3.2. Difference between thermodynamic and mechanical pressure

Mechanical pressure (p bar) is negative one-third of the sum of three normal stresses: $\tau_{xx}, \tau_{yy}, \tau_{zz}$. This can be expressed as:

p_{bar} = \lambda + \frac{2}{3}\mu \nabla \cdot V$$ where $\lambda$ and $\mu$ are fluid properties.

Mechanical pressure is a specific measure of pressure that relates to the internal stresses in a fluid. It is calculated by taking the average of three types of stresses that act normal to the surface of the fluid element. This means it considers how much force is applied per unit area due to these stress components. The equation shows that mechanical pressure depends on both a constant called $\lambda$ and a term that involves the divergence of the velocity field, which affects the fluid's motion. Understanding this is crucial because it helps us differentiate between mechanical pressure and thermodynamic pressure in fluid dynamics.

Imagine trying to measure how hard you can push against a balloon. The 'mechanical pressure' tells you how the air inside is pushing back at your fingers based on its shape and how it's deforming. Just like in our equation, the pressure will change based on the balloon's material properties (like elasticity) and how much you're pushing on it!

Mechanical pressure and thermodynamic pressure can be equal under specific conditions:

1. $\lambda + \frac{2}{3}\mu = 0$
2. $\nabla \cdot V = 0$

The first condition is known as the Stokes Hypothesis, and the second is common in incompressible flow, like water.

For both mechanical and thermodynamic pressures to be the same, certain conditions must be fulfilled. The Stokes Hypothesis posits that the relationship between the constants $\lambda$ and $\mu$ must be very specific, which isn't often the case in real fluids. Additionally, for conditions that denote incompressibility, like most liquids, the divergence of the velocity must equal zero. This simplifying assumption allows engineers and scientists to work with a more straightforward model while analyzing fluid behavior in various situations.

Think of a scenario where you're trying to find the temperature of the sea. Under certain stable conditions, the surface temperature (which is like thermodynamic pressure) and the temperature at deeper levels (akin to mechanical pressure) can be nearly the same. If the water is calm (equivalent to the divergence being zero), the temperatures equalize. However, as you go deeper and face different pressures (like wind creating waves), these temperatures can start to vary.

The Stokes Hypothesis is rarely satisfied because $\lambda$ is usually positive. Therefore, it's uncommon to find situations where $\lambda + \frac{2}{3}\mu = 0$.

This hypothesis is an important concept in fluid mechanics, but its practical applicability is limited. Since $\lambda$ represents a physical property related to fluid compressibility, it's predominantly positive, leading to the conclusion that finding instances where the condition holds true is quite rare. This realization emphasizes that while mathematical models are useful, very few real-world situations can perfectly align with simplified assumptions like those of the Stokes Hypothesis.

Consider a car on different terrains. On a flat highway, the signs (like the Stokes Hypothesis) may apply quite easily. However, when you hit a bumpy off-road path, the situation changes drastically. The simple rules you believed applied start to break down because the conditions change, reflecting how rarely the Stokes Hypothesis is met in actual fluid situations.

--