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.

Incompressible fluids

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

w⃗≡Dv⃗Dt=∂v⃗∂t+(v⃗⋅∇)v⃗\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:

w⃗m=w⃗ρdV=f∗⃗dV\begin{aligned} \va{w} m = \va{w} \rho \dd{V} = \va{f^*} \dd{V} \end{aligned}

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

ρw⃗=f∗⃗\begin{aligned} \rho \va{w} = \va{f^*} \end{aligned}

Next, we want to find another expression for f∗⃗\va{f^*}. We know that the overall force F⃗\va{F} on an arbitrary volume VV of the fluid is the sum of the gravity body force F⃗g\va{F}_g, and the pressure contact force F⃗p\va{F}_p on the enclosing surface ∂V\partial V. Using the divergence theorem, we then find:

F⃗=F⃗g+F⃗p=∫Vρg⃗dV−∮∂VpdS⃗=∫V(ρg⃗−∇p)dV=∫Vf∗⃗dV\begin{aligned} \va{F} = \va{F}_g + \va{F}_p = \int_V \rho \va{g} \dd{V} - \oint_{\partial V} p \dd{\va{S}} = \int_V (\rho \va{g} - \nabla p) \dd{V} = \int_V \va{f^*} \dd{V} \end{aligned}

Where p(r⃗,t)p(\va{r}, t) is the pressure field, and g⃗(r⃗,t)\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:

f∗⃗=ρw⃗=ρg⃗−∇p\begin{aligned} \va{f^*} = \rho \va{w} = \rho \va{g} - \nabla p \end{aligned}

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

w⃗=Dv⃗Dt=g⃗−∇pρ\begin{aligned} \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 ∂V\partial V. Then, by the divergence theorem:

0=∮∂Vv⃗⋅dS⃗=∫V∇⋅v⃗dV\begin{aligned} 0 = \oint_{\partial V} \va{v} \cdot \dd{\va{S}} = \int_V \nabla \cdot \va{v} \dd{V} \end{aligned}

Since VV is arbitrary, the integrand must vanish by itself, leading to the continuity relation:

∇⋅v⃗=0\begin{aligned} \nabla \cdot \va{v} = 0 \end{aligned}

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

Dv⃗Dt=g⃗−∇pρ∇⋅v⃗=0\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}

Compressible fluids

If the fluid is compressible, the condition ∇⋅v⃗=0\nabla \cdot \va{v} = 0 no longer holds, so to update the equations we demand that mass is conserved: the mass evolution of a volume VV is equal to the mass flow through its boundary ∂V\partial V. Applying the divergence theorem again:

0=ddt∫VρdV+∮∂Vρv⃗⋅dS⃗=∫Vdρdt+∇⋅(ρv⃗)dV\begin{aligned} 0 = \dv{}{t}\int_V \rho \dd{V} + \oint_{\partial V} \rho \va{v} \cdot \dd{\va{S}} = \int_V \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) \dd{V} \end{aligned}

Since VV is arbitrary, the integrand must be zero. The new continuity equation is therefore:

0=dρdt+∇⋅(ρv⃗)=dρdt+v⃗⋅∇ρ+ρ∇⋅v⃗=DρDt+ρ∇⋅v⃗\begin{aligned} 0 = \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) = \dv{\rho}{t} + \va{v} \cdot \nabla \rho + \rho \nabla \cdot \va{v} = \frac{\mathrm{D} \rho}{\mathrm{D} t} + \rho \nabla \cdot \va{v} \end{aligned}

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 pp is affected by compression, and the fundamental thermodynamic relation reduces to dE=−pdV\dd{E} = - p \dd{V}.

Then the pressure is given by a thermodynamic equation of state p(ρ,T)p(\rho, T), which depends on the system being studied (e.g. the ideal gas law p=ρRTp = \rho R T). However, the quantity in control of the dynamics is not pp, but the internal energy EE. Dividing the fundamental thermodynamic relation by m Dtm \: \mathrm{D}t, where mm is the mass of dV\dd{V}:

DeDt=−pDvDt\begin{aligned} \frac{\mathrm{D} e}{\mathrm{D} t} = - p \frac{\mathrm{D} v}{\mathrm{D} t} \end{aligned}

With ee and vv the specific (i.e. per unit mass) internal energy and volume. Using that ρ=1/v\rho = 1 / v, and substituting the above continuity relation:

DeDt=−pDDt(1ρ)=pρ2DρDt=−pρ∇⋅v⃗\begin{aligned} \frac{\mathrm{D} e}{\mathrm{D} t} = - p \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{1}{\rho} \Big) = \frac{p}{\rho^2} \frac{\mathrm{D} \rho}{\mathrm{D} t} = - \frac{p}{\rho} \nabla \cdot \va{v} \end{aligned}

It makes sense to see a factor −∇⋅v⃗-\nabla \cdot \va{v} here: an incoming flow increases ee. This gives us the time-evolution of ee due to compression, but its initial value is another equation of state e(ρ,T)e(\rho, T).

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

Dv⃗Dt=g⃗−∇pρDρDt=−ρ∇⋅v⃗DeDt=−pρ∇⋅v⃗\begin{aligned} \boxed{ \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} \qquad \quad \frac{\mathrm{D} \rho}{\mathrm{D} t} = - \rho \nabla \cdot \va{v} \qquad \quad \frac{\mathrm{D} e}{\mathrm{D} t} = - \frac{p}{\rho} \nabla \cdot \va{v} } \end{aligned}

What happens if the fluid is actually incompressible, so ∇⋅v⃗=0\nabla \cdot \va{v} = 0 holds again? Clearly:

Dv⃗Dt=g⃗−∇pρDρDt=0DeDt=0\begin{aligned} \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} \qquad \quad \frac{\mathrm{D} \rho}{\mathrm{D} t} = 0 \qquad \quad \frac{\mathrm{D} e}{\mathrm{D} t} = 0 \end{aligned}

So ee is constant, which is in fact equivalent to saying that ∇⋅v⃗=0\nabla \cdot \va{v} = 0. The equation for ρ\rho enforces conservation of mass for inhomogeneous fluids, i.e. fluids that are “lumpy”, but where the size of the lumps is conserved by incompressibility.

References

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