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If I have a fluid in motion at a given time and I know its pressure $p$ everywhere, I know its dynamic viscosity $\mu$ and I know its velocity field $\vec V$.

This system must hold some intrinsic potential energy $E$.

I know that in a fluid at rest the pressure is a measurement of potential energy, so if the fluid is either static or has uniform velocity then I should just get:

$$\Psi(x) = p(x)$$

I.e. the energy density at a given point is just the pressure.

However, if the velocity gradient is not uniform, then there must be potential energy dependent on both the velocity gradient at a point and the viscosity of the fluid.

What should the energy density be in this case?

Kyle Kanos
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Makogan
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1 Answers1

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You're looking for an equation of state (EOS); for instance, an ideal gas follows, $$p=\rho e(\gamma-1)$$ where $p$ is the pressure, $\rho$ the density, $e$ the specific internal energy and $\gamma$ the adiabatic index.

As far as I know, there isn't an EOS that uses the viscosity, which should make sense since it is not a state variable. It only plays a role in the dynamic evolution via the Navier-Stokes equations.

Kyle Kanos
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  • Is there not a way I could express the dynamic evolution as an equation of state nonetheless?

    From a high level, it is clear that if one looks at a specific time, the fluid in motion has energy contained into it, and that this energy is dependent on it;s internal velocity field and viscoscity. As time passes this energy will dissipate as heat. So if the fluid has energy at a given time one should be able to describe the energy, no?

     – Makogan May 05 '23 at 07:30
  • No, the "dynamic evolution" I mention is time-evolution of the fluid system as solved by the Navier-Stokes PDE. The EOS relates pressure, density, temperature, energy, etc. While you use the EOS as a closure condition, the two are separate things. – Kyle Kanos May 05 '23 at 13:45