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fountain-pen-ink.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Flow of Ink in a Fountain Pen</title>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<style>
body {
font-family: Arial, sans-serif;
max-width: 800px;
margin: 0 auto;
padding: 20px;
line-height: 1.6;
}
h1 {
color: #333;
text-align: center;
}
p, ul {
font-size: 1.1em;
}
</style>
</head>
<body>
<h1>Deriving the Relation Governing Ink Flow in a Fountain Pen</h1>
<p>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.</p>
<h2>Key Variables</h2>
<ul>
<li><b>\( \gamma \)</b> = Surface tension of the ink (N/m)</li>
<li><b>\( \eta \)</b> = Viscosity of the ink (Pa·s)</li>
<li><b>\( R \)</b> = Radius of the capillary in the pen (m)</li>
<li><b>\( L \)</b> = Length of the capillary (m)</li>
<li><b>\( \theta \)</b> = Contact angle of the ink with the pen material</li>
<li><b>\( Q \)</b> = Flow rate of ink (m³/s)</li>
</ul>
<h2>Step 1: Capillary Pressure</h2>
<p>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:</p>
$$
\Delta P = \frac{2 \gamma \cos \theta}{R}
$$
<p>This pressure difference is responsible for pushing the ink through the narrow capillary channel in the pen.</p>
<h2>Step 2: Hagen-Poiseuille Flow for Viscous Resistance</h2>
<p>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:</p>
$$
Q = \frac{\Delta P \cdot \pi R^4}{8 \eta L}
$$
<h2>Step 3: Substitute Capillary Pressure</h2>
<p>Now, substituting \( \Delta P = \frac{2 \gamma \cos \theta}{R} \) from Step 1 into the expression for \( Q \), we get:</p>
$$
Q = \frac{\left( \frac{2 \gamma \cos \theta}{R} \right) \cdot \pi R^4}{8 \eta L}
$$
<p>Simplifying the expression:</p>
$$
Q = \frac{2 \gamma \cos \theta \cdot \pi R^3}{8 \eta L}
$$
<p>or:</p>
$$
Q = \frac{\pi \gamma R^3 \cos \theta}{4 \eta L}
$$
<h2>Final Relation</h2>
<p>The final relationship governing the flow of ink in a fountain pen as a function of the ink’s viscosity and surface tension is:</p>
$$
Q = \frac{\pi \gamma R^3 \cos \theta}{4 \eta L}
$$
<h2>Conclusion</h2>
<p>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.</p>
</body>
</html>