# Microscopic definition of dynamic pressure of fluids

So we have a moving fluid and we know from Bernouli's equation that there is a term called dynamic pressure(not to be confused with the hydrostatic pressure of the fluid). So,what exactly is it and how can it be explained microscopically? Note:do not involve relativity please!

Dynamic pressure of fluids is the kinetic energy per unit volume of the fluid. Its unit are the same as pressure, and Bernoulli's equation,

$0.5\rho v^2 + \rho gy + p = constant$

can be written as

$P_{dynamic} + \psi_{gravity} + P = constant$, where $P_{dynamic} = 0.5\rho v^2$ and $\psi_{gravity}$ is the force potential of gravity.

As for microscopic explanation, it is the total kinetic energy per unit volume of a fluid (average sum of KE of particles per unit volume).

Update: Microscopic explanation in more detail

Consider all the particles in a (tiny) region with constant flow speed $v$. Consider one particle. This particle has an arbitrary velocity, $v_i$, and mass $m_i$. Hence, the kinetic energy of this particle is $0.5mv_i^2$. The total kinetic energy in the region is

$\Sigma 0.5m_iv_i^2 = 0.5\Sigma m_iv_i^2 = 0.5 \Sigma m_iv_i \cdot v_i = 0.5\Sigma m_iv_i \cdot \Sigma v_i = 0.5m_{total}v_{cm} \cdot v_{flow} = 0.5m_{total}v_{cm}^2$, using the fact that the flow velocity is equal to the center of mass velocity in a region of constant flow velocity (so, any tiny region in a fluid).

Hence, the total kinetic energy per unit volume of a fluid is equal to the dynamic pressure term, explaining the motivation for the dynamic pressure term. Note: This term comes from an energy analysis, not a force analysis.