Continuum hypothesis

A dilute gas mixture is assumed to behave as a continuum when the mean free path of the molecules is much smaller than the characteristic dimensions of the problem geometry[5,42,43, 68, 129, 154, 164]. A relevant dimensionless group of variables, the Knudsen number, Kn, is defined as  

By use of the typical numerical value for the mean free path (2.611) for a gas at temperature 300 K and pressure 101325 Pa, this condition indicates that the continuum assumption is valid provided that the characteristic dimension of the apparatus is L 0.001 cm. Nevertheless, it is noted that at low gas pressures, say 10 Pa, the mean free path is increased and might be comparable with the characteristic dimensions of the apparatus. In particular, for low pressures and small characteristic dimensions we might enter a flow regime where the continuum assumption cannot be justified. 

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Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by colhsions between molecules. An approximate expression based on Brown, et al. J. Appl. Phys., 17, 802-813 [1946]) for the mean free path is... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space 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 co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Continuous web dryers, 9 119-120 Continuous weighing, 26 248 Continuum hypothesis, of flow phenomena, 11 735-736... 

Two examples where the continuum hypothesis may be invalid are low pressure gas flow in which the mean free path may be comparable to a linear dimension of the equipment, and high speed gas flow when large changes of properties occur across a (very thin) shock wave. 

The scale in chaotic laminar mixing goes down from the machine scale to a scale where the continuum hypothesis breaks down and phenomena are dominated by physical effects, due to intermolecular forces, such as van der Waals. Danescu and Zumbrunnen (21) and Zumbrunnen and Chibber (22) took advantage of this and devised an ingenious device to create controlled three-dimensional chaotic flows, with which they were able to tailor the morphology and properties of blend films and composites. 

More or less as a spin-off result of the foregoing analysis determining the transport coefficients, a rigorous procedure deriving the governing equations of fluid dynamics from first principles was established. It is stressed that in classical fluid dynamics the continuum hypothesis is used so that the governing... 

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

The development of convenient and usable forms of the basic conservation principles and the role of the constitutive models and boundary conditions in a continuum mechanics framework occupy the remaining sections of this chapter. In the remainder of this section, we discuss the foundations and consequences of the continuum hypothesis in more detail. 

In other words, it must be possible to choose an averaging volume that is arbitrarily small compared with the macroscale L while still remaining very much larger than the microscale 8. Although the condition (2 2) will always be sufficient for validity of the continuum hypothesis, it is unnecessarily conservative because of the use of volume averaging in the definition (2-1) rather than the more fundamental ensemble average definition of macroscopic variables. Nevertheless, the preceding discussion is adequate for our present purposes. 

A second similar consequence of the continuum hypothesis is an uncertainty in the boundary conditions to be used in conjunction with the resulting equations for motion and heat transfer. With the continuum hypothesis adopted, the conservation principles of classical physics, listed earlier, will be shown to provide a set of so-called field equations for molecular average variables such as the continuum point velocity u. To solve these equations, however, the values of these variables or their derivatives must be specified at the boundaries of the fluid domain. These boundaries may be solid surfaces, the phase boundary between a liquid and a gas, or the phase boundary between two liquids. In any case, when viewed on the molecular scale, the boundaries are seen to be regions of rapid but continuous variation in fluid properties such as number density. Thus, in a molecular theory, boundary conditions would not be necessary. When viewed with the much coarser resolution of the macroscopic or continuum description, on the other hand, these local variations of density (and other molecular variables) can be distinguished only as discontinuities, and the continuum (or molecular average) variables such as u appear to vary smoothly on the scale L, right up to the boundary where some boundary condition is applied. 

Once we adopt the continuum hypothesis and choose to describe fluid motions and heat transfer processes from a macroscopic point of view, we derive the governing equations by invoking the familiar conservation principles of classical continuum physics. These are conservation of mass and energy, plus Newton s second and third laws of classical mechanics. 

This all relates to the fact that even though there are an infinite number of rational and irrational numbers, the infinite number of irrationals is in some sense greater than the infinite number of rationals. To denote this difference, mathematicians refer to the infinity of rationals as Nq the infinity of irrationals as C, which stands for continuum. There is a simple relationship between C and Nq- K C =2 o. The continuum hypothesis states that C = N j however, the question of whether or not C truly equals N, is considered undecidable. In other words, great mathematicians such as Kurt Godel proved that the hypothesis was a consistent assumption in one branch of mathematics. However, mathematician Paul Cohen proved that it was also consistent to assume the continuum hypothesis is false ... 

Incidentally, 10 would belong to the same level of infinity. You should be able to convince yourself from the diagonal argument used above that the real numbers can be represented as a power set of the counting numbers. Therefore, we can set c = Ki. The continuum hypothesis presumes that there is no intermediate cardinality between bfo and Hi. Surprisingly, the ttuth this proposition is undecidable, neither it nor its negation contradicts the basic assumptions of Zermelo-Fraenkel set theory, on which the number system is based. 

The next higher transfinite number, assuming the continuum hypothesis is tme, would be the cardinality of the power set of h5i, namely H2 = This might represent the totality of possible functions or of geometrical curves. 

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. 

The examples of non-Newtonian microchannel flows cited in the present article so far inherently assume that the continuum hypothesis is not disobeyed, so far as the description of the basic governing equations is concerned. This, however, ceases to be a valid consideration in certain fluidic devices, in which the characteristic system length scales are of the same order as that of the size of the macromolecules being transported. Fan et al. [10], in a related study, used the concept of finitely extended nonlinear elastic (FENE) chains to model the DNA molecules and employed the dissipative particle dynamics (DPD) approach to simulate the underlying flow behavior in some such representative cases. From their results, it was revealed that simple DPD fluids essentially behave as Newtonian fluids in Poiseuille flows. However, the velocity profiles of FENE... 

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