Fluid flow Knudsen number

We begin the comparison of experimental data with predictions of the conventional theory for results related to flow of incompressible fluids in smooth micro-channels. For liquid flow in the channels with the hydraulic diameter ranging from 10 m to 10 m the Knudsen number is much smaller than unity. Under these conditions, one might expect a fairly good agreement between the theoretical and experimental results. On the other hand, the existence of discrepancy between those results can be treated as a display of specific features of flow, which were not accounted for by the conventional theory. Bearing in mind these circumstances, we consider such experiments, which were performed under conditions close to those used for the theoretical description of flows in circular, rectangular, and trapezoidal micro-channels. 

For single-phase fluid flow in smooth micro-channels of hydraulic diameter the Poiseuille number is independent of the Reynolds number. For single-phase gas flow in micro-channels of hydraulic diameter 16.6 to 4,010 pm, Knudsen number of Kn = 0.001-0.38, Mach number of Ma = 0.07-0.84, the experimental friction factor agrees quite well with theoretical one predicted for fully developed laminar flow. 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

For all practical purposes, isothermal flows with Ma < 0.2 can be treated as incompressible, i.e., density variations in the fluid around the particle are negligible. Compressibility effects become important as Ma is increased, especially for Ma approaching and in excess of unity. The Knudsen number is defined as the ratio of the molecular mean free path in the fluid to some characteristic particle dimension. For a spherical particle... 

The Navier-Stokes equations are valid when A is much smaller than the characteristic flow dimension L. When this condition is violated, the flow is no longer near equilibrium and the linear relations between stress and rate of strain and the no-slip velocity condition are no longer valid. Similarly, the linear relation between heat flux and temperature gradient and the no-jump temperature condition at a solid-fluid interface are no longer accurate when A is not much smaller than L. The different Knudsen number regimes are delineated in Fig. 2. 

Another study, [17], used helium as their working fluid and carried out the experiments in 51.25 X 1.33 micrometer microchannels. They showed that, as long as the Knudsen number is in the slip flow range, the Navier-Stokes equations are still applicable and the discontinuities at the boundaries need to be represented by the appropriate boundary conditions. They obtained the following formula for the mass flow rate including the slip effects... 

Ft, Thermal accommodation coefficient K, Thermal conductivity Kn, Knudsen number M, Mass of the fluid Ma Mach number m, Mass flow rate... 

The scientific program starts with an introduction and the state-of-the-art review of single-phase forced convection in microchannels. The effects of Brinkman number and Knudsen numbers on heat transfer coefficient is discussed together with flow regimes in microchannel single-phase gaseous fluid flow and flow regimes based on the Knudsen number. In some applications, transient forced convection in microchannels is important. 

In most cases one is interested in fluid flows at scales that are much larger than the distance between the molecules. The value of the molecular mean free path in air at room temperature and 1 atm of pressure is A = 6.7 x 10-8 m and in water A = 2.5 x 10-10 m. When the Knudsen number - defined as the ratio of the molecular mean-free-path to a characteristic length scale of the flow (e.g. the size of the smallest eddies) - is small, the fluid can be described as a continuous medium in motion. In this continuum approximation the flow can be characterized by the velocity field v(x, t) representing the instantaneous velocity of infinitesimal fluid elements at time t and at position x. Fluid elements represent small volumes of fluid that are much smaller than the smallest characteristic scale of the flow, but sufficiently large to contain a large number of molecules so that a well defined local velocity exists and molecular fluctuations can be neglected. 

The measurements by Harley, Pfahler, and Urbanek not only provide a solid basis for modeling fluid flows in small ducts but also raise a question about the nature of that flow at elevated temperatures. The lower-temperature data justify the use of hydrodynamic theory in simple ducts. Whether this will hold in more complex flow structures needs further study. For gas flow in ducts where the Knudsen number is 0.05 or greater, slip flow is observed. Urbanek s data suggest that there may be increased wall interactions as the temperature approaches the boiling point. A more definitive study is needed to clarify this point. 

The standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference comoving with the fluid. When gas flow in microreactors at high temperature or low pressure is considered, this assumption may break down. The principle quantity determining the flow regime of gases and deviations from the standard continuum description is the Knudsen number, defined as... 

The slip flow near the boundary surface can be analyzed based on the type of fluids, i.e., gas and Newtonian and non-Newtonian liquids. The sUp flow in gases has been derived based on Maxwell s kinetic theory. In gases, the concept of mean free path is well defined. Slip flow is observed when characteristic flow length scale is of the order of the mean free path of the gas molecules. An estimate of the mean free path of ideal gas is /m 1/(Vlna p) where p is the gas density (here taken as the number of molecules per unit volume) and a is the molecular diameter. The mean free path / , depends strongly on pressure and temperature due to density variation. Knudsen number is defined as the ratio of the mean free path to the characteristic length scale... 

For micro- and nanoscale gaseous flows with large Knudsen number, say 0.1, the equilibrium assumption inherent in the Navier-Stokes equations becomes invalid, and a discrete approach has to be used. This is also true for liquids, although the Knudsen number is not an appropriate parameter to measure deviation from continuum for liquid flows. Given the observations made from macroscaled devices and the known fluid properties, the Reynolds numbers for the flows in micro- and nanoscaled devices are very small due to the small length scales. 

In order to simulate fluid flow, heat transfer, and other related physical phenomena over various length scales, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to the fluid dynamics research community are governed by the principles of continuum conservation and are expressed in terms of first- or second-order partial differential equations that mathematically represent these principles (within the restrictions of a continuum-based firamework). However, in case the requirements of continuum hypothesis are violated altogether for certain physical problems (for instance, in case of high Knudsen number rarefied gas flows), alternative formulations in terms of the particle-based statistical tools or the atomistic simulation techniques need to be resorted to. In this entry, we shall only focus our attention to situations in which the governing differential equations physically originate out of continuum conservation requirements and can be expressed in the form of a general differential equation that incorporates the unsteady term, the advection term, the diffusion term, and the source term to be elucidated as follows. 

Reynolds number) is in general smaller than unity, and the ratio of the inertial force to surface tension (Weber number) is also small, so that accurate microfluidic metering is somewhat more difficult than metering for macroscopic flows. For gas flows, the effect of the Knudsen number, defined as the ratio of the molecular mean free path to the channel size, is high and cannot be ignored for microscale channels, and hence the fluid metering discussed here will be for liquids only. The Reynolds number Re, the Weber number We, and the Knudsen number Kn are expressed as follows ... 

Knudsen number is the ratio of the mean free path of fluid molecules to a typical dimension of gas flow. 

Knudsen number (Kn) is the ratio of mean free path I of fluid molecules to a t3rpical dimension of gas flow a, i.e., Kn = Ija. Rarefaction parameter 8 is the inverse Knudsen number. Velocity distribution function is defined so that the quantity /(f, r, v) dr dv is the number of particles in the phase volume dr dv near the point (r, v) at the time t. 

The main parameter determining the gas rarefaction is the Knudsen number Kn = Ha, where I is the mean free path of fluid molecules and a is a typical dimension of gas flow. If the Knudsen number is sufficiently small, say Kn < 10 , the Navier-Stokes equations are applied to calculate gas flows. For intermediate and high values of the Knudsen number, the Navier-Stokes equations break down, and the implementation of rarefied gas dynamics methods is necessary. In practical calculations usually the rarefaction parameter defined as the inverse Knudsen number, i.e.. 

Sharipov F, Cumin LMG, Kalempa D (2004) Plane Couette flow of binary gaseous mixture in the whole range of the Knudsen number. Eur J Mech B/Fluids 23 899-906... 

Wang M, Li ZX (2007) An Enskog based Monte Carlo method for high Knudsen number non-ideal gas flows. Comput Fluid 36(8) 1291-1297... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

Matsuda Y, Mori H, Niimi T, Uenishi H, Hirako M (2007) Development of pressure sensitive molecular film applicable to pressure measurement for high Knudsen number flows. Exp Fluids 42 543-550... 

For a Newtonian liquid flow, this assumption is justified if the microchannel has a hydraulic diameter larger than 1 pm. In fact, for liquids the typical mean free path X of molecules under ambient conditions is 0.1-1 nm. Since the fluid velocity tends to evidence a slip at the microchannel walls for Knudsen numbers, defined as Kn = XjD, larger than 0.001, the no-slip botmdary condition has to be abandoned only when the hydraulic diameter of the microchannel becomes less than 1 pm. 

Transport processes involving aerosol particles are frequently classified according to the particle Knudsen number [Ref.5.3, Chap.l]. When a particle is much larger than a gas molecular mean free path (Kn l), the particle is said to be in the continuum regime since the usual hydrodynamic equations for fluid flow pertain. To the extent that it need be addressed directly, this regime is not discussed in this chapter. The principal focus here is upon particles for which Kn > 1. 

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