[Paper Review] Spinning rough disk moving in a rarefied medium

Alexander Plakhov, Tatiana Tchemisova|ArXiv.org|Apr 17, 2008

Adhesion, Friction, and Surface Interactions10 references10 citations

TL;DR

This paper investigates the Magnus effect in a two-dimensional rarefied gas where a spinning rough disk experiences a transverse force due to multiple elastic collisions within surface cavities, rather than non-elastic friction. Using billiard dynamics and optimal mass transport, it derives the resistance force and moment as functions of cavity shape, showing that even with purely elastic collisions, a nonzero transverse force arises, leading to trajectory deflection—offering a new mechanism for the reverse Magnus effect in zero-temperature, free molecular flow.

ABSTRACT

A spinning rough disk moves through a rarefied medium on the plane. The roughness is formed by small cavities on the disk boundary. The medium is so rare that mutual interaction of particles can be neglected. All collisions of particles with the disk are perfectly elastic; there may happen multiple collisions in the cavities. We calculate the force of resistance acting on the body and examine how it depends on the kind of roughness (shape of the cavities). We show that the nonzero transversal component of the force generally appears, resulting in deflection of the disk trajectory. In several simple cases the trajectory is determined. We compare our results with the ones known in the literature. It is known that there is a transversal force acting on a spinning body (most often a sphere or a cylinder) moving in a rarefied gas, due to nonelastic interaction of gas particles with the body. We propose another mechanism of creating the transversal force, resulting from multiple reflections of particles from the body.

Motivation & Objective

• To analyze the resistance force and moment acting on a spinning rough disk in a rarefied, zero-temperature gas.
• To investigate how the shape of surface cavities (roughness) influences the transverse component of resistance.
• To establish a mechanism for the Magnus effect based solely on multiple elastic reflections in cavities, independent of tangential friction.
• To derive equations of motion for the disk and analyze its trajectory under different cavity geometries.
• To compare the new mechanism with existing models based on non-elastic particle-surface interactions.

Proposed method

• Model the rough disk as a limit of discrete, rotationally symmetric sets $B_m$ with small cavities, converging to a smooth circle as $m \to \infty$.
• Use billiard dynamics to describe elastic, specular reflections of point particles from the cavity surfaces.
• Define the resistance force $\vec{R}(B, \omega, \vec{v})$ and moment $R_I(B, \omega, \vec{v})$ as limits of forces on $B_m$ as $m \to \infty$.
• Apply optimal mass transport theory to compute the force components, particularly the transverse force.
• Derive equations of motion: $M \frac{d\vec{v}}{dt} = \vec{R}(B, \omega, \vec{v})$ and $I \frac{d\omega}{dt} = R_I(B, \omega, \vec{v})$.
• Analyze specific cavity shapes (e.g., triangular, equilateral) and compute the function $\alpha(\lambda, \nu)$ governing the transverse force.

Experimental results

Research questions

• RQ1 Can a transverse force arise in a spinning rough disk moving in a purely elastic, rarefied gas flow, even without tangential friction?
• RQ2 How does the shape of surface cavities affect the magnitude and direction of the resistance force and its moment?
• RQ3 What is the functional dependence of the transverse force on the dimensionless rotation parameter $\lambda = \omega r / v$?
• RQ4 How does the new mechanism of the Magnus effect compare quantitatively with classical models based on non-elastic particle interactions?
• RQ5 Can the reverse Magnus effect be generated solely by multiple reflections in surface cavities under elastic collision conditions?

Key findings

• A nonzero transverse component of the resistance force emerges due to multiple elastic reflections in surface cavities, even with zero-temperature, perfectly elastic collisions.
• The transverse force is proportional to $\frac{1}{2} \alpha(\lambda, \nu) M_g \omega v$, where $\alpha(\lambda, \nu)$ depends on cavity geometry $\nu$ and rotation parameter $\lambda = \omega r / v$.
• For certain cavity shapes, $\alpha(\lambda, \nu)$ can exceed 2, significantly larger than the maximum value of 1 in non-elastic models.
• The function $\alpha(\lambda, \nu)$ varies substantially with $\lambda$, showing a dependence of up to twofold variation for fixed $\nu$.
• The time-averaged transverse force for periodic cavity structures (e.g., regular polygons) matches the steady-state formula, confirming consistency.
• The reverse Magnus effect is shown to be possible via purely elastic, cavity-mediated momentum transfer, independent of thermalization or tangential friction.