Изследвано е цилиндрично течение на Кует за разреден газ между два въртящи се цилиндъра. Получени са профилите на налягането, скоростта и температурата по метода на прякото статистическо моделиране (DSMC) и чрез числено решаване на уравненията на Навие-Стокс за свиваем флуид. Резултатите сочат много добро съвпадение за малки числа на Кнудсен Kn = Показано е, че при различни кинематични гранични условия, газът изостава или избързва спрямо скоростта на стената, или има поведение на твърдо еластично тяло. Получените резултати са важни при решаването на неравнинни, задачи от микрофлуидиката с отчитане на ефектите на кривината. 189