Euler's equation and Bernoulli's equation are key concepts in fluid mechanics. They help us understand how fluids move and behave under different conditions. These equations are simplified versions of more complex fluid dynamics, making them useful for many real-world applications.

Euler's equation describes inviscid fluid motion, while Bernoulli's equation relates pressure, velocity, and elevation in steady, incompressible flow. Together, they form the foundation for analyzing fluid systems, from simple pipe flows to complex aerodynamic problems.

Euler's Equation

Derivation of Euler's equation

• Begin with the Navier-Stokes equations for incompressible flow, which describe the motion of fluid substances $\rho \frac{D\vec{V}}{Dt} = -\nabla p + \rho \vec{g} + \mu \nabla^2 \vec{V}$
• Simplify by assuming inviscid flow, where the fluid has no viscosity $(\mu = 0)$, reducing the equation to $\rho \frac{D\vec{V}}{Dt} = -\nabla p + \rho \vec{g}$
• Expand the material derivative $\frac{D\vec{V}}{Dt}$, which represents the rate of change of velocity following a fluid particle, into local and convective acceleration terms $\rho (\frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla)\vec{V}) = -\nabla p + \rho \vec{g}$
• Rearrange the terms to obtain Euler's equation, which describes the motion of an inviscid fluid $\rho \frac{\partial \vec{V}}{\partial t} + \rho (\vec{V} \cdot \nabla)\vec{V} = -\nabla p + \rho \vec{g}$

Physical interpretation of Euler's equation

• Local acceleration term $\rho \frac{\partial \vec{V}}{\partial t}$ represents the rate of change of velocity with respect to time at a fixed point in space (unsteady flow)
• Convective acceleration term $\rho (\vec{V} \cdot \nabla)\vec{V}$ represents the rate of change of velocity due to the change in position of the fluid particle (flow with spatial velocity variations)
• Pressure gradient term $-\nabla p$ represents the force per unit volume acting on the fluid particle due to the pressure difference in the flow field
• Body force term $\rho \vec{g}$ represents the force per unit volume acting on the fluid particle due to external forces like gravity (weight of the fluid)

Bernoulli's Equation

Bernoulli's equation from Euler's equation

• Begin with Euler's equation for inviscid flow $\rho \frac{\partial \vec{V}}{\partial t} + \rho (\vec{V} \cdot \nabla)\vec{V} = -\nabla p + \rho \vec{g}$
• Simplify by assuming steady flow, where the velocity does not change with time $(\frac{\partial \vec{V}}{\partial t} = 0)$, reducing the equation to $\rho (\vec{V} \cdot \nabla)\vec{V} = -\nabla p + \rho \vec{g}$
• Further simplify by assuming incompressible flow, where the density $\rho$ is constant, yielding $(\vec{V} \cdot \nabla)\vec{V} = -\frac{1}{\rho}\nabla p + \vec{g}$
• Apply the vector identity $(\vec{V} \cdot \nabla)\vec{V} = \nabla(\frac{V^2}{2}) - \vec{V} \times (\nabla \times \vec{V})$ to the left-hand side of the equation $\nabla(\frac{V^2}{2}) - \vec{V} \times (\nabla \times \vec{V}) = -\frac{1}{\rho}\nabla p + \vec{g}$
• Assume irrotational flow, where the curl of the velocity is zero $(\nabla \times \vec{V} = 0)$, simplifying the equation to $\nabla(\frac{V^2}{2}) = -\frac{1}{\rho}\nabla p + \vec{g}$
• Integrate along a streamline from point 1 to point 2, representing the path a fluid particle follows in the flow field $\int_1^2 \nabla(\frac{V^2}{2}) \cdot d\vec{s} = -\int_1^2 \frac{1}{\rho}\nabla p \cdot d\vec{s} + \int_1^2 \vec{g} \cdot d\vec{s}$
• Evaluate the integrals to obtain Bernoulli's equation, which relates velocity, pressure, and elevation along a streamline $\frac{V_2^2}{2} + \frac{p_2}{\rho} + gz_2 = \frac{V_1^2}{2} + \frac{p_1}{\rho} + gz_1$

Applications of Bernoulli's equation

• Bernoulli's equation relates velocity, pressure, and elevation along a streamline in steady, incompressible, and inviscid flow $\frac{V^2}{2} + \frac{p}{\rho} + gz = constant$
• Apply Bernoulli's equation between two points along a streamline to solve for unknown variables (velocity, pressure, or elevation) $\frac{V_1^2}{2} + \frac{p_1}{\rho} + gz_1 = \frac{V_2^2}{2} + \frac{p_2}{\rho} + gz_2$
• Use the continuity equation for incompressible flow in pipes and ducts to relate velocities and cross-sectional areas $A_1V_1 = A_2V_2$ (mass conservation)
• Combine Bernoulli's equation and the continuity equation to analyze fluid flow in pipes, ducts, and open-channel flows (rivers, canals)
  • Determine flow rates, pressure drops, or changes in elevation
  • Design piping systems, nozzles, or flow measurement devices (Venturi meters, orifice plates)
• Modify Bernoulli's equation to account for energy losses due to friction (major losses), minor losses (valves, fittings), or pumps/turbines (energy added or removed)
  • Include head loss terms or pump/turbine work in the energy balance along the streamline
  • Analyze the performance of hydraulic systems, turbomachinery, or flow control devices