Categories: Fluid dynamics, Fluid mechanics, Physics.

Euler equations

The Euler equations are a system of partial differential equations that govern the movement of ideal fluids, i.e. fluids without viscosity. There exist several forms, depending on the surrounding assumptions about the fluid.

Incompressible fluid, uniform density

In a fluid moving according to the velocity vield \(\va{v}(\va{r}, t)\), the acceleration felt by a particle is given by the material acceleration field \(\va{w}(\va{r}, t)\), which is the material derivative of \(\va{v}\):

\[\begin{aligned} \va{w} \equiv \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v} \end{aligned}\]

This infinitesimal particle obeys Newton’s second law, which can be written as follows:

\[\begin{aligned} \va{w} \dd{m} = \va{w} \rho \dd{V} = \va{f^*} \dd{V} \end{aligned}\]

Where \(\dd{m}\) and \(\dd{V}\) are the particle’s mass volume, and \(\rho\) is the fluid density, which we assume, in this case, to be constant in space and time. Then the effective force density \(\va{f^*}\) represents the net force-per-particle. By dividing the law by \(\dd{V}\), we find:

\[\begin{aligned} \rho \va{w} = \va{f^*} \end{aligned}\]

Next, we want to find another expression for \(\va{f^*}\). We know that the overall force \(\va{F}\) on an arbitrary volume \(V\) of the fluid is the sum of the gravity body force \(\va{F}_g\), and the pressure contact force \(\va{F}_p\) on the enclosing surface \(S\). Using Gauss’ theorem, we then find:

\[\begin{aligned} \va{F} = \va{F}_g + \va{F}_p = \int_V \rho \va{g} \dd{V} - \oint_S p \dd{\va{S}} = \int_V (\rho \va{g} - \nabla p) \dd{V} = \int_V \va{f^*} \dd{V} \end{aligned}\]

Where \(p(\va{r}, t)\) is the pressure field, and \(\va{g}(\va{r}, t)\) is the gravitational acceleration field. Combining this with Newton’s law, we find the following equation for the force density:

\[\begin{aligned} \va{f^*} = \rho \va{w} = \rho \va{g} - \nabla p \end{aligned}\]

Dividing this by \(\rho\), we get the first of the system of Euler equations:

\[\begin{aligned} \boxed{ \va{w} = \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} } \end{aligned}\]

The last ingredient is incompressibility: the same volume must simultaneously be flowing in and out of an arbitrary enclosure \(S\). Then, by Gauss’ theorem:

\[\begin{aligned} 0 = \oint_S \va{v} \cdot \dd{\va{S}} = \int_V \nabla \cdot \va{v} \dd{V} \end{aligned}\]

Since \(S\) and \(V\) are arbitrary, the integrand must vanish by itself everywhere:

\[\begin{aligned} \boxed{ \nabla \cdot \va{v} = 0 } \end{aligned}\]

Combining this with the equation for \(\va{w}\), we get a system of two coupled differential equations: these are the Euler equations for an incompressible fluid with spatially uniform density \(\rho\):

\[\begin{aligned} \boxed{ \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} \qquad \quad \nabla \cdot \va{v} = 0 } \end{aligned}\]

Incompressible fluid, variable density

The above form is straightforward to generalize to incompressible fluids with non-uniform spatial densities \(\rho(\va{r}, t)\). In other words, these fluids are “lumpy” (variable density), but the size of their lumps does not change (incompressibility).

To update the equations, we demand conservation of mass: the mass evolution of a volume \(V\) is equal to the mass flow through its boundary \(S\). Applying Gauss’ theorem again:

\[\begin{aligned} 0 = \dv{t} \int_V \rho \dd{V} + \oint_S \rho \va{v} \cdot \dd{\va{S}} = \int_V \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) \dd{V} \end{aligned}\]

Since \(V\) is arbitrary, the integrand must be zero. This leads to the following continuity equation, to which we apply a vector identity:

\[\begin{aligned} 0 = \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho + \rho (\nabla \cdot \va{v}) \end{aligned}\]

Thanks to incompressibility, the last term disappears, leaving us with a material derivative:

\[\begin{aligned} \boxed{ 0 = \frac{\mathrm{D} \rho}{\mathrm{D} t} = \dv{\rho}{t} + (\va{v} \cdot \nabla) \rho } \end{aligned}\]

Putting everything together, Euler’s system of equations now takes the following form:

\[\begin{aligned} \boxed{ \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} \qquad \nabla \cdot \va{v} = 0 \qquad \frac{\mathrm{D} \rho}{\mathrm{D} t} = 0 } \end{aligned}\]

References

  1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.

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