A non-isothermal Couette slip gas flow

by Prof. Nevena Stevanovic (University of Belgrade, Faculty of Mechanical Engineering), Snezana Milicev (University of Belgrade, Faculty of Mechanical Engineering)

Wednesday 1 Dec 2010, 18:00 → 19:00 Europe/Belgrade

301 F (Mathematical Institute)

Micro-Couette flow as a shear-driven flow is encountered often in Micro-Electro-Mechanical-Systems (MEMS), such as microcomb mechanisms, microbearings and micromotors. Moreover, the most frequently conditions in these systems correspond to the slip flow regime (Kn<0.1), so the results in that region are very useful. As the thermal effects are very often present in MEMS, many authors have been investigated the heat transfer in Couette flow. The microscopic approach is used mostly, either by solving kinetic Boltzmann equation or by direct simulation Monte-Carlo DSMC. In this paper micro-Couette slip gas flow for moderately high and low Reynolds numbers is analyzed by macroscopic approach analytically and numerically. The gas flow between parallel plates that is preserved in relative motion is considered in two cases: the temperatures of the walls are equal and constant, i.e. the isothermal walls, and the temperatures of the walls are different, but constant. The Knudsen number is Kn≤0.1, which corresponds to the slip flow and continuum. The flow is defined by continuity, Navier-Stokes and energy continuum equations, along with the velocity slip and the temperature jump first order boundary conditions. The gas flow is subsonic and the ratio Ma2/Re is taken to be of the order of a small parameter. An analytical solution for velocity and temperature is obtained by developing a perturbation scheme. The first approximation corresponds to the continuum flow conditions, while the others represent the contribution of the rarefaction effect. Moreover, the same system of continuum governing equations with corresponding first order boundary conditions is solved numerically, by Runge-Kutta method, in order to verify presented analytical solutions. The influences of the viscosity and conductivity dependence on temperature, the dissipation and the rarefaction on the velocity and temperature profiles are explored. It is shown, in the case of isothermal walls, that temperature influence on viscosity and conductivity is insignificant on the velocity and temperature fields. Then, the exact analytical solution for constant viscosity and conductivity is found. It is shown that, although very simple, it is complete substitution to the exact numerical solution for the isothermal walls case. The results for the velocity and temperature fields are also compared with some numerical and analytical results of other authors and good agreement is achieved. - - - - - - - - - - Kuetovo strujanje gasa je često prisutno u mikro-elektro-mehaničkim sistemima (MEMS) kao što su mikročešljevi, mikroležaji ili mikromotori. U tim sistemima uslovi strujanja su najčešće takvi da odgovaraju režimu strujanja gasa sa klizanjem (Kn<0.1). Osim toga često je potrebno uzeti u obzir i neizotermnost ovih strujanja. U literaturi se uglavnom sreću rezultati dobijeni mikroskopskim pristupom koji se baziraju na rešavanju Bolcmanove kinetičke jednačine ili na direktnoj simulaciji Monte-Karlo (DSMC). U ovom radu je makroskopskim pristupom analizirano mikro Kuetovo strujanje gasa pri malim i umereno velikim vrednostima Rejnoldsovog broja. Dobijeno je rešenje za strujanje gasa između paralelnih ploča koje se kreću u suprotnim smerovima kada je temperatura zidova konstantna i jednaka (izotermski zidovi) i kada su temperature zidova različite ali konstantne. Analizirano je strujanje sa klizanjem kada je Knudsenov broj manji od 0,1. Kuetovo strujanje gasa se ovde opisuje jednačinama kontinuuma (jednačina kontinuiteta, Navije-Stoksova i jednačina energije) zajedno sa graničnim uslovima klizanja i temperaturskog skoka na zidu. Pretpostavljeno je da je Mahov broj mali, pa se definiše mali parametar kao odnos kvadrata Mahovog broja i Rejnoldsovog broja. Analitičko rešenje za polje brzine i temperature se nalazi perturbacionom metodom. Rešenje za prvu aproksimaciju odnosi se na strujanje bez klizanja, dok ostale aproksimacije sadrže korekciju zbog efekta razređenosti. U cilju verifikacije dobijenog analitičkog rešenja, sistem osnovnih jednačina kontinuuma zajedno sa graničnim uslovima klizanja prvog reda, rešen je i numerički metodom Runge–Kuta. Analiziran je uticaj zavisnosti koeficijenta viskoznosti i provođenja od temperature, kao i uticaj disipacije i klizanja na polje brzine i temperature. Pokazano je da je u slučaju izotermskih zidova zanemarljiv uticaj zavisnosti koeficijenta viskoznosti i provođenja toplote od temperature na polje brzine i temperature, pa je za konstantnu vrednost viskoznosti i provođenja toplote određeno tačno analitičko rešenje. Pokazano je da su rešenja dobijena njegovom primenom u slučaju izotermskih zidova zadovoljavajuće tačnosti. Dobijeni rezultati za polje brzine i temperature takođe su verifikovani poređenjem sa rezultatima drugih autora.