In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Kinetic Theory and Fluid Dynamics

THE difficulty of reconciling line spectra with the kinetic theory of gases, has been referred to by Prof. Fitzgerald (NATURE, January 3, p. 221). The following considerations show that it is possible under certain suppositions to have a number of spectral rays with a very restricted number of degrees of freedom. Most of us, I believe, now accept a definite atomic charge of electricity, and if each charge is imagined to be capable of moving along the surface of an atom, it would represent two degrees of freedom. If a molecule is capable of sending out a homogeneous vibration, it means that there must be a definite position of equilibrium of the “electron.” If there are several such positions, the vibrations may take place in several periods. Any one molecule may perform for a certain time a simple periodic oscillation about one position of equilibrium, and owing to some impact the electron may be knocked over into a new position. The vibrations under these circumstances would not be quite homogeneous, but if the electron oscillates about any one position sufficiently long to perform a few thousand oscillations, we should hardly notice the want of homogeneity. Each electron at a given time would only send out vibrations which in our instruments would appear as homogeneous. Each molecule could thus successively give rise to a number of spectral rays, and at any one time the electron in the different molecules would, by the laws of probability, be distributed over all possible positions of equilibrium, so that we should always see all the vibrations which any one molecule of the gas is capable of sending out. The probability of an electron oscillating about one of its positions of equilibrium need not be the same in all cases. Hence a line may be weak not because the vibration has a smaller amplitude, but because fewer molecules give rise to it. The fact that the vibrations of a gas are not quite homogeneous, is borne out by experiment. If impacts become more frequent by increased pressure, we should expect from the above views that the time during which an electron performs a certain oscillation is shortened; hence the line should widen, which is the case. I have spoken, for the sake of simplicity, as if an electron vibrating about one position of equilibrium could only do so in one period. If the forces called into play, by a displacement, depend on the direction of the displacement, there would be two possible frequencies. If the surface is nearly symmetrical, we should have double lines.

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The Kinetic Theory of Gases

Effective diffusivity is the most important key parameter needed in the analysis, design and optimization of heat and mass transfer during food drying process. Generally, two types of effective diffusivities are used to develop the mathematical modelling of food drying namely, moisture dependent effective diffusivity (MDED) and temperature dependent effective diffusivity (TDED). However, no study has extensively investigated which effective diffusivity is more accurate in predicting drying kinetics. The main goal of this study is to determine the appropriate effective diffusivity for predicting the drying kinetics. Drying models were developed for different fruits and vegetables based on moisture dependent and temperature dependent effective diffusivities. COMSOL Multiphysics, a finite element based engineering simulation software is used to solve the coupled heat and mass transfer equations. 3D moisture profiles were developed to investigate the spatial moisture distribution during drying. Extensive experimental investigation on five types of fruit and vegetables were conducted and results were compared with the simulated results. The experiments were repeated three times, and the average of the moisture content at each value was used for constructing the drying curves. Close agreement between experimental and simulated results validates the models developed. It was observed that the moisture profile and temperature profile in case of MDED were more closely fitted with the experimental results. For all fruits and vegetables, the moisture ratio with MDED was significantly lower than moisture ratio with TDED. This finding confirms that the MDED is more accurate for predicting kinetics in food drying. Moreover, the moisture ratio of apple was lowest whereas pear showed the highest moisture ratio. On the hand, carrot showed a considerably lower moisture ratio compared to potato.

Determination of appropriate effective diffusivity for different food materials

Acoustic streaming can be generated in microchannels by low-frequency acoustic transducer in the vicinity of sharp structures. Close to the tip, the strong curvature induces bent trajectories on the time-periodic acoustic flow, locally enhancing the streaming-generating force. In this study, we investigate the influence of the sharp structure and vibration velocity on the streaming flow. The vibration velocities are characterized by directly visualizing the displacement of tracing particles and the generated acoustic streaming is observed using particle image velocimetry, under various operating conditions. By measuring the concentration of a fluorescence dye, we evaluate the mixing performance for different values of tip angle, vibration amplitude, and flow rate through the microchannel. Our results confirm that intense streaming is generated under low-frequency (2.5 kHz) acoustic condition when the local curvature of the boundary is close to or smaller than the viscous boundary-layer thickness. It is shown that the sharpest the edge tip, the largest the vortices size and the spatial extent of the induced streaming, therefore greatly enhancing the mixing between two miscible liquids. The mixing index, linearly characterizing the mixing degree between 1 (totally separated) and 0 (perfectly mixed), jumps from 0.73 (without acoustic excitation) to 0.38 (with acoustic excitation), resulting in a highly mixed homogeneous fluid just after the sharp edge. This emphasizes the promising potential of acoustic streaming to enhance mass transfer inside microchannels which is usually limited by the laminar flow conditions.

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Acoustic streaming near a sharp structure and its mixing performance characterization

Preface Boltzmann Equation Highly Rarefied Gas: Free Molecular Gas and Its Correction Slightly Rarefied Gas: Asymptotic Theory of the Boltzmann System for Small Knudsen Numbers Simple Flows Flows Induced by Temperature Fields Flows with Evaporation and Condensation Bifurcation in the Half-Space Problem of Evaporation and Condensation Ghost Effect and Bifurcation I: Benard and Taylor-Couette Problems Ghost Effect and Bifurcation II: Ghost Effect of Infinitesimal Curvature and Bifurcation of the Plane Couette Flow Appendix A. Supplement to the Boltzmann Equation Appendix B. Methods of Solution Appendix C. Some Data Bibliography List of Symbols Index

Molecular gas dynamics : theory, techniques, and applications

The Knudsen pump (KP) is a kind of micro-pump that can form thermally induced flows induced by temperature fields in rarefied gas environments. It has the advantages of having no moving parts, simple structure, easy construction and extension, a wide range of energy sources, and low energy consumption. With the development of Micro/Nano Electro Mechanical Systems (MEMS/NEMS), extensive studies have been conducted on KPs, and the applications of KPs have widened. In order to obtain efficient flow fields in KPs, it is necessary to adopt modern computational methods for simulation and analysis. In many circumstances, the simulation and experimental results have good agreement. However, there seems to be no comprehensive review on KPs at present. In this paper, KPs are first defined and classified according to the flow mechanisms of the thermally induced flows. Then, the three aspects of configurations, performance, and applications of KPs in the current state of research are reviewed and analyzed. Finally, the current problems of KP are discussed, and some suggestions are provided for future research and applications. Knudsen pumps are micro-sized, non-mechanical gas pumps that can be useful in a range of biomedical and mechanical fields. Zhang and his team from China’s North-eastern University analyzed Knudsen pumps’ configurations, performance, applications, and current problems. In Knudsen pumps, gas flow is induced by temperature gradients along the walls of the device. They benefit from low energy consumption and ease of manufacture. Different configurations, realized through the placement of variably heated walls, offer multiple mechanisms to transport rarefied gases. Knudsen pumps can be used, amongst other applications, to generate a vacuum, deliver fluids, and separate gases. Micro-channels play a key role in Knudsen pump efficiency and gas flow operation; however, there is a lack of research into efficiency which the authors say should be addressed in future research.

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Knudsen pumps: a review

Sensitivity analysis of intermittent microwave convective drying based on multiphase porous media models

Numerical simulation of thermal edge flow in ratchet-like periodically patterned micro-channels

A Knudsen pump operates under the thermal transpiration effect or the thermal edge effect on the micro-scale. Due to the uneven temperature distribution of the walls in the channel axis direction or the constant temperature of the tips on the walls, directional thermally-induced flow is generated. In this paper the Direct Simulation Monte Carlo (DSMC) method is applied for N2–O2 gas mixtures in the ratios of 4:1, 1:1, and 1:4 with different Knudsen numbers in a classic rectangular Knudsen pump to study the flow characteristics of the gas mixtures in the pump. The results show that the changing in the gas physical properties does not affect the distribution of the velocity field, temperature fields, or other fields in the Knudsen pump. The thermal creep effect is related to the molecular mass of the gas. Even in N2 and O2 gas mixtures with similar molecular masses, N2 can be also found to have a stronger thermal creep effect. Moreover, the lighter molecular weight gas (N2) can effectively promote the motion of the heavier gas (O2).

Study of Flow Characteristics of Gas Mixtures in a Rectangular Knudsen Pump.

Knudsen force generated by thermally driven gas flow in a microscale structure has been used for gas detection and has shown immeasurable potential in the field of microelectromechanical system (MEMS) gas sensors due to its novel sensing characteristics. In this article, the performances of three kinds of Knudsen force gas sensors with improved isosceles triangular shuttle arm structures were studied. In the first design, the top side and right side lengths were equal; in the second, the top side and bottom side lengths were equal; and for the third, the bottom side and right side lengths were equal. A detailed investigation including gas flow, thermal characteristics, Knudsen force, and coupling effects between the shuttle-heater pairs was conducted using the direct simulation Monte Carlo (DSMC) method and the main mechanisms for gas flow presented were almost the same in this work. However, the second design returned the highest Knudsen force performance. The value increased by 42.9% (P = 387 Pa) compared to the Knudsen force of the original square shuttle arm. The results also demonstrate that the coupling effects become weak toward the right with an increase in the number of shuttle-heater pairs.

https://www.mdpi.com/2072-666X/11/7/634/pdf

Xiaowei Wang

Papers

Knudsen pumps: a review

Sensitivity analysis of intermittent microwave convective drying based on multiphase porous media models

Numerical simulation of thermal edge flow in ratchet-like periodically patterned micro-channels

Study of Flow Characteristics of Gas Mixtures in a Rectangular Knudsen Pump.

Impact of Improved Design on Knudsen Force for Micro Gas Sensor