Knudsen layer

In fhe slip flow regime, fhe Navier-Sfokes equations are applieable exeepf in fhe layer nexf fo fhe surfaee, fhe Knudsen layer as illusfrafed in Figure 2. To use fhe Navier Sfrokes equations fhroughouf... 

The commonly used slip boundary conditions are called Maxwellian boundary conditions [1]. Since they are first order in accuracy, an extended set of boundary conditions was proposed by [2], which can be used in early transition of the slip flow regime. To do so, fhe velocity and temperature gradients at the boundary are written in terms of the Taylor expansion of the gradients within the layer one mean free path away from the boundary (called the Knudsen Layer). 

In the Knudsen layer, the Maxwellian velocity slip boundary condition approximates the true gas velocity at the boundary by the velocity that the molecules would have if a linear velocity gradient existed as shown in figure 1 [18-19], In other words, the magnitude of the slip is calculated from the velocity gradient evaluated at y = X,... 

Evaluating Mwaii = FmMm at the Knudsen layer boundary, X, we obtain... 

R. Kelly and R.W. Dreyfus, On the effect of Knudsen-layer formation on studies of vaporization, sputtering, and desorption. Surf. Sci. 1988, 198, 263-276. 

Slip flow corresponds to 0.01 < Aw < 0.1 (here the no-slip boundary conditions no longer apply and a moving layer known as the Knudsen layer builds up between the bulk fluid and the surface) ... 

Physically, gas flows in the transition regime, where Kn is 0(1), are characterized by the formation of narrow, highly nonequilibrium zones (Knudsen layers) of thickness of the order of the molecular mean free path X the flow structure is then determined by the fast kinetic processes. Moreover, in the case of unsteady flows, an initial Knudsen time interval is of the order Tq = X/v, where v is the molecular velocity. Thus, the Knudsen layer can be computed accurately only by directly solving the Boltzmann equation. 

The Knudsen layer plays a fundamental role in the slip flow regime. This thin layer, between one and two molecular mean free paths in thickness, is a region of local nonequilibrium which is observed in any gas flow near a surface. In nonrarefied flows, the Knudsen layer is too thin for having any significant influence, but in the slip flow regime, it should be taken into account. 

Although the Navier-Stokes equations are not valid in the Knudsen layer, due to a nonlinear stress/strain rate behavior in this small layer [4], their use with appropriate boundary velocity slip... 

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 ... 

A corrective coefficient has been calculated by different authors [6] who have shown that a better prediction of the flow out of the Knudsen layer would be obtained with this corrective coefficient slightly different from unity, unlike as initially proposed by Maxwell. Its value depends on the collision model for example, = 1.11 for a HS model [7]. Equation 12 is called first-order slip boundary condition, because it involves the Knudsen number (9(Kn)) and the first derivative du Jdn ) . 

Lockerby DA, Reese JM, Gallis MA (2005) Capturing the Knudsen layer in continuum-fluid models of nonequilibrium gas flows. AIAA J 43(6) 1391-1393... 

Velocity Slip Boundary Conditions The commonly used slip boxmdary conditions are called Maxwellian boxmdary conditions [5]. The first-order approximation for the velocity slip within the Knudsen layer, which does not take into accoxmt thermal creep, is given by... 

Zhang et al. [14, 15] developed a new slip model that considered the effect of relative position on the slip surface in the Knudsen layer ... 

The large surface-to-volume ratio in microchannels tends to enhance several factors that were neglected in macroscale flow and heat transfer, such as surface frictiOTi, compressibility, surface roughness, viscous, channel surface geometry, surface electrostatic charges, axial wall heat conduction, and measurement errors. Velocity slip flow has been widely studied accurately to model the characteristics of gas flow in the Knudsen layer for aU flow regimes. A review of the slip velocity can be fotmd in Zhang et al. [4]. 

Lilley CR, Sader JE (2008) Velocity profile in the Knudsen layer according to the Boltzmann equation. Proc R Soc A 464 2015-2035... 

Dongari N, Barber RW, Emerson DR, Stefanov SK, Zhang Y, Reese JM (2012) The effect of Knudsen layers on rarefied cylindrical Couette gas flows. Microfiuid Nanofiuid 14 31-43... 

Comubert R, d Humieres D, Levermore D (1991) A Knudsen layer theory for lattice gases. Phys D 47 241-259... 

Zhang Y-H, Gu X-J, Barber RW, Emerson DR (2006) Capturing Knudsen layer phenomena using a lattice Boltzmann method. Phys Rev E 74 1-7... 

Making use of the preceding transient behavior, we construct a large number of steady solutions, from which the behavior of steady evaporation is clarified. That is, we pursue the time development until a uniform state ahead of the Knudsen layer develops enough and confirm that g and h there correspond to those of the Maxwellian if necessary, we introduce the following cut and patch process several times replace the contact layer etc. ahead of the nearly uniform region by a uniform state and pursue the time development. Since we are interested only in a steady solution, we don t have to follow the time development accurately, and the time consuming characteristic coordinate method is unnecessary. Instead, in order to avoid to miss any possible stable steady solution, we try various nonuniform Initial conditions. 

Knudsen layer + contact layer + shock wave,... 

Knudsen layer + expansion wave L + contact layer... 

After each element is separated enough from each other, the Knudsen layer, practically steady, corresponds to the steady solution in Sec. IB, and the uniform regions on both sides of shock wave or expansion wave or contact layer are confirmed numerically to be related by the classical gas dynamic relation of shock wave or simple... 

Various flow problems involving evaporation and condensation phenomena are quite common in ordinary circumstances and have aroused an interest of scientists not only in the field of fluid dynamics but also of kinetic theory. The reason for this is that the ordinary continuum-based fluid dynamics cannot describe qualitatively correctly the process of evaporation and condensation occurring at the interface even in ihe continuum limit because of the existence of a nonequilibrium region, the thickness of which is of the order of the molecular mean free path, in the close vicinity of the interface between the condensed phase and the gas phase. Such a nonequilibrium region is called the Knudsen layer, in which collisions between molecules are not so frequent that the momentum and energy exchanges between the molecules leaving the interface... 

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