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Functions
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
Hilbert was one of the earliest champions of Cantor's work on the infinite. At the height of the controversy surrounding Cantor's theory, he wrote: ‘No one shall drive us from the paradise which Cantor has created for us'. cannot be proved to be true either using the usual axioms about sets. Therefore the truth or falsity of the continuum hypothesis is undecidable in axiomatic set theory. One can therefore choose whether to assume its truth or its falsity. More precisely, we can choose to include the continuum hypothesis or its negation as an additional axiom of set theory. This seems somewhat paradoxical. It means that there are two different versions of set theory—one where the continuum hypothesis is ‘true' and one where it is ‘false'. (In fact, it is now known that there are other statements of this type so there are many different versions of set theory! Fortunately they only differ in rather esoteric aspects -the properties of sets developed in chapter 3 are common to all the different set theories.)
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The continuum hypothesis conjectures that there does not exist a set with a cardinal number between ℵ0 $ \aleph _0 $ and c $ \mathsf c $ . This has led to an occasional use of the notation c=ℵ1 $ \mathsf c = \aleph _1 $ .
Elements of Continuum Mechanics
Published in Clement Kleinstreuer, Biofluid Dynamics, 2016
Approximate Methods (Kn≤0.5). In micro-scales the characteristic dimension of the flow conduits are comparable to the mean-free-path of the gas media they operate in. Under such conditions the “continuum hypothesis” may break down. Constitutive laws that determine stress tensors and heat flux vectors for continuum flows have to be modified in order to incorporate the rarefaction effects. The very well known “no-slip” boundary conditions for velocity and temperature of the fluid on the walls are subject to modifications in order to incorporate the reduction of momentum and energy exchange of the molecules with the surroundings. Deviation from continuum hypothesis is identified by the Knudsen number, which is the ratio of the mean-free-path of the molecules to a characteristic length scale, i.e., Kn=λ/L. This length scale L should be chosen in order to include the gradients of density, velocity, and temperature within the flow domain. For example for external flows boundary layer thickness, and for internal fully developed flows channel half thickness should be used as the characteristic length scale. According to the Knudsen number the flow can be divided into various regimes. These are: continuum, slip, transition, and free-molecular flow regimes.
Novel CFD-DEM approach for analyzing spherical and non-spherical shape particles in spouted fluidized bed
Published in Particulate Science and Technology, 2023
CFD consists of computational strategies to solve governing equations dictating fluid flow. As shown in Equations (1) and (2), the fundamental equations of mass and momentum are solved considering the continuum hypothesis. The SIMPLE algorithm in Fluent 2021R1 was employed to solve the Navier-Stokes equation. The k-epsilon turbulence model was used to consider turbulence because of the high density of solid particulates. where is fluid volume fraction, is the fluid density, is the fluid phase velocity vector, is stress tensor, represents momentum source term, and is shared pressure. Interested readers can refer in detail about multiphase modeling (Yadigaroglu and Hewitt 2018).
Transient multiphysics coupled model for multiscale droplet condensation out of moist air
Published in Numerical Heat Transfer, Part A: Applications, 2023
Shao-Fei Zheng, Zi-Yi Wu, Yi-Ying Gao, Yan-Ru Yang, Bengt Sundén, Xiao-Dong Wang
The transient multiphysics coupled model is used to numerically study the growth dynamic of an individual droplet in a moist air environment with the total pressure pma = 1 bar and the temperature Tma = 303.15 K. The relative humidity of moist air ranges from 60% to 94% with the NCG mass concentration varying from 97.4% to 95.8%. Depending on different dew points of moist air, the surface subcooling varies from 2 K to 6 K. Various functional surfaces, from hydrophilic surfaces to superhydrophobic surfaces, have been widely utilized to promote dropwise condensation [1]. In this work, three representative contact angles (θ = 60°, 90°, 120°) are considered. During dropwise condensation, the minimum droplet size is commonly at the nucleation nanometer. Considering the limitation of the continuum hypothesis, a minimum droplet radius of 10 μm is used in this work. The gravity-driven droplet departure is usually located at the millimeter scale [1]. Thus, this work considers a maximum droplet radius of 1000 μm. Considering the dynamic and coupling characteristics for droplet condensation in the NCG environment, the condensation behaviors are firstly analyzed under different working conditions. Subsequently, a pure steam ambient and the pure conduction regime within the droplet are investigated to better understand the dynamic and coupling effects.
Estimation of characteristic vortex structures in complex flow
Published in Journal of Turbulence, 2021
Kaustav Chaudhury, Chandranath Banerjee, Swapnil Urankar
Perhaps, the most popular approach for defining a vortex region is based on the velocity gradient tensor (VGT). Note that the VGT is well defined anywhere in the flow field, provided the continuum hypothesis is obeyed. The VGT based analysis of motion near a critical point (location with zero velocity and indeterminant slopes of the streamlines) [1, 3] transforms the problem into an eigenvalue problem. The eigenvalues and the invariants of the VGT may be used to describe the swirling nature of the trajectories and the vortical structures [1, 3, 5]. For example, the second invariant of VGT, Q, defines a vortex region where Q>0. However, this description undermines the zones with combined rotation and straining [5]. The discriminant, Δ, of the characteristic equation for the eigenvalue problem of VGT, can differentiate regions with pure straining without rotation () and the regions with rotation with/without straining (). However, none of these parameters provide a quantitative assessment of the swirling motion near the critical point.