Momentum accommodation

When the fractions of molecules reflected specularly and diffusively are known, the slip length can be determined, as shovm by Maxwell. Maxwell introduced a tangential momentum accommodation coefficient defined as... 

The average incident tangential momentum is muh while the average scattered tangential momentum is muf. If the gas molecule equilibrates with the surface and the scattered momentum is zero, we have Knudsen cosine scattering and complete accommodation of the incident gas molecule with the surface. On the other extreme, if specular reflection occurs, the incident momentum is retained upon scattering and mut = muf. The momentum accommodation coefficient, / is introduced to describe the type of scattering that does occur, and it is defined by... 

The pressure dependence of effective viscosity obviously depends upon the value of the momentum accommodation coefficient. Momentum accommodation data are relatively rare, but some representative data are given in Table 1. Note that all values are relatively close to unity. Because of this observation, momentum accommodation coefficients are normally assumed to be unity in applications... 

The reflecting boundary condition, due to Maxwell, can be written in terms of the momentum accommodation coefficient ac as (4)... 

As mentioned earlier, the factor a is the thermal accommodation coefficient and am the momentum accommodation or reflection coefficient. From the data of Rosenblatt and LaMer (1946), Schmitt (1959), and Keng and Orr (1966), as a first approximation a value of 1.25 seems reasonable for Cm, whereas for Ct a value of 2 is a good approximation (Brock, 1962b). These numbers then imply values of 0.89 for am and 0.97 for at. 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

In this relation, Fm, the tangential momentum accommodation coefficient, is a function of the interaction between gas molecules and the surface. If the surface is smooth and reflects the molecules specularly, F , will be zero. For diffuse reflections F =l. This means that all the tangential momentum is lost at the wall. Diffuse reflection results from the penetration of the molecules into interstices in the surface where multiple impacts occur before the molecules depart. 

Accommodation coefficients may be significantly different from unity for light atoms and closer to unity for heavy atoms. As shown experimentally in [11], Fm values for slip flow of argon, nitrogen, and carbon dioxide fell between 0.75 and 0.85. The results also showed that Fm is independent of pressure. Their channels are not isolated from contamination to obtain realistic values. Experimental mass flow rate values agree well with the analjdical predictions using the slip boundary condition and experimentally determined momentum accommodation coefficients. 

Heat convection for gaseous flow in a circular tube in the slip flow regime with uniform temperature boundary condition was solved in [23]. The effects of the rarefaction and surface accommodation coefficients were considered. They defined a fictitious extrapolated boundary where the fluid velocity does not slip by scaling the velocity profile with a new variable, the shp radius, pj = l/(l + 4p.,Kn), where is a function of the momentum accommodation coefficient, and defined as p, =(2-F,j,)/F,j,. Therefore, the velocity profile is converted to the one used for the... 

More reeently, [26] has eonfirmed the need to include the second order slip condition at higher Kn number values. Their work was both theoretical and experimental using nitrogen and helium in a silicon channels. They used the second order slip approximation to obtain the equation for the volumetric flow rate and related it to the ratio of inlet to outlet pressure. It was shown that when using the Navier-Stokes equation, the boundary conditions must be modified to include second order slip terms as the Knudsen number increases. They also studied in depth the accommodation coefficient Fv and verified the need for further study. It was shown that as the Knudsen number increases, the momentum accommodation value deviates further and further from unity for instance Kn -0.5 yields Fv 0.8 for helium. The values found for nitrogen were quite similar. The measurements agreed with past studies such as [11] for lower Kn. 

Also, the tangential momentum accommodation coefficients obtained experimentally varied in range from 0.3 to 0.7, quite different from the conventional value of 1. This value range agreed with the earlier analytical work [11]. 

Fm, Tangential momentum accommodation coefficient K Molecular mean free path... 

Arkilic, E. B., Breuer, K. S., Sehmidt, M. A., (2001) Mass Flow and Tangential Momentum Accommodation in Silicon Micromachined Channels, Journal of Fluid Mechanics, Vol.437, pp.29-43. 

In Eq. (1), Fm is the momentum accommodation factor and has a value close to unity for the gas-solid couples used most commonly in engineering, and is also taken so in this work. In Eq. (2), Ts is the temperature of the fluid molecules at the wall, T is the wall temperature, y is the ratio of the specific heats of the fluid, and Ft is the thermal accommodation factor. Ft may take a value in the range O.O-I.O, depending on the gas and solid surface, the gas temperature and pressure, the temperature difference between the gas and the surface, and is determined experimentally. 

E.B. Arkilic, K.S. Breuer, and M.A. SchmidL Mass Flow and Tangential Momentum Accommodation in Silicon Micromachined Channels, J. Fluid Mech., V. 437, pp. 29-43 (2001). 

Gp for the CL scattering kernel obtained in [5] are given in Table 2. In all regimes the influence of the energy accommodation coefficient on the flow rate is weak, while the momentum accommodation coefficient ott affects significantly the flow rate in the transition (8 = 1) and near the free molecular (5 = 0.01) regimes. 

For an isothermal wall, when a simulated particle collides with the wall, a diffuse-reflection model is used to determine the result of reflection, whereby the outgoing velocity is randomly assigned according to a half-range Maxwellian distribution determined by the wall temperature. This is also known as the full thermal and momentum accommodation method. 

Nanoscale gas transport Shear-driven flow Surface force effects Tangential momentum accommodation coefficient... 

Overall the results show that the wall force field penetration depth is an additional length scale for gas flows in nano-channels, breaking dynamic similarity between rarefied and nanoscale gas flows solely based on the Knudsen and Mach numbers. Hence, one should define a new dimensionless parameter as the ratio of the force field penetration depth to the characteristic channel dimension, where wall effects cannot be neglected for large values of this dimensionless parameter. Additionally, the calculated tangential momentum accommodation coefficients for a specific gas-surface couple were found to be constant regardless of different base pressure, channel height, wall velocity, and Knudsen number. Results of different gas-surface couples reveal that TMAC is only dependent on the gas-surface couple properties and independent of the Knudsen number. 

In this equation, a is the tangential momentum accommodation coefficient, equal to unity for perfectly diffuse molecular reflection and zero for purely specular reflection. In Maxwell s model, MsUp overestimates the real velocity at the wall but leads to a rather good prediction of the velocity out of the Knudsen layer, as represented in Fig. 2. After non-dimensionalization with the characteristic length L, a reference velocity uo, and a reference temperature Tq, Eq. 10 is written as follows ... 

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