The Euler equations are a system of partial differential equations that govern the movement of ideal fluids, i.e. fluids without viscosity.
In a fluid moving according to the velocity field , the acceleration felt by a particle is given by the material acceleration field , which is the material derivative of :
This infinitesimal particle obeys Newton’s second law, which can be written as follows:
Where and are the particle’s mass and volume, and is the fluid density, which we assume to be constant in space and time in this case. Now, the effective force density represents the net force-per-particle. By dividing the law by , we find:
Next, we want to find another expression for . We know that the overall force on an arbitrary volume of the fluid is the sum of the gravity body force , and the pressure contact force on the enclosing surface . Using the divergence theorem, we then find:
Where is the pressure field, and is the gravitational acceleration field. Combining this with Newton’s law, we find the following equation for the force density:
Dividing this by , we get the first of the system of Euler equations:
The last ingredient is incompressibility: the same volume must simultaneously be flowing in and out of an arbitrary enclosure . Then, by the divergence theorem:
Since is arbitrary, the integrand must vanish by itself, leading to the continuity relation:
Combining this with the equation for , we get a system of two coupled differential equations: these are the Euler equations for an incompressible fluid with spatially uniform density :
If the fluid is compressible, the condition no longer holds, so to update the equations we demand that mass is conserved: the mass evolution of a volume is equal to the mass flow through its boundary . Applying the divergence theorem again:
Since is arbitrary, the integrand must be zero. The new continuity equation is therefore:
When the fluid gets compressed in a certain location, thermodynamics states that the pressure, temperature and/or entropy must increase there. For simplicity, let us assume an isothermal and isentropic fluid, such that only is affected by compression, and the fundamental thermodynamic relation reduces to .
Then the pressure is given by a thermodynamic equation of state , which depends on the system being studied (e.g. the ideal gas law ). However, the quantity in control of the dynamics is not , but the internal energy . Dividing the fundamental thermodynamic relation by , where is the mass of :
With and the specific (i.e. per unit mass) internal energy and volume. Using that , and substituting the above continuity relation:
It makes sense to see a factor here: an incoming flow increases . This gives us the time-evolution of due to compression, but its initial value is another equation of state .
Putting it all together, Euler’s system of equations now takes the following form:
What happens if the fluid is actually incompressible, so holds again? Clearly:
So is constant, which is in fact equivalent to saying that . The equation for enforces conservation of mass for inhomogeneous fluids, i.e. fluids that are “lumpy”, but where the size of the lumps is conserved by incompressibility.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.