What are the differences between Euler's and Bernoulli's equations for fluid flow?

The Euler equation, named after Swiss mathematician Leonhard Euler, is a fundamental equation in calculus that describes the relationship between a function, its first derivative, and its independent variable. In the context of differential equations, the Euler equation typically refers to a second-order linear homogeneous ordinary differential equation of the form (a(x)y'' + b(x)y' + c(x)y = 0), where (y) is the unknown function, (a(x)), (b(x)), and (c(x)) are given functions of (x), and (y''), (y') represent the second and first derivatives of (y) with respect to (x), respectively.

Euler's equation, a fundamental principle in fluid dynamics, describes the relationship between the pressure, velocity, and elevation of a fluid moving in a steady, inviscid flow. Mathematically, it states that the total mechanical energy per unit mass of the fluid remains constant along a streamline. An everyday example illustrating Euler's equation is the flow of water through a pipeline. As water flows steadily through the pipe, its pressure, velocity, and elevation change, but the total mechanical energy per unit mass, including kinetic and potential energy, remains constant along the flow path.

One interesting aspect is that the Euler equations assume no viscosity, which simplifies calculations but requires additional considerations for real-world applications where viscosity plays a role in boundary layers and turbulence. Additionally, solutions to the Euler equations can exhibit phenomena such as shock waves, which are crucial in designing supersonic and hypersonic vehicles.

Bernoulli's equation, named after Swiss mathematician Daniel Bernoulli, is a fundamental principle in fluid dynamics that describes the relationship between pressure, velocity, and elevation along a streamline in an inviscid, incompressible flow. It states that the total mechanical energy per unit mass of a fluid particle remains constant along a streamline, neglecting friction, viscosity, and external forces such as gravity. Mathematically, Bernoulli's equation is expressed as shown above. Bernoulli's equation finds applications in various fields, including aerodynamics, hydrodynamics, and engineering.

Bernoulli's principle is also crucial in understanding phenomena like stall and the formation of vortices. Stall occurs when the airflow over the wing becomes disrupted, typically due to exceeding a critical angle of attack. Vortices are generated at wingtips and can affect aircraft following behind, influencing safety and separation requirements.

Bernoulli's equation, a fundamental principle in fluid dynamics, describes the relationship between pressure, velocity, and elevation in a fluid flow. It states that the total mechanical energy per unit mass of a fluid remains constant along a streamline. Mathematically, it is expressed as P + (1/2)ρv^2 + ρgh = constant, where P is pressure, ρ is density, v is velocity, g is gravitational acceleration, and h is height. A real-life example is the flight of an airplane: as air velocity increases over the wings (lowering pressure), lift is generated, allowing the plane to fly.

Euler's equation describes the motion of fluid particles along a streamline and considers both kinetic and potential energy changes. It's applicable to inviscid, irrotational flow. Bernoulli's equation, derived from Euler's, applies to steady, incompressible, inviscid flow along a streamline, assuming no energy loss due to friction. It relates fluid pressure, velocity, and elevation along the streamline, emphasizing conservation of mechanical energy. For instance, while Euler's equation helps analyze airflow around an airfoil, Bernoulli's equation explains lift generation on an airplane wing due to pressure differences.

Euler's equation describes the motion of inviscid fluids, accounting for changes in velocity and pressure along streamlines based on the conservation of momentum, while Bernoulli's equation describes the conservation of mechanical energy along a streamline in an inviscid, incompressible flow, relating pressure, velocity, and elevation terms based on the conservation of energy. Euler's equation is widely used in aerodynamics, hydrodynamics, and engineering to predict fluid flow behavior, while Bernoulli's equation finds applications in analyzing pressure changes, velocity variations, and elevation effects in various fluid flow scenarios, such as pipe flow and hydraulic systems.

Euler basically assume no viscosity and are based on conservation laws (mass, momentum, energy) whereas Bernoulli, assumes steady flow, incompressibility (for incompressible fluids), and no friction. Every aero engineering student knows they got these things wrong Atleast a few times, there’s no shame in trying to understand the same concept a number of times.

These are both simplified abstractions from the Navier-Stokes equation, and as such they represent specific fluid flow geometry types and flow conditions. As simplications, they are of practical value in pedagogy and design of very simple engineering systems, but should not be viewed as a complete representation of flow behaviour. The Navier-Stokes equations are derived from classical physics assumptions, so are limited where molecular or atomistic interactions become significant. Boltzmann equations then must be adopted. Most commercial codes for modelling fluid dynamics are based on solving the Navier-Stokes equations, including simplified forms of those equations. Recently Lattice Boltzmann codes have also become available.

Euler equations are crucial for analyzing the flow dynamics and heat transfer during hypersonic flight. Bernoulli's principle is applied to measure airspeed based on pressure differences between a pitot tube (facing the airflow) and a static port (side-facing).

When viscosity effects are negligible or can be approximated as such. Bernoulli's principle offers a simpler approach to understanding fluid behavior under ideal conditions, without the complexity of solving full Euler equations.

When dealing with supersonic or hypersonic flows where the effects of viscosity are minimal. Euler equations are appropriate because they assume inviscid flow and can accurately predict shock waves and other complex flow phenomena.