[Paper Review] Body Motion in a Resistive Medium at Temperature T

Mario I. Molina|ArXiv.org|Jul 9, 2002

Geophysical and Geoelectrical Methods1 references3 citations

TL;DR

This paper presents a closed-form analytical solution for the resistive drag force on a macroscopic body moving through a one-dimensional ideal gas at finite temperature T. Using a simplified model with variable inelasticity via a restitution coefficient ε, it derives an exact expression showing that drag transitions from linear (Stokes' law) at low speeds (V ≪ V_T) to quadratic (Newton's law) at high speeds (V ≫ V_T), with the thermal speed V_T = √(kT/m) as the critical crossover scale.

ABSTRACT

We consider a macroscopic body propagating in a one-dimensional resistive medium, consisting of an ideal gas at temperature $T$. For a whole family of collisions with varying degree of inelasticity, we find an exact expression for the effective force on the moving body as a function of the body's speed and the value of the restitution coefficient. At low and high speeds it reduces to the well-known Stoke's and Newton's law, respectively.

Motivation & Objective

• To model the resistive drag force on a macroscopic body moving through a one-dimensional ideal gas at finite temperature T.
• To extend a previous zero-temperature model by incorporating thermal motion of medium particles and finite-temperature effects.
• To derive a closed-form expression for the effective drag force as a function of body speed V and restitution coefficient ε.
• To identify the crossover regime between Stokes’ and Newton’s drag laws, governed by the thermal speed V_T.
• To analyze the stopping distance behavior, contrasting finite-T (finite stopping distance) with zero-T (logarithmic divergence) cases.

Proposed method

• Model the medium as a one-dimensional ideal gas at thermal equilibrium with particle speed distribution g(v) = (1/√(2π))·(1/V_T)·exp[−½(v/V_T)²].
• Treat the body as a massive, partially absorbing wall with restitution coefficient ε ∈ [0,1], so momentum transfer per collision is ∝ (1+ε)|v−V|.
• Compute momentum flux from left and right using velocity-dependent collision rates: dn_L ∝ Θ(v−V)(v−V)dt and dn_R ∝ Θ(V−v)(V−v)dt.
• Integrate over all particle speeds to compute net force: F = F_left − F_right, leading to an integral expression involving error functions and exponentials.
• Use the Gaussian speed distribution and perform exact integration to derive the closed-form force expression (Eq. 6) involving Erf and exponential terms.
• Analyze asymptotic limits (V ≪ V_T and V ≫ V_T) to recover Stokes’ and Newton’s laws respectively, confirming the physical consistency of the model.

Experimental results

Research questions

• RQ1 How does the resistive drag force on a macroscopic body depend on its speed V in a one-dimensional resistive medium at finite temperature T?
• RQ2 What is the role of the restitution coefficient ε in determining the effective drag force, and how does it affect the transition between linear and quadratic drag regimes?
• RQ3 How does the thermal speed V_T = √(kT/m) act as a crossover scale between Stokes’ and Newton’s drag laws in this model?
• RQ4 What is the stopping distance behavior of the body in a finite-temperature medium, and how does it differ from the zero-temperature case?
• RQ5 Can a closed-form analytical solution be derived for the effective drag force in this simplified many-body system with inelastic collisions?

Key findings

• The effective drag force is exactly solvable in closed form and depends on body speed V, thermal speed V_T, and restitution coefficient ε.
• For V ≪ V_T, the force scales linearly with speed: F ≈ −ρ(1+ε)√(8mkT/π)·V, confirming Stokes’ law in this regime.
• For V ≫ V_T, the force scales quadratically with speed: F ≈ −mρ(1+ε)V², confirming Newton’s law in this regime.
• The crossover between linear and quadratic drag occurs near V ≈ V_T, with a smooth transition governed by the error function and exponential terms in the exact solution.
• At finite temperature T, the stopping distance is finite due to the linear drag regime at low speeds, in contrast to the logarithmic divergence seen at zero temperature.
• The degree of inelasticity (via ε) only renormalizes the effective density of the medium and does not alter the functional form of the drag law in the asymptotic regimes.