Slip-flow regime

In order to obtain a qualitative view of how the transition regime differs from the continuum flow or the slip flow regime, it is instructive to consider a system close to thermodynamic equilibrium. In such a system, small deviations from the equilibrium state, described by thermodynamic forces X, cause thermodynamic fluxes J- which are linear functions of the (see, e.g., [15]) ... 

In the case of one-sided illumination of a strongly absorbing sphere, the photophoretic force for the slip-flow regime is given by (Reed, 1977)... 

The continuum flow assumption will only be valid when Kn < 10. As Kn increases, the flow enters the slip flow regime (10 < Kn < 10 ), transition flow regime (10 < Kn < 10), and eventually the free-moleeular flow regime (Kn > 10). These four regimes are illustrated in Figure 1. 

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... 

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... 

In early fhe transition regime, the DSMC method also requires a large number of particles, which makes it expensive in terms of computational time and memory requirements. Therefore, until recently, the advances in the gaseous flow regime in micro channels were in fhe slip flow regime [9]. 

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). 

The present lecture summarizes some of tiie most recent joint research results from tiie cooperation between the Federal University of Rio de Janeiro, Brasil, and tiie University of Miami, USA, on tiie fransient analysis of both fluid flow and heat transfer within microchannels. This collaborative link is a natural extension of a long term cooperation between the two groups, in the context of fimdamental work on transient forced convection, aimed at tiie development of hybrid numerical-analytical techniques and tiie experimental validation of proposed models md methodologies [1- 9]. The motivation of this new phase of tiie cooperation was thus to extend the previously developed hybrid tools to handle both transient flow and transient convection problems in microchannels within the slip flow regime. 

In this section, the results will be presented in tabular and graphical forms, for Nusselt number for both constant wall temperature and constant wall heat flux cases with variable Kn, Br, Pe values to investigate the effects of rarefaction, viscous dissipation, and axial conduction in the slip-flow regime for microtubes. Table 1 presents the effect of rarefaction on laminar flow fully developed Nu values for constant wall temperature (Nut) and constant wall heat flux (Nuq) cases, where viscous dissipation and axial... 

In this lecture, a variety of results for convective heat transfer in microtubes and microchannels in the slip flow regime under different conditions have been presented. Both constant wall temperature and constant wall heat flux cases have been analyzed in microtubes, including the effects of rarefaction, axial conduction, and viscous dissipation. In rough microchannels the abovementioned effects have also been investigated for the constant wall temperature boundary condition. Then, temperature-variable dynamic viscosity and thermal conductivity of the fluid were considered, and the results were compared with constant property results for microchannels, with the effects of rarefaction and viscous dissipation. 

The asymptotic values of the flowrate for duct flows at high Knudsen number are constants depending on the duct aspect ratio. This offers the possibility of obtaining a model for the rarefaction coefficient Cr Kn) and in particular the coefficient a. The objective is to construct a unified expression for a Kn) that represents the transition of a from zero in the slip flow regime to its asymptotic constant value in the free-molecular flow regime. 

The general slip boundary condition gives the correct nondimensional velocity profile, where the normalization is obtained using the local channel averaged velocity. This eliminates the flowrate dependence in modeling the velocity profile. For channel flows, we obtain b = - m the slip flow regime. Evidence based on comparisons of the model with the DSMC and Boltzmann solutions shows that b = - is valid in the entire Knudsen regime. 

Therefore, the unified model employs two empirical parameters a and P) and two known parameters b = -I and cTo- Although this empiricism is not desired, the a value in varies from zero in the slip flow regime to an order-one value of o as oo. Finally, the model is adapted to the finite aspect ratio rectangular ducts using a standard aspect ratio correction given in Eq. (7). 

The applicability of slip flow is not well accepted to date by the academic community. One of the problems is the small length scale of slip flow regime (if present) in comparison to the length scale of the flow or system. The hydrodynamic boundary condition appears to be one of no slip, unless the flow is examined on a length scale comparable to the slip length. Hence, very accurate techniques with high spatial resolution capable of interfacial flow measurements are required to detect the effects of slip. Some of the experimental techniques for quantification of liquid slip are presented in the following sections. 

Rarefied flows in the slip flow regime, rarefaction increases the hydrodynamic and thermal entrance lengths owing to slip at the walls (see Pressure-Driven Single-Phase Gas Flows ). 

Free-molecular regime Internal rarefied gas flows Slip flow regime Transitional regime... 

A moderate gas rarefaction, say 5 < 8 < 100, can be taken into account applying the Navier-Stokes equation with the velocity slip boundary ccmditions. This is the so-called slip flow regime. In practice, the slip boundary conditions mean that the bulk velocity is not equal to zero on the wall, but its tangential component Mj is proportional to its normal gradient... 

---