Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Bernoulli’s Principle

As the speed of a fluid increases, its pressure decreases. This principle is a cornerstone of aerodynamic analysis.
📜

The statement of the theorem

Bernoulli's Principle is a direct consequence of the conservation of energy applied to an ideal fluid flow. Consider a steady, incompressible, and inviscid flow (ρ=constant\rho = constant, μ=0\mu = 0) along a streamline L\mathcal{L} in a conservative gravitational field. The principle states that the total mechanical energy per unit volume, PtotalP_{total}, remains constant along L\mathcal{L}.\n\nFormally, if v\mathbf{v} is the fluid velocity, PP is the static pressure, ρ\rho is the fluid density, and gg is the acceleration due to gravity, the conservation of energy dictates:\n\nPρ+12∣v∣2+gh=Cstreamline\frac{P}{\rho} + \frac{1}{2} |\mathbf{v}|^2 + g h = C_{streamline} \n\nwhere hh is the elevation (potential energy per unit weight), and CstreamlineC_{streamline} is a constant value for all points on the streamline. \n\nFor two points, 1 and 2, along the streamline, the principle is expressed as:\n\n(P1+12ρ∣v1∣2+ρgh1)=(P2+12ρ∣v2∣2+ρgh2)\left(P_{1} + \frac{1}{2} \rho |\mathbf{v}_{1}|^2 + \rho g h_{1}\right) = \left(P_{2} + \frac{1}{2} \rho |\mathbf{v}_{2}|^2 + \rho g h_{2}\right) \n\nThis formulation requires the flow to satisfy the Euler equations and the continuity equation, ensuring that the work done by pressure forces and gravity is balanced by the change in kinetic energy.