The flow of ink in a fountain pen is largely governed by the balance between capillary action and the resistive forces due to viscosity. We can derive the relation for the flow rate \( Q \) of ink in terms of the ink’s surface tension \( \gamma \), viscosity \( \eta \), and other parameters of the pen.
The capillary pressure difference \( \Delta P \) driving the ink flow is due to the surface tension of the ink and can be given by the Young-Laplace equation:
$$ \Delta P = \frac{2 \gamma \cos \theta}{R} $$This pressure difference is responsible for pushing the ink through the narrow capillary channel in the pen.
The resistive force to the flow of ink is due to the viscosity of the ink moving through the capillary. According to Hagen-Poiseuille’s law, the volumetric flow rate \( Q \) for a fluid through a cylindrical tube is given by:
$$ Q = \frac{\Delta P \cdot \pi R^4}{8 \eta L} $$Now, substituting \( \Delta P = \frac{2 \gamma \cos \theta}{R} \) from Step 1 into the expression for \( Q \), we get:
$$ Q = \frac{\left( \frac{2 \gamma \cos \theta}{R} \right) \cdot \pi R^4}{8 \eta L} $$Simplifying the expression:
$$ Q = \frac{2 \gamma \cos \theta \cdot \pi R^3}{8 \eta L} $$or:
$$ Q = \frac{\pi \gamma R^3 \cos \theta}{4 \eta L} $$The final relationship governing the flow of ink in a fountain pen as a function of the ink’s viscosity and surface tension is:
$$ Q = \frac{\pi \gamma R^3 \cos \theta}{4 \eta L} $$This derived relation shows that the flow rate \( Q \) of ink in a fountain pen depends directly on the surface tension \( \gamma \) and inversely on the viscosity \( \eta \). Higher surface tension and a larger capillary radius \( R \) increase the flow rate, while higher viscosity \( \eta \) and capillary length \( L \) reduce it.