We primarily denote derivatives by some variable primarily through subscripts
but because temporal derivatives are so common we will also denote temporal derivatives by dot notation
will be used to represent Euclidean positions and will repsresnt general coordinates. and will both be the velocity field of particles .
For a diffeomorphism of a domain by some function we define incompressibility as
for volume defined in the usual way by
The volume of after being advected is given by
Obviously if incompressibility is maintained
The differential form of this statement is that
Usually one starts with some PDE
and incompressibility is added by the gradient of some scalar field that guarantees :
There are a few perspectives for what pressure is that are valuable to understand:
This is perhaps the most fundamental geometric perspective for incompressibility. Ignoring the fancy notation, all this really depends on is the fact that differentiation is a linear map and that there’s a nice orthogonal decomposition of arbitrary (but sufficiently smooth) vector fields using different differential operators. What this boils down to is that making a vector field incompressible is the removal of the part of it that has divergence, which is a projection.
Recall the Hodge-deRham decomposition of a -form into three orthogonal components
for a 0-form, a 2-form, and harmonic.
Now note that , , and .
Therefore, so long as for the decomposition
the gradient will be
so long as the vector field is divergence free. By the definition of harmonic vector fields we see that is orthogonal to the kernel of and so being incompressible is simply setting . The required pressure is therefore just and pressure projection is
where can be computed by
Variational energy minimization
The variational perspective for pressure is that its the vector field that its the arugment for the minimal kinetic energy necessary to be divergence-free:
This idea comes from orthogonoality in the sense that it’s finding a that removes as much from as possible. The key advantage of this formulation is that it does not depend on splitting as projection does. That is, the previous formulation is more amenable for solving for and simultaneously (as well as other things if there is a need).
By doing some arithmetic we see
Consider the Lagrangian mechanics perspective of dynamics where the general positions and general velocities are defined by minimizing some Lagrangian and the Euler-Lagrange equations
Here we assume that represents positions and densities in a piece of geometry, which implies that represents a diffeomorphism of the geometry itself.
Incompressibility adding the constraint for some linaer operator that satisfies
where is the diffeomorphism induced by the general coordinates .
Note that the adjoint operator, given the condition that or for any point along the boundary, can be defined by
Augmenting this Lagrangian for incompressibility is therefore done by
Necessary conditions for optimality dictate that
where the final condition is simply a restating of the incompresibility condition.
On hte other hand we can the adjoint operator from before
This parameter is what is commonly called pressure.
An example with viscous fluids
Recall that incompressible Navier-Stokes can be written as
where is such that
If we ignore the term (which can be dealt with by an ODE integrator) and solve for the future implicitly we obtain
Where the first and second rows represent and respectively for
which, is the augmented lagrangian for the energy
By applying Shur’s complement method to solve for one obtains from the above matrix
which simplifies to
In the inviscid case, or when we split viscosity and lump it into , this naturally turns into
our standard Poisson problem.
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