# Continuity Between Statistical Mechanics and Fluid Dynamics

1. Dec 3, 2008

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Imagine a jet of fluid (perhaps air) impinging on a flat plate. It could be said that the jet has a slightly higher mean velocity in the direction normal to the flat surface (we'll arbitrarily call this X).

From a classical thermodynamic point of view it could be said that the gas has a higher temperature (I guess the normal distribution of gas velocities is now centred around a slightly higher median velocity in the X direction), although the distribution of particle velocities is no longer isotropic as one would expect for a static fluid.

Using the laws of classical thermodynamics, shouldn't heat (molecular kinetic energy) be transferred from the fluid to the plate as the molecules in the fluid are on average travelling faster than the molecules in the plate (in the x direction at least).

I imagine the amount of heat transferred to the plate would be relatively small for a jet of say 20m/s given a plate and fluid initially at room temperature, as the average particle velocities at room temperature are in the order of hundreds of meters per second.

From an fluid dynamics point of view, the power dissipated by the jet on the plate should be some function of the drag force on the plate and the velocity of the jet.

From a thermodynamic point of view, shouldn't this power dissipated by the jet acting on the plate be equal to the heat transferred to the plate by virtue of the disparity between the jet's molecules' kinetic energy and the plate's molecules' kinetic energy (temperature), since all of the work done by the fluid on the plate must result in the heating of the plate.

Shouldn't there be some sort of direct coherence between the shear force applied to the plate (determined using fluid dynamics) and the heat transferred to the plate.

The fluid should heat the plate through frictional effects (shear between the fluid and plate), but what I'm getting at is; (I think) the heating calculated through frictional shearing should be equal to the heat transferred to the plate if analysed using statistical mechanics methods.

If so, could one use statistical mechanics to ascertain the drag coefficient of a body without having to empirically test a model, by solving equations for the heat transferred to the plate (using statistical mechanics) and the power dissipated by drag(0.5*rho*A*V^3Cd)

I guess without knowledge of the collisional efficiency between the fluid's and plate's molecules and the velocity distribution it could be difficult to arrive at figure for the heat transferred to the plate, however I don't see why these properties can't be calculated, at least for simple geometries.

Fundamentally both approaches describe are the same process. The shear forces on a plate incurred by a jet impinging on it are derived from macrostate descriptions of the fluid molecules interacting with the plate, whereas a statistical mechanics analysis is derived from a microstate description of the molecules interacting with the plate (this analysis looks more like classical heat transfer than some hydrodynamic phenomenon like drag)

Fundamentally, the molecular processes are the same (they have to be right!). I'm hypothesising whether the drag coefficient of a body be analytically derived by approaching the analysis of the frictional effects by considering them to be on a molecular scale?

The shearing frictional process between a fluid and immovable body is afterall very similar (on a molecular level) to a conventional heat transfer process. Friction occurs through the collision of molecules in the fluid with molecules in the plate.

The only difference I can see here is that the molecules have some aggregate velocity rather than an isotropic velocity distribution (which would be the case if a static fluid at a higher temperature were transferring heat conventionally to the plate).

It may be that the modelling of these statistical parameters is incredibly difficult (moreso than conventional macrostate fluid modelling techniques, e.g. CFD).

2. Jun 4, 2009

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Anyone? I'd like to edit the post to contain more question marks and be a bit more concise, but I can't seem to anymore.