Fluid Flow

PDEs · Axiom Academy

How Laplace's equation governs the flow of air around aircraft wings and water around ship hulls The Hidden Mathematics of Flow When an airplane soars through the sky or a ship glides through water, the fluid flows around these objects in elegant patterns. These patterns aren't random—they're governed by fundamental partial differential equations, particularly Laplace's equation . In this module, we'll explore potential flow theory , a powerful framework that uses PDEs to predict and analyze fluid behavior in countless real-world applications, from designing aircraft wings to optimizing ship hulls. Let's dive into the mathematics that makes flight and efficient naval transport possible! In potential flow theory, we describe fluid motion using a velocity potential function φ(x, y, z). The key insight is that the fluid velocity at any point is the gradient of this potential: For incompressible, irrotational flow (which describes many real-world situations), the velocity potential satisfies Laplace's equation : This is the same Laplace's equation you've seen before! It appears everywhere in physics—from electrostatics to heat conduction to fluid dynamics. Fluid density remains constant. This is an excellent approximation for liquids and low-speed air flow. Fluid particles don't rotate as they move. Valid away from boundaries and in inviscid (non-viscous) flow. Stream Functions and Streamlines

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