What is the difference between mechanical and thermodynamic pressure?

The difference has to do with the fact that when you sum the normal stress on each face of a differential fluid element using the Newtonian constitutive law, you get something different from the thermodynamic pressure, which is what you normally think of as "pressure". There's a good explanation of it in Viscous Fluid Flow by Frank White.

So the constitutive law for a fluid (or any continuum) is what connects the stress to the strain. For a Newtonian fluid, the constitutive law is:

$$\tau_{ij} = -p\delta_{ij}+\mu(u_{i,j}+u_{j,i}) + \delta_{ij}\lambda u_{k,k}$$

Where $\mu$ is the dynamic viscosity and $\lambda$ is the bulk viscosity, both properties of the fluid. When you sum this over all the faces of the fluid element, you get:

$$\tau_{ii}=-3(p-u_{i,i}(\frac{2}{3}\mu+\lambda))$$

Divide by -1/3 to get:

$$p_{mech} = p_{therm}-u_{i,i}(\frac{2}{3}\mu+\lambda)$$

The original pressure term was the thermodynamic pressure and I added a subscript to make it a little clearer in the last equation. These two pressures are different by the product of the divergence of the velocity and a term related to the material properties. If you're talking about incompressible flow, then there's no difference at all because the divergence of the velocity is zero. If you're talking about compressible flow, then the difference is still small, but depends on how compressible the fluid is and how big this fluid property term is. Stokes basically assumed away the problem by saying that $$\frac{2}{3}\mu+\lambda=0$$ Which is "Stokes Hypothesis"

I don't think I quite understand your question, but I'll do my best.

In Thermodynamics, pressure is defined in a bevy of ways. If we look at the Thermodynamic Identity: $$ dU = TdS - PdV + \mu dN$$ (where $U$ is the Energy, $T$ is the Temperature, $S$ is the Entropy, $P$ is the Pressure, $V$ is the Volume, $\mu$ is the Chemical Potential, and $N$ is the Number of Particles) we can see that pressure is: $$ P = -\left( \dfrac{dU}{dV} \right)_{\text{constant } S,N} = T \left( \dfrac{dS}{dV} \right)_{\text{constant } U,N} = \mu \left( \dfrac{dN}{dV} \right)_{\text{constant } S,U}.$$

However, there are even more identities for pressure (derived in the same way) if we use the Helmholtz Free Energy: $$ F = U - TS \to dF = -S dT - PdV + \mu dN. $$

Mechanical Pressure - at least in the way I think you're thinking of it - is pretty simple, at least relatedly. Pressure is just: $$ P = \dfrac{F}{A}, $$ Force per unit Area.

I'm not the person to ask about pressure in fluid flow. I don't know much about fluid dynamics.

I do know this though, the equation you mentioned: $$ P = \rho RT $$ is called the (monatomic) Ideal Gas Law, and is derived under the assumption that the gas is at equilibrium and is non-interacting, - along with a few other assumptions that I don't remember - so you generally can't apply it to dynamic fluids (though, as others have pointed out, there are various situations in which you can apply it). The pressure in the equation is the Pressure the gas exerts on it's surroundings (i.e. the pressure the gas inside a balloon exerts outwards on the balloon).

Basically we know pressure as force over area: $$p = \dfrac {F}{A} = \dfrac {F x}{A x} = \dfrac {Work}{Volume} = \dfrac{Energy}{Volume}.$$

In continuum mechanics, pressure will be evaluated through the derivation of the strain energy functional with respect to Jacobian of the deformation gradient as: $$ p = \dfrac{\partial(\psi)}{ \partial J}, $$ where $\psi$ is the strain energy functional and $J=det(F)$. This pressure could be a function of $J$ and also temperature, depending on the energy functional of the employed constitutive model.

$\mathbf{Fluid~wise~ talking}$, I think the best way to distinguish different definitions for pressure is to look at Bernoulli equation regarding the fact this is valid only for incompressible fluids as reads:

$$\dfrac{p(static)}{\gamma} + \dfrac{1}{2} \rho v^2 (dynamic~pressure) + Z (related ~to ~hydrostatic ~pressure) = Cte,$$

where $\gamma = \rho g$.

• Static pressure: Pressure at any given point of a fluid (either compressible or incompressible)

• Hydrostatic pressure: The pressure at any given point of a non-moving (static) >incompressible< fluid. For instance, in a Barotropic fluid, the static pressure and hydrostatic pressure are the same.

• Piezometric (head) or Hydraulic pressure: $$h= Z + \dfrac{p(static)}{\gamma}$$ for incompressible fluids.

• Stagnation pressure: The pressure that fluid exerts when it is forced to stop moving:
  $$p_0 = p (static pressure) + \dfrac{1}{2} \rho v^2 (dynamic~ pressure)$$

• Mechanical(total) pressure: $$p(mech) = p (static) + \dfrac{1}{2} \rho v^2$$

• Thermodynamic pressure: The definition of this pressure depends on whether the flow is incompressible (divergence free) or compressible.

---> incompressible: $$p(mech) = p(thermo)$$

---> compressible: $$p(mech) = p(thermo) + \nabla \cdot v * A,$$

where $A$ is a term related to material properties of the flow like bulk and shear modulus.

Also the pressure can evaluated through equation of state (EOS) and in general, it is the rate of internal energy with respect to volume as is: $$p = \dfrac{\partial U}{\partial V}$$

It is clear from the above formulation that the equivalency of thermodynamic and mechanical pressure emerges when the flow is divergence free ($\nabla \cdot v=0$)