The Fountain Pen Mystery · AI Generator #GEN-B7F37401

Physics · fluid-mechanics

Problem

A vintage fountain pen works by having ink flow from an internal reservoir through a narrow feed channel to the nib. The key to its operation is that ink only flows when the pen touches paper, creating a capillary bridge. Consider a simplified model: the ink reservoir is a cylinder of diameter D=8.0  mmD = 8.0\;\text{mm} and height h0=40  mmh_0 = 40\;\text{mm}, initially full of ink. The feed channel is a thin tube of radius r=0.15  mmr = 0.15\;\text{mm} and length L=12  mmL = 12\;\text{mm}. When writing, the capillary bridge at the nib tip creates a pressure difference that drives flow. The ink has viscosity η=3.2×10−3  Pa⋅s\eta = 3.2 \times 10^{-3}\;\text{Pa}\cdot\text{s} and density ρ=1100  kg/m3\rho = 1100\;\text{kg/m}^3. When writing, the pressure at the nib tip is effectively Pnib=Patm−ΔPcapP_{\text{nib}} = P_{\text{atm}} - \Delta P_{\text{cap}}, where ΔPcap=850  Pa\Delta P_{\text{cap}} = 850\;\text{Pa} is the pressure reduction due to capillary effects. (a) Find the initial flow rate of ink when the pen begins writing with a full reservoir. (b) As ink flows out, the ink level in the reservoir drops, creating an air space above the ink. Assuming this air space remains at atmospheric pressure, find the height hstoph_{\text{stop}} of ink remaining in the reservoir when the pen stops flowing.
Intuitive explanation

The heart of this problem is understanding how pressure differences drive viscous flow through a narrow channel. In part (a), you'll use the Hagen-Poiseuille equation, which relates flow rate to the pressure gradient along the tube—here, that gradient comes from the hydrostatic pressure of the full reservoir plus the capillary suction at the nib. In part (b), as ink drains and the reservoir empties, the hydrostatic pressure decreases; flow stops when the weight of the remaining ink column can no longer overcome the capillary pressure pulling ink out, creating an equilibrium condition.

Intuitive explanation

The heart of this problem is understanding how pressure differences drive viscous flow through a narrow channel. In part (a), you'll use the Hagen-Poiseuille equation, which relates flow rate to the pressure gradient along the tube—here, that gradient comes from the hydrostatic pressure of the full reservoir plus the capillary suction at the nib. In part (b), as ink drains and the reservoir empties, the hydrostatic pressure decreases; flow stops when the weight of the remaining ink column can no longer overcome the capillary pressure pulling ink out, creating an equilibrium condition.

Related theory

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