Friction pressure loss in micro channel rarefied gas flows

Strujanje gasa kroz mikrokanale prisutno je u mikro elektro mehaničkim sistemima (MEMS). Dimenzije mikrokanala su reda veličine µm, pa dužina slobodnog puta molekula nije zanemarljivo mala. Efekat razređenosti dolazi do izražaja, pa je potrebno uzeti u obzir granične uslove klizanja na zidu. U ovom radu dobijena su rešenja za dvodimenzijsko, izotermsko, stišljivo strujanje gasa kroz mikrokanale sporo promenljivog poprečnog preseka, pri veoma malim vrednostima Mach-ovog broja. Korišćeni su granični uslovi klizanja drugog reda, što je uslovilo korišćenje i jednačina količine kretanja drugog reda, tj. Burnettovih jednačina. Pokazalo se da se za pomenute uslove strujanja one svode na Navier-Stokes-ove. U radu je prikazano analitičko rešenje za slučajeve strujanja gasa kada je vrednost Reynolds-ovog broja veća, pa osim razređenosti do izražaja dolazi i uticaj inercije. Za takve uslove strujanja dobijena su rešenja za polje pritiska i brzine, kao i analitički izrazi za određivanje faktora trenja duž kanala i njegove srednje vrednosti. Iz njih se vidi da pri strujanju razređenog gasa faktor trenja zavisi i od Reynolds-ovog i od Knudsen-ovog broja.Gas flows through micro-channels are encountered in many applications of Micro-Electro-Mechanical Systems (MEMS) Dimensions of the MEMS are within µm range, which means that rarefaction must be considered. It is common to use slip conditions at the wall and continuum equations for solving these problems. In this paper isothermal, compressible and subsonic gas flows through micro channels with slowly varying cross sections are analyzed. In order to provide a higher accuracy, the second order boundary conditions are used. This approach requires the higher order momentum equation, i.e. the Burnett equation. Solutions are obtained for higher Reynolds number values when the inertia effect is important together with the rarefaction effect. For such flow conditions analytical relations for pressure and velocity fields are presented. Also, analytical expressions for the friction factor change along a micro channel with constant cross section and for the average friction factor are obtained. The derived relations show that the friction factor depends not only on the Reynolds number, but also on the Knudsen number

Friction pressure loss in micro channel rarefied gas flows

Strujanje gasa kroz mikrokanale prisutno je u mikro elektro mehaničkim sistemima (MEMS). Dimenzije mikrokanala su reda veličine µm, pa dužina slobodnog puta molekula nije zanemarljivo mala. Efekat razređenosti dolazi do izražaja, pa je potrebno uzeti u obzir granične uslove klizanja na zidu. U ovom radu dobijena su rešenja za dvodimenzijsko, izotermsko, stišljivo strujanje gasa kroz mikrokanale sporo promenljivog poprečnog preseka, pri veoma malim vrednostima Mach-ovog broja. Korišćeni su granični uslovi klizanja drugog reda, što je uslovilo korišćenje i jednačina količine kretanja drugog reda, tj. Burnettovih jednačina. Pokazalo se da se za pomenute uslove strujanja one svode na Navier-Stokes-ove. U radu je prikazano analitičko rešenje za slučajeve strujanja gasa kada je vrednost Reynolds-ovog broja veća, pa osim razređenosti do izražaja dolazi i uticaj inercije. Za takve uslove strujanja dobijena su rešenja za polje pritiska i brzine, kao i analitički izrazi za određivanje faktora trenja duž kanala i njegove srednje vrednosti. Iz njih se vidi da pri strujanju razređenog gasa faktor trenja zavisi i od Reynolds-ovog i od Knudsen-ovog broja.Gas flows through micro-channels are encountered in many applications of Micro-Electro-Mechanical Systems (MEMS) Dimensions of the MEMS are within µm range, which means that rarefaction must be considered. It is common to use slip conditions at the wall and continuum equations for solving these problems. In this paper isothermal, compressible and subsonic gas flows through micro channels with slowly varying cross sections are analyzed. In order to provide a higher accuracy, the second order boundary conditions are used. This approach requires the higher order momentum equation, i.e. the Burnett equation. Solutions are obtained for higher Reynolds number values when the inertia effect is important together with the rarefaction effect. For such flow conditions analytical relations for pressure and velocity fields are presented. Also, analytical expressions for the friction factor change along a micro channel with constant cross section and for the average friction factor are obtained. The derived relations show that the friction factor depends not only on the Reynolds number, but also on the Knudsen number

A microbearing gas flow with different walls' temperatures

An analytical solution for the non-isothermal 2-D compressible gas flow in a slider microbearing with different temperatures of walls is presented in this paper. The slip flow is defined by the continuity, Navier-Stokes and energy continuum equations, along with the velocity slip and the temperature jump first order boundary conditions. Knudsen number is in the range of 10-3-10-1, which corresponds to the slip flow. The ratio between the exit microbearing height and the microbearing length is taken to be a small parameter. Moreover, it is assumed that the microbearing cross-section varies slowly, which implies that all physical quantities vary slowly in x-direction. The model solution is treated by developing a perturbation scheme. The first approximation corresponds to the continuum flow conditions, while the second one involves the influence of rarefaction effect. The analytical solutions of the pressure, velocity, and temperature for moderately high Reynolds numbers are presented here. For these flow conditions the inertia, convection, dissipation, and rate at which work is done in compressing the element of fluid are presented in the second approximation, also

Influence of transport coefficients' dependence on temperature for gas flow in microbearing

The paper presents an analytical solution for the non-isothermal compressible gas flow in a slide microbearing with different temperatures of walls. The gas flow is defined by the Navier-Stokes-Fourier system of the continuum equations and first order boundary conditions. Knudsen number corresponds to the slip and continuum flow (Kn lt = 10(-1)) and Reynolds number is moderately high, so inertia needs to be included. The solution is obtained by perturbations with the first approximation that relates to the continuum flow and the second one that involves second-order effects: the rarefaction, inertia, convection, dissipation, and rate at which work is done in compressing the element of fluid. The presented model analyzes the influence of the dependence of transport coefficients on temperature. The obtained analytical solution for the pressure, velocity, and temperature is approved by a comparison with the results of other authors. The microbearings can often be a part of MEMS, so the presented method and the obtained analytical solution can serve for solving similar non-isothermal shear-driven or pressure-driven problems. The paper gives an estimation about the error in values for microbearing mass flow and load capacity if the dependence of transport coefficients on temperature are neglected

A constant wall temperature microbearing gas flow

U radu je analizirano neizotermsko dvodimenzijsko stišljivo strujanje gasa u mikroležaju konstantnih i jednakih temperatura zida. Vrednost Knudsenovog broja je između 10-3 i 10-1, što odgovara režimu strujanja sa klizanjem. Ovaj režim strujanja definiše se jednačinama kontinuuma: Navije-Stoksovom i jednačinom energije i graničnim uslovom klizanja i temperaturskog skoka na zidu. Strujanje gasa je dozvučno, pa se mali parametar definiše kao ε = κM2/Re. Osim toga pretpostavljeno je se poprečni presek mikrokanala menja sporo, što dovodi do spore promene svih veličina u pravcu strujanja. Rešenje je dobijeno perturbacionom metodom. Prva aproksimacija predstavlja rešenje za slučaj strujanja gasa bez klizanja, dok se u drugoj aproksimaciji javlja uticaj klizanja i inercije. Dobijena su analitička rešenja za raspodelu pritiska, brzine i temperature u mikroležaju pri umereno velikim vrednostima Rejnoldsovog broja.A non-isothermal two-dimensional compressible gas flow in a slider microbearing with constant and equal wall temperature is investigated in this paper analytically. The slip flow is defined by the Navier-Stokes and energy continuum equations along with the velocity slip and the temperature jump first order boundary conditions. Knudsen number is in the range of 10-3 to 10-1, which corresponds to the slip flow. The gas flow is subsonic and the ratio κM2/Re is taken to be a small parameter. Moreover, it is assumed that the microbearing cross-section varies slowly, which implies that all physical quantities vary slowly in x-direction. The model solution is treated by developing a perturbation scheme. The first approximation corresponds to the continuum flow conditions, while the second one involves the influence of inertia as well as rarefaction effect. The analytical solutions of the pressure, velocity and temperature for moderately high Reynolds numbers are obtained

Navier-stokes-fourier analytic solutions for non-isothermal couette slip gas flow

The explicit and reliable analytical solutions for steady plane compressible non-isothermal Couette gas flow are presented These solutions for velocity and temperature are developed by macroscopic approach from Navier-Stokes-Fourier system of continuum equations and the velocity slip and the temperature jump first order boundary conditions. Variability of the viscosity and thermal conductivity with temperature is involved in the model. The known result for the gas flow with constant and equal temperatures of the walls (isothermal walls) is verified and a new solution for the case of different temperature of the walls is obtained Evan though the solution for isothermal walls correspond to the gas flow of the Knudsen number (Kn lt = 0.1), i. e. to the slip and continuum flow, it is shown that the gas velocity and related shear stress are also valid for the whole range of the Knudsen number. The deviation from numerical results for the same system is less than 1%. The reliability of the solution is confirmed by comparing with results of other authors which are obtained numerically by microscopic approach. The advantage of the presented solution compared to previous is in a very simple applicability along with high accuracy

An exact analytical solution for the second order slip-corrected Reynolds lubrication equation

Izvedena je opšta Rejnoldsova jednačina podmazivanja za uslove strujanja sa klizanjem primenom opštih graničnih uslova klizanja drugog reda i pokazano je da ona ima tačno analitičko rešenje. Rešenje je dobijeno tansformacijom zavisno promenljive što je dovelo do mogućnosti analitičkog proračuna raspodele pritiska i masenog protoka u mikroležaju. Na taj način, poređenjem sa datim analitičkim rešenjem omogućena je provera eksperimentalnih i numeričkih rezultata za strujanje gasa u mikroležajima.We derive a general slip-corrected compressible Reynolds lubrication equation, valid for any choice of the slip velocities, and show that it possesses the exact analytical solution. It is obtained by a suitable transformation of the dependent variable, and it yields both the pressure distribution in the bearing and the mass flow rate through it. It can be usefully applied for testing the other, experimental or numerical results obtained under the same or similar physical conditions, against this solution

Synthesis, characterization, antimicrobial and cytotoxic effects of Mn(II), Cu(II), Zn(II) and Bi(III) complexes with hydrazone derivatives of girard’s reagent P and T

U ovom radu opisana je sinteza, karakterizacija i kristalna struktura kompleksa 1, nastalog između jona Zn(II) i (E)-1-(2-okso-2-(2-(hinolin-2-ilmetilen)hidrazinil)etil)piridin- -ijum-hlorida (HL1Cl) i kompleksa 2, 3, 4 i 5, nastalih između jona Cu(II), Mn(II), Zn(II) i Bi(III) sa (E)-N,N,N-trimetil-2-okso-2-(2-(1-(tiazol-2-il)etiliden)hidrazinil)etan-1-aminijum hloridom (HL2Cl). Svi kompleksi okarakterisani su rendgenskom strukturnom analizom, elementalnom analizom i IC spektroskopijom. U slučaju kompleksa 1 ligand HL1Cl koordinovan je u deprotonovanom obliku preko hinolinskog atoma azota, azometinskog atoma azota i karbonilnog atoma kiseonika. Ligand HL2Cl u kompleksima 2–4 koordinovan je u deprotonovanom formalno neutralnom cviterjonskom obliku preko NNO donorskog seta atoma, dok u kompleksu 5 ostaje u protonovanoj formi. Ispitana je antimikrobna aktivnost, test na račićima Artemia salina i DPPH test na svim kompleksima. Antimikrobna aktivnost je ispitana na pet sojeva Gram-pozitivnih i pet sojeva Gram-negativnih bakterija, dva soja kvasaca i jednom soju gljivica. Određena je i citotoksična aktivnost prema pet malignih ćelijskih linija (HeLa, A375, MCF7, PC-3 i A549) i jednoj normalnoj ćelijskoj liniji (HaCaT). Nije ispitivana citotoksična aktivnost kompleksa 5 usled njegove hidrolize u rastvoru DMSO. Kompleks 1 pokazao je značajnu antibakterijsku aktivnost posebno prema Gram-negativnim bakterijama, sa aktivnošću sličnoj hloramfenikolu, dok je dinuklearni kompleks 3 pokazao antifungalnu aktivnost sličnog intenziteta sa amfotericinom B. Kompleksi 2 i 3 pokazali su značajnu citotoksičnu aktivnost, pri čemu je aktivnost kompleksa 3 tek neznatno slabija od one koju pokazuje cisplatin prema ćelijama MCF7.In this work synthesis, characterization and crystal structures of 1, Zn(II) complex ([ZnL1(NCS)2]), with (E)-1-(2-oxo-2-(2-(quinolin-2-ylmethylene)hydrazinyl)ethyl)pyridin- -1-ium chloride (HL1Cl) and 2, 3, 4 and 5, Cu(II), Mn(II), Zn(II) and Bi(III) complexes with (E)-N,N,N-trimethyl-2-oxo-2-(2-(1-(thiazol-2-yl)ethylidene)hydrazinyl)ethan-1-aminium chloride (HL2Cl) are represented. All complexes are characterized by X-ray crystallographic analysis, elemental analysis and IR spectroscopy. In case of complex 1 hydrazone ligand HL1Cl is coordinated in deprotonated form through the quinoline nitrogen, azomethine nitrogen and carbonyl oxygen atoms. Hydrazone ligand HL2Cl in complexes 2–4 is coordinated in deprotonated formally neutral zwitter-ionic form via NNO donor set atoms, while in complex 5 it remains in protonated form. Antimicrobial activity, brine shrimp assay and DPPH radical scavenging activity of all complexes was evaluated. Antimicrobial activity was tested against a five Gram-negative and five Gram-positive bacteria, two yeasts and one fungal strain. Cytotoxic activity against five malignant cancer cell lines (HeLa, A375, MCF7, PC-3 and A549) and normal cell line HaCaT was tested as well. Due to hydrolysis that is occurring in DMSO solution of complex 5, this complex was not tested for cytotoxic activity. Complex 1 showed a significant antibacterial activity especially towards Gram-negative bacteria, with intensity similar to chloramphenicol, while the binuclear Mn(II) complex (3) showed antifungal activity of similar intensity as amphotericin B. Complexes 2 and 3 showed a significant cytotoxic activity. The activity of Mn(II) complex (3) is only slightly weaker than that of cisplatin against breast cancer MCF7 cells

An exact analytical solution for the second order slip-corrected Reynolds lubrication equation

Izvedena je opšta Rejnoldsova jednačina podmazivanja za uslove strujanja sa klizanjem primenom opštih graničnih uslova klizanja drugog reda i pokazano je da ona ima tačno analitičko rešenje. Rešenje je dobijeno tansformacijom zavisno promenljive što je dovelo do mogućnosti analitičkog proračuna raspodele pritiska i masenog protoka u mikroležaju. Na taj način, poređenjem sa datim analitičkim rešenjem omogućena je provera eksperimentalnih i numeričkih rezultata za strujanje gasa u mikroležajima.We derive a general slip-corrected compressible Reynolds lubrication equation, valid for any choice of the slip velocities, and show that it possesses the exact analytical solution. It is obtained by a suitable transformation of the dependent variable, and it yields both the pressure distribution in the bearing and the mass flow rate through it. It can be usefully applied for testing the other, experimental or numerical results obtained under the same or similar physical conditions, against this solution

The exact analytical solution for the gas lubricated bearing in the slip and continuum flow regime

The exact analytical solution for the compressible two-dimensional gas flow in the microbearing is presented. The general slip-corrected Reynolds lubrication equation is derived and it is shown that it possesses an exact analytical solution. It is obtained by a suitable transformation of the independent variable, and it provides the pressure distribution in the bearing as well as the mass flow rate through it. By neglecting the rarefaction effect, this solution is also applicable to the continuum gas flow in the bearing, which also does not exist in the open literature. The obtained analytical solution can be usefully applied for testing the other, experimental or numerical results