Three methods for deriving equations of motion for perfect fluids, which are fluids with no dissipation and thus no viscosity or heat transfer:

• Hamiltonian Principle
  • Macroscopic analogy to motion of systems of particles
• Kinetic Approach
  • Average motion of large # of interacting particles
• Force balance on a small volume of fluid using conservation laws
  • Least insight but shows how equations follow naturally from a small # of postulates
  • Natural approach as Newton’s Laws tells us the effect of a force on a fluid parcel

An similar, but related concept is that of an ideal fluid which is a perfect fluid with the additional requirement that the fluid’s equation of state is the ideal gas law. The ideal fluid corresponds to the an ideal gas in statistical equilibrium.

Eulerian Approach

Fixed spatial location that measures how the fluid properties change at that point.

The field point of view

Lagrangian Approach

Follows an individual parcel as it moves. Labels are moving with this parcel.

The fluid parcel point of view

We can always map between the location space and the label space (i.e. between the two approaches). Need a mapping to switch between the two approaches for modeling fluids.

Hamiltonian Approach

Particle Labeling and Continuum Limit

Fluids (as a model) are continuous and can be divided into ‘blobs’ that can interact and fill the space it is contained. They can deform and change shape.

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Postulate 3.1:

Fluid particles are open sets (i.e. they don’t contain boundaries) and is essential for a continuous fluid without discrete parts. Thus fluid particles are not points but small volumes.

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Postulate 3.2:

Fluid particles can be identified and remain distinct over some finite time interval between $T_1$ and $T_2$. Therefore we can label and track fluid particles over any finite time interval (in theory).

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These postulates imply that a fluid cannot break

In the continuum limit, particle labels become continuous variables.

Additionally, a constraint we will see emerge when we approach this from kinetic theory is that each fluid particle is infinitesimal in the sense that it is small compared to the scale of the fluid system but large compared to the size of the constituent particles that actually make up the fluid. When this is the case, there is a reasonable number of particles so that the macroscopic variables have good statistics and behave in a smooth and continuous manner. However, if a fluid scale is too small or too sparse, then the continuum limit can fail to be suitable. If a system is too sparse than we can’t break the fluid into fluid particles that are small compared to the size of the fluid system. At scales close to the particle scale, quantum effects and the discrete number of states also becomes appreciable. Essentially this boils down to the idea that discrete many particle systems appear continuous at large scales.

Eulerian vs. Lagrangian Descriptions

Labeling scheme: Define a set of time independent labels $\bold a = (a,b,c)$ to identify each particle. Often this is chosen as the position of the center of mass of the parcels at some initial time. In this case, each value of $\bold a$ describes the initial location of a different fluid parcel. Overtime the locations of these parcels may change, but the labels remain fixed. So $\bold a=(1,1,1)$ will always refer to the same fluid parcel at anytime.

The positions of each particle is then given as a function of the labels and time $\tau$