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This paper presents an analytical solution for steady rarefied compressible viscous gas flow in the microchannels and nanochannels with constant or slowly varying cross section. It covers both all gas rarefaction regimes (from continuum to free molecular gas flow) and all Mach number regimes (from subsonic to supersonic). The solutions for the velocity and pressure distribution in the channels is obtained by the macroscopic approach, using the one-dimensional model of continuum equations. A specially modeled friction factor is attained by an approach that includes both the general velocity slip boundary condition, and the dynamic viscosity generalized by the rarefaction correction parameter. This method spreads the application of the solution to the entire range of Knudsen numbers. Moreover, inclusion of the inertia effect into the governing equations allows the application of the solution to both subsonic and supersonic gas flows. The presented solution confirms the existence of the Knudsen minimum in the diverging, converging and microchannels and nanochannels with constant cross section.

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.

The analytical solution for steady viscous pressure-driven compressible isothermal gas flow through micro- and nanochannels with variable cross section for all Knudsen and all Mach number values is presented in this paper. The continuum one-dimensional governing equations are solved using the friction factor that is established in a special way to provide solutions for mass flow rate, pressure, and velocity distribution through the microchannels and nanochannels in the entire rarefaction regime. The friction factor, defined by the general boundary condition and generalized diffusion coefficient proposed by Beskok and Karniadakis (1999, "A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales," J. Microscale Thermophys. Eng., 3, pp. 43-77), spreads the solution application to all rarefaction regimes from continuum to free molecular flow. The correlation between the product of friction factor and Reynolds number (Poiseuille number) and Knudsen number is established explicitly in the paper. Moreover, the obtained solution includes the inertia effect, which allows the application of the solution to both subsonic and supersonic gas flows, which was not shown earlier. The presented solution confirms the existence of the Knudsen minimum in the diverging, converging, and microchannels and nanochannels with constant cross section. The proposed solution is verified by comparison with experimental, analytical, and numerical results available in literature.

In this paper, pressure-driven gas flow through a microtube with constant wall temperature is considered. The ratio of the molecular mean free path and the diameter of the microtube cannot be negligible. Therefore, the gas rarefaction is taken into account. A solution is obtained for subsonic as well as slip and continuum gas flow. Velocity, pressure, and temperature fields are analytically attained by macroscopic approach, using continuity, Navier-Stokes, and energy equations, with the first order boundary conditions for velocity and temperature. Characteristic variables are expressed in the form of perturbation series. The first approximation stands for solution to the continuum flow. The second one reveals the effects of gas rarefaction, inertia, and dissipation. Solutions for compressible and incompressible gas flow are presented and compared with the available results from the literature. A good matching has been achieved. This enables using proposed method for solving other microtube gas flows, which are common in various fields of engineering, biomedicine, pharmacy, etc. The main contribution of this paper is the integral treatment of several important effects such as rarefaction, compressibility, temperature field variability, inertia, and viscous dissipation in the presented solutions. Since the solutions are analytical, they are useful and easily applicable.

A steady compressible isothermal slip gas flow in a microtube is analyzed for low Reynolds numbers. The microtube is with varying cross-section, which from the standpoint of geometry gives three considered cases: convergent microtube, divergent microtube and microtube with constant radius. The gas flow is caused by the pressure difference between the inlet and the outlet cross-section. The solution for pressure and mass flow rate is obtained by macroscopic approach from Navier-Stokes equations with the second order velocity slip boundary condition. Analysis of the order of dimensionless terms in governing equations is possible with the usage of exact relation between Reynolds, Mach and Knudsen numbers. The solution procedure for this flow model is based on perturbation approach, where significant variables are assumed in the form of perturbation series. The first approximation represents the solution for the continuum flow conditions, and the second approximation incorporates the effect of gas rarefaction. The obtained solution is compared with verified results of numerical experiment found in literature and good agreement is achieved. Hence, the reliability of obtained solution and presented method is confirmed.

In this paper the authors considered compressible isothermal steady subsonic slip gas flow through microtube at low Reynolds numbers. The dimensions of the components in microelectromechanical systems (MEMS) are very small (the order of magnitude is 1 µm), and because of that the effect of gas rarefaction cannot be neglected, so it should be taken into account. In the analyzed problem gas flow occurs due to the pressure difference at the inlet and outlet section of the circular microtube, with the variable cross-section (convergent case, divergent case, case with constant radius). The solving approach in the paper is macroscopic, so the fluid flow is defined by the continuity and Navier-Stokes equations. The rarefaction effect is taken into account by the velocity slip boundary condition at the walls. The solution process is based on a perturbation analysis where the order of the terms in the governing equations and boundary conditions is determined by using the exact relation between Mach, Reynolds and Knudsen numbers. Therefore, with this problem-solving procedure, two approximations for the pressure and velocity fields are obtained. The first approximation of the characteristic variables corresponds to the conditions of continuum flow, while the second approximation shows the effects of rarefied gas flow.

U radu se proučava osnosimetrično, izotermsko, stišljivo strujanje gasa sa klizanjem, pri malim vrednostima Rejnoldsovog broja. Problem je rešen primenom jednačine kontinuiteta i Navije-Stoksovih jednačina, zajedno sa Maksvelovim graničnim uslovom prvog reda. Analitički rezultati su dobijeni primenom perturbacione metode. Dobijena rešenja se dobro slažu sa poznatim eksperimentalnim rezultatima drugih autora., Low Reynolds number, axisymmetric, isothermal, compressible, slip gas flow in microtubes is investigated in this paper. The problem is solved by the continuum equations, continuity and Navier-Stokes, along with Maxwell first order boundary condition. The analytical results are obtained by perturbation method. The solutions show a good agreement with experimental results.

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 analytical solution for the two-dimensional non-isothermal compressible gas flow in a slider micro-bearing is presented in this paper. It is shear-driven subsonic slip gas flow. In view of the fact that a fluid flow in a micro system is rarefied, the model takes into account this influence. The slip gas flow is defined by the continuity, Navier-Stokes and energy continuum equations, along with the velocity slip and the temperature jump Maxwell-Smoluchowski first order boundary conditions. Knudsen number is in the range of 0-10-1, which corresponds to the continuum and slip gas flow. The analytical solutions of the pressure, velocity and temperature are obtained. The results are of the order of Navier-Stokes equations, i.e., O(Kn) and they agree well with those found in the open literature. The solutions enable analysis of the relevant parameters influences on the load carrying capacity of the micro-bearing. Hence, the influences of the flow conditions expressed by Mach, Reynolds and Knudsen numbers, the ratio of the inlet to outlet heights, bearing number, as well as temperature field are investigated. Moreover, obtained analytical solutions comprise second order effects for gas flows in micro-systems such as inertia and dissipation which enabled the assessment of their impact on the load carrying capacity of the micro-bearing, i.e. estimation of the calculations safety when these effects are neglected.

A non-isothermal pressure-driven compressible subsonic slip gas flow in microchannels with variable cross section (convergent, divergent and constant high channel) is analyzed in this paper analytically. Since the flows correspond to microspaces, the rarefaction effect must be taken into account. Solutions are obtained for gas flow with equal temperatures of the walls. The slip gas flow is defined by macroscopic approach, using the continuity, Navier-Stokes and energy continuum equations, along with the velocity slip and temperature jump Maxwell-Smoluchowski first order boundary conditions. The Knudsen number is supposed for the slip gas flow, and Mach number for the subsonic flow. Fully developed flow is considered, which assumes a slow variation of all physical quantities in the streamwise direction. These assumptions, along with the known relationship between the Mach, Reynolds and Knudsen numbers, are used to derive a precise estimation of each term's contribution in the non-dimensional governing equations, as well as in the boundary conditions. This evaluation also makes a distinction between low and moderately high Reynolds numbers flows, which are solved separately. The solutions correspond to the slip and continuum flows. Finally, the analytical solutions for the pressure, velocity , and temperature are obtained. The results are in a good agreement with those obtained by some experiments found in the open literature.

A steady plane subsonic compressible non-isothermal Couette gas flow is analyzed for moderately high and low Reynolds numbers. The flow channel is formed by two plates in relative motion. Two cases are considered: (a) isothermal walls where the temperatures of the plates are equal and constant and (b) with constant but different plate temperatures. The Knudsen number is Kn a (c) 1/2 0.1, which corresponds to the slip and continuum flow. 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. 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. In addition, a numerical solution of the problems is given to confirm the accuracy of the analytical results. The exact analytical solution, for constant viscosity and conductivity is found for the isothermal walls case as well. It is shown that it is entirely a substitution to the exact numerical solution for the isothermal walls case.

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.