17. Falling book cover

17. Falling Book Cover

A book is placed horizontally, and the partially open cover is allowed to fall. Investigate the motion of the falling book cover.

A thin flat plate (book cover, sheet of paper, playing card) dropped horizontally can exhibit four distinct falling regimes depending on the dimensionless inertia I* and Reynolds number Re. Understanding these requires combining rigid-body dynamics with unsteady aerodynamics.

Dimensionless Parameters

• Dimensionless inertia: I* = ρ_s h / (ρ_f a) — ratio of plate inertia to fluid inertia
• Reynolds number: Re = ρ_f v a / μ
• Froude number: Fr = v / √(g a)

All four regimes can be mapped on the I*–Re phase diagram (Andersen et al., 2005).

Falling Regimes

Equations of Motion (Andersen–Pesavento–Wang Model)

Three degrees of freedom (x, y, θ) with added mass:

(m + m_a) ẍ = F_x,aero
(m + m_a) ÿ = −mg + F_y,aero
(I_s + I_a) θ̈ = M_aero

Added mass: m_a = ρ_f π a²/4, added moment: I_a = ρ_f π a⁴/128.

Aerodynamic Force Model

F_L = ρ_f a (A₁ sin2α + A₂ cos2α) |v_rel|²/2
F_D = ρ_f a (C_D0 + C_D2 cos²α) |v_rel|²/2
M = ρ_f a² (B₁ sin2α + B₂ α̇a/|v_rel|) |v_rel|²/2

where α = θ − arctan(ẏ/ẋ) is the effective angle of attack and A₁, A₂, B₁, B₂ are fitted from experiments.

Flutter Frequency Scaling

f_flutter ≈ (1/2π) √(g/(a · I*))

This scaling predicts that lighter sheets flutter faster, and larger sheets flutter slower — consistent with paper vs cardboard comparisons.

References

• Andersen A., Pesavento U., Wang Z.J. (2005). J. Fluid Mech. 541, 91
• Mahadevan et al. (1999). Phys. Fluids 11, 1
• Belmonte et al. (1998). Phys. Rev. Lett. 81, 345
• Field et al. (1997). Nature 388, 252

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