Momentum accommodation coefficient

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)... 

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... 

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... 

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. 

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 ... 

Prabha SK, Sathian SP (2012) Molecular-dynamics study of Poiseuille fiow in a nanochannel and calculation of energy and momentum accommodation coefficients. Phy Rev E 85 1-10... 

Here a22 is the tangential momentum accommodation coefficient, a2g and are "second-order" coefficients, is the Chapman-Enskog first-order approximation to the viscosity coefficient. Equation (2.68) is, of course, valid for general gas-surface scattering kernels (2.49). Equation (2.68) reduces approximately to... 

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