PX285 - H1 - fluids

definition

• fundamentally, fluid is a substance which fills up a vessel with no fixed shape
• the molecules in a fluid are not fixed in a lattice like in solids, and they are able to move freely relative to each other

methods of describing fluids

considering every molecule

• each molecule has a velocity, ${\overrightarrow{v}}_i$
• kinematic equations from the newton's second law give the trajectories of the molecules
• it works if there are a small number of molecules

kinetic approach

• using a distribution function, $f(\overrightarrow{r},\overrightarrow{v},t)$, that gives the number of molecules at a certain location (a sufficiently small volume) with a certain velocity (a small interval of velocities) at a certain time
• the boltzmann equation such a function, but it is a 7D problem

continuous matter approach - fluid/hydro-dynamics

• introducing density:

• more rigorously:

• from quantum mechanics, it is known that $\mathrm{\Delta }V$ cannot approach 0
• as $\mathrm{\Delta }V\to 0$, there are fluctuations below a certain volume, $\mathrm{\Delta }{V}^{\ast }$, in $\rho$ due to discrete nature

• for water, air, etc, in atmospheric pressure, $(\mathrm{\Delta }{V}^{\ast }{)}^{1/3}\simeq {10}^{-9}$ mm

• for scales smaller than ${10}^{-9}$ mm, fluid mechanics works

• defining the knudsen number:

where, $\lambda$ is the mean free path, and $L$ is the size of the system of interest

• mean free path is the average distance that a molecule travels between two collisions

• if $k_n\ll 1$, fluid approach is justified

• at atmospheric pressure, $\lambda \simeq 0.1\mu$m

• at $100$ km height above the ground, $\lambda \simeq 1$ cm