Drag force and transport property of a small cylinder in free molecule flow: A gas-kinetic theory analysis

Abstract

Analytical expressions are derived for aerodynamic drag force on small cylinders in the free molecule flow using the gas kinetic theory. The derivation considers the effect of intermolecular interactions between the cylinder and gas media. Two limiting collision models, specular and diffuse scattering, are investigated in two limiting cylinder orientations with respect to the drift velocity. The earlier solution of Dahneke (B. E. Dahneke, Journal of Aerosol Science 4, 147, 1973) is shown to be a special case of the current expressions in the rigid-body limit of collision.Drag force expressions are obtained for cylinders that undergo Brownian rotation and for those that align with the drift velocity. The validity of the theoretical expressions is tested against experimental mobility data available for carbon nanotubes.

Theory and Experiment of Binary Diffusion Coefficient of n-Alkanes in Dilute Gases

“…The derivation considered the intermolecular potential of interaction and assumed that the drag force is caused by momentum exchange due to gas-particle collisions. Following a similar approach, we recently derived a generalized expression for the drag coefficient of a small cylinder in free molecular flow, considering detailed momentum transfer and the potential energy of interactions between the cylinder and gas, integrated over the Boltzmann energy distribution and a full-range of impact parameters . The theory is shown to predict the experimental measurements of the electric mobility of carbon nanotubes accurately.…”

“…To the best of our knowledge, this approach is the most rigorous found in the literature. According to Liu et al, the drag force, F ⊙ , on a freely rotating, perfect cylinder of the aspect ratio L / R ≫ 1, where L is the cylinder length and R is the cylinder radius, is where V̅ is the drift velocity, c ’s are the drag coefficients, and φ is the momentum accommodation function. , In the above equation, the subscripts s and d denote the limiting specular elastic and diffuse inelastic collisions, respectively, and ⊥ and ∥ indicate whether the cylinder axis is perpendicular or parallel to the drift velocity, respectively. Following the Chapman-Enskog assumption, we assume the collision is specular elastic locally and thus φ  0.…”

“…It has been shown that the drag coefficient on the side wall of a cylinder is where Ω s,⊥ (1,1)* is the reduced collision integral of the cylindrical section in the specular scattering limit with the cylinder axis perpendicular to the drift velocity. In the above equation, the radius R will be replaced by an effective collisional diameter after a potential function is defined, as discussed below.…”

“…Combining eq with eqs –, we obtain Liu et al showed that Ω s ,⊥ (1,1)* may be given by the following equation: where is the reduced velocity, and g is the velocity of the cylinder relative to gas, and Q s,⊥ is the collision cross section: where b is the impact factor, and χ is the scattering angle given as We treat Φ­( r ) based on a simple pair interaction potential of n -alkane molecules of arbitrary size with typical bath gases, including He and N 2 , as will be discussed in section . The values of Ω s,⊥ (1,1)* are numerically calculated and tabulated in the same section.…”

“…The purpose of the present work is to propose a general expression for the binary diffusion coefficient involving a long-chain molecule and a roughly spherical mass (e.g., N 2 and He), taking advantage of a recently derived, generalized transport theory of aerodynamic drag, electric mobility, and diffusion on the basis of a rigorous gas-kinetic theory analysis for slender bodies in dilute gases and the free molecule regime . The theory was tested against experimental electric mobility of carbon nanotubes over wide ranges of tube diameter and length.…”