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 approximately 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.
Highlights
Thermophoresis refers to the force and, potentially, motion experienced by solid particles or surfaces exposed to gas under rarefied conditions and in the presence of a temperature gradient
We investigated theoretically an instance of the phenomenon of thermophoresis, a term that refers to the forces on and motions of objects caused by temperature gradients when these objects are exposed to rarefied gases
We considered the problem of thermophoresis of a spherical particle, obtaining an analytical solution by solving the regularized 13-moment equations (R13) moment equations, a model that provides a macroscopic description of rarefied gas flows up to the transition regime for Knudsen numbers smaller than one
Summary
Thermophoresis refers to the force and, potentially, motion experienced by solid particles or surfaces exposed to gas under rarefied conditions and in the presence of a temperature gradient. This phenomenon seems to have been first noted by Tyndall (1870) while observing the spatial redistribution of ambient dust in the proximity of a heated surface. The thermophoretic force has been assumed to point from the hot to the cold region, that is, opposite to the temperature gradient. This condition has been labelled as ‘positive’ thermophoresis. Young (2011) presented a comprehensive examination of the various theories on particle thermophoresis at arbitrary Knudsen numbers under the light of experimental data, and concluded that the accuracy of the measurements and the interval of Knudsen numbers explored are such that confirmation of the validity of the theories could not be achieved
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