Drag Force on a Flat Plate Moving Through a Rarefied Gas

Problem Statement

The air resistance on artificial satellites moving in near-vacuum space, or on tiny dust particles falling through a rarefied gas, is understood not through everyday fluid mechanics but through the microscopic accumulation of collisions based on the kinetic theory of gases. In this problem, we rigorously derive this drag force from molecular-level collisions.

Consider a rarefied gas of single-atom molecules with mass $m$m and number density $n$n. All molecules move at the same speed $v_0$v0​ relative to the space, with directions that are completely isotropic (uniformly distributed in all directions). External forces such as gravity are negligible.

A thin flat plate of cross-sectional area $S$S moves at constant velocity $V$V in the $+z$+z direction (perpendicular to the plate face), through this gas. It is assumed that $0 < V < v_0$0<V<v0​.

Gas molecules undergo perfectly elastic collisions with the plate. Derive the drag force $F$F on the plate in the direction opposing its motion.

The face of the plate facing the direction of motion is called the "front face," and the opposite face is the "rear face."

Use spherical coordinates with $\theta$θ ($0 \le \theta \le \pi$0≤θ≤π) as the angle between the molecular velocity vector and the positive $z$z-axis, and $\phi$ϕ ($0 \le \phi < 2\pi$0≤ϕ<2π) as the angle of the $xy$xy-projection with the $x$x-axis.

Note: For an isotropic velocity distribution, the fraction of molecules moving within an infinitesimal solid angle $d\Omega = \sin\theta\, d\theta\, d\phi$dΩ=sinθdθdϕ is $\frac{d\Omega}{4\pi}$4πdΩ​.

Constraints

• Molecular mass: $m = 5.0 \times 10^{-26} \text{ kg}$m=5.0×10−26 kg
• Number density: $n = 6.0 \times 10^{20} \text{ m}^{-3}$n=6.0×1020 m−3
• Molecular speed: $v_0 = 400 \text{ m/s}$v0​=400 m/s
• Plate velocity: $V = 100 \text{ m/s}$V=100 m/s
• Plate cross-sectional area: $S = 2.0 \text{ m}^2$S=2.0 m2

Input Format

Find the magnitude of the net drag force $F$F in $\text{N}$N and give the answer as a positive integer equal to the value multiplied by $10$10.