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Thermophoresis of a spherical particle : modelling through moment-based, macroscopic transport equations
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Padrino-Inciarte, Juan C., Sprittles, James E. and Lockerby, Duncan A. (2019) Thermophoresis of a spherical particle : modelling through moment-based, macroscopic transport equations. Journal of Fluid Mechanics, 862 . pp. 312-347. doi:10.1017/jfm.2018.907 ISSN 0022-1120.
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Official URL: https://doi.org/10.1017/jfm.2018.907
Abstract
We consider the linearized form of the regularized 13-moment equations (R13) to model the slow, steady gas dynamics surrounding a rigid, heat-conducting sphere when a uniform temperature gradient is imposed far from the sphere and the gas is in a state of rarefaction. Under these conditions, the phenomenon of thermophoresis, characterized by forces on the solid surfaces, occurs. The R13 equations, derived from the Boltzmann equation using the moment method, provide closure to the mass, momentum, and energy conservation laws in the form of constitutive, transport equations for the stress and heat flux that extend the Navier-Stokes-Fourier model to include non-equilibrium effects. We obtain analytical solutions for the field variables that characterize the gas dynamics and a closed-form expression for the thermophoretic force on the sphere. We also consider the slow, streaming flow of gas past a sphere using the same model resulting in a drag force on the body. The thermophoretic velocity of the sphere is then determined from the balance between thermophoretic force and drag. The thermophoretic force is compared with predictions from other theories, including Grad’s 13-moment equations (G13), variants of the Boltzmann equation commonly used in kinetic theory, and with recently published experimental data. The new results from R13 agree well with results from kinetic theory up to a Knudsen number (based on the sphere’s radius) of about 0.1 for the values of solid-to-gas heat conductivity ratios considered. However, in this range of Knudsen numbers, where for a very high thermal conductivity of the solid the experiments show reversed thermophoretic forces, the R13 solution, which does result in a reversal of the force, as well as the other theories predict significantly smaller forces than the experimental values. For Knudsen numbers between 0.1 and 1 approximately, the R13 model of thermophoretic force qualitatively shows the trend exhibited by the measurements and, among the various models considered, results in the least discrepancy.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QC Physics T Technology > TJ Mechanical engineering and machinery |
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| Divisions: | Faculty of Science, Engineering and Medicine > Engineering > Engineering Faculty of Science, Engineering and Medicine > Science > Mathematics |
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| Library of Congress Subject Headings (LCSH): | Gas flow -- Mathematical models, Microfluidics, Nanofluids | |||||||||||||||
| Journal or Publication Title: | Journal of Fluid Mechanics | |||||||||||||||
| Publisher: | Cambridge University Press | |||||||||||||||
| ISSN: | 0022-1120 | |||||||||||||||
| Official Date: | 10 March 2019 | |||||||||||||||
| Dates: |
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| Volume: | 862 | |||||||||||||||
| Page Range: | pp. 312-347 | |||||||||||||||
| DOI: | 10.1017/jfm.2018.907 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Re-use Statement: | This article has been published in a revised form in Journal of Fluid Mechanics [http://doi.org/XXX]. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © copyright holder. | |||||||||||||||
| Access rights to Published version: | Open Access (Creative Commons) | |||||||||||||||
| Date of first compliant deposit: | 26 November 2018 | |||||||||||||||
| Date of first compliant Open Access: | 16 January 2019 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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