# Euler equations for ideal fluids, approximations

• Niles
In summary, the Euler equations for ideal compressible flow can be written as \partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p and \partial_t \rho + \nabla \cdot(\rho v) = 0. When written in terms of small-value expansions, the equations become \partial_t v = -\frac{1}{\rho_0}\nabla \delta \rho and \partial_t (\delta \rho) = -\rho_0 \nabla \cdot v. The reason for this change is that \delta \rho \nabla \cdot v is negligible compared to \rho_0
Niles

## Homework Statement

The Euler equations for ideal compressible flow are given by
$$\partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p \\ \partial_t \rho + \nabla \cdot(\rho v) = 0$$
In my book these are written in terms of the small-value expansions $\rho = \rho_0 + \delta \rho$, $p = p_0 + \delta p$ and the equations become
$$\partial_t v = -\frac{1}{\rho_0}\nabla \delta \rho \\ \partial_t (\delta \rho) = -\rho_0 \nabla \cdot v$$

In the second equation, I don't understand why the RHS becomes $\rho_0 \nabla \cdot v$ instead of $(\rho_0+\delta \rho) \nabla \cdot v$?

$\delta \rho \nabla \cdot v$ is negligible compared to $\rho_0 \nabla \cdot v$ because $\delta \rho << \rho_0$

## 1. What are the Euler equations for ideal fluids?

The Euler equations for ideal fluids are a set of equations that describe the behavior of a fluid in motion, based on the principles of conservation of mass, momentum, and energy. They are written in the form of partial differential equations and are used to model the flow of incompressible fluids.

## 2. What is the difference between ideal and real fluids?

Ideal fluids are theoretical fluids that are assumed to have no viscosity or internal friction, and therefore do not experience any energy losses during flow. Real fluids, on the other hand, have viscosity and experience energy losses, making their behavior more complex and difficult to model.

## 3. What is the significance of the Euler equations in fluid dynamics?

The Euler equations are important in fluid dynamics because they provide a fundamental understanding of the behavior of fluids in motion. They are used to model a wide range of phenomena, from the flow of air over an airplane wing to the motion of ocean currents.

## 4. What are some common approximations made in the Euler equations?

One common approximation made in the Euler equations is the assumption of incompressibility, which means that the fluid's density remains constant. Another common approximation is the neglect of external forces, such as gravity or friction, in order to simplify the equations and make them easier to solve.

## 5. How are the Euler equations solved for practical applications?

The Euler equations can be solved using computational methods, such as finite difference or finite element methods, to approximate the solutions numerically. In some cases, analytical solutions can also be derived for simplified versions of the equations, but these may not be applicable to more complex real-world scenarios.

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