We shall be considering to be the perturbation that is introduced
The BGK collision operator is defined as:
In order to determine as a function of , we first need to evaluate the temperature and density distributions by computing the zeroth, and 2nd moments of the distribution function
Since
Now, evaluating the taylor series expansion of
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Here we determine the conditions necessary to ensure that initially pressure is constant throughout the domain
If I expand this now:
The background quantities are taken to be at unity
NOTE: The 2D and 3D background distributions are assumed to have zero bulk velocities.
Determining the linearized collision operator in the case of 2D perturbed systems:
We shall be considering to be the perturbation that is introduced
The BGK collision operator is defined as:
In order to determine as a function of , we first need to evaluate the temperature and density distributions by computing the zeroth, and 2nd moments of the distribution function
Since
Determining the linearized collision operator in the case of 3D perturbed systems:
We shall be considering to be the perturbation that is introduced
The BGK collision operator is defined as:
In order to determine as a function of , we first need to evaluate the temperature and density distributions by computing the zeroth, and 2nd moments of the distribution function
Since