3.1. Introduction to Navier–Stokes equations

So we will talk about the difference between the thermodynamic and mechanical pressure. Mechanical pressure derived during the Navier–Stokes equations is not the same as thermodynamic pressure. The mechanical pressure p_bar is negative one-third of the sum of three normal stresses: p_bar = -1/3 (τ_xx + τ_yy + τ_zz) or p_bar = λ + 2/3 μ divergence of V.

The section begins by clarifying the distinction between thermodynamic and mechanical pressure. Thermodynamic pressure is a conventional measure of pressure based on thermodynamic equilibrium, while mechanical pressure is calculated using stress components within a fluid. In a fluid at rest, the mechanical pressure can be expressed as the average of the normal stresses, which represent the internal forces acting per unit area. These stresses include τ_xx, τ_yy, and τ_zz, which are normal stresses acting in the x, y, and z directions respectively. The derived relation shows that mechanical pressure is influenced by the fluid's mechanical properties (λ and μ) and the flow characteristics (divergence of velocity).

Consider a balloon. The air inside the balloon exerts pressure on the balloon walls (mechanical pressure), while the temperature and volume conditions relate to its thermodynamic pressure. If the balloon is perfectly elastic, the mechanical pressure we measure is notably different from the temperature-affected pressure we might calculate based purely on thermal dynamics.

If we want mechanical and thermodynamic pressure to be the same, two conditions can apply: either λ + 2/3 μ = 0 or divergence of V = 0, commonly seen in incompressible flow assumptions.

In order for mechanical pressure to equate to thermodynamic pressure, certain mathematical conditions must be satisfied. The Stokes Hypothesis states that if the fluid is considerably incompressible (divergence of velocity equals zero) or if certain parameters of the fluid (λ and μ) hold specific values, the pressures can align. This is often encountered in typical applications in hydraulic engineering where incompressibility is a standard simplification because it significantly simplifies calculations and analysis, particularly in water systems. However, the assumption that λ + 2/3 μ = 0 is rarely satisfied according to experimental data.

Think of a pipe filled with water flowing steadily. If the water is assumed incompressible, the pressure readings observed at various points along the pipe will reflect the same mechanical pressure as determined under equilibrium conditions, adhering to Stokes’ conditions. This makes the analysis of water flow manageable, aligning practical observations with theoretical predictions.

The Navier–Stokes equations describe the motion of a fluid. The desired momentum equation is derived from combining the deformation law with Newton’s Law, yielding the famous Navier–Stokes equation.

The section introduces the Navier–Stokes equations, which are crucial in fluid dynamics for predicting fluid flow. They originate from the fundamental principles of mechanics and account for forces acting within a fluid. To arrive at these equations, one takes the deformation law for a viscous fluid and merges it with Newton’s law of motion. The resultant equations encapsulate how velocity, pressure, density, and external forces correlate in fluid dynamics under various conditions.

Imagine a swimming pool where you’re moving water by pushing it with your hand. The Navier–Stokes equations would allow you to predict not just the immediate effect of your hand movement on water's velocity and direction but also how that affects other regions in the pool over time. This predictive power is vital in numerous applications from engineering to meteorology.

For incompressible flow (divergence of V = 0), if we assume μ is constant, we can simplify the Navier–Stokes equations for practical hydraulic applications.

In compressible fluids, changes in density can affect flow significantly. However, in incompressible flow—like most water applications—and when the dynamic viscosity (μ) is treated as constant, simplifications to the Navier–Stokes equations can be made. The divergence of the velocity field being zero simplifies the equations and directly impacts how pressure and velocity change as fluid flows through different sections of a system. This constant viscosity assumption is common in hydraulic engineering since many fluids behave almost ideally under typical conditions.

Think of a water hose with a consistent flow. The pressure and velocity of the water when flowing through this hose can be predicted accurately since we’re treating the water as incompressible. This mirrors how hydraulic engineers design systems by using simplified Navier–Stokes equations, ensuring performance is as expected without dealing with the complexity of compressible flows.

For inviscid flow (viscous terms negligible), the Navier–Stokes equations reduce to Euler's equation, which describes the motion of an ideal fluid.

When friction and viscous effects can be neglected in fluid dynamics, the Navier–Stokes equations simplify to Euler's equation. This is particularly useful for high-velocity flows where body forces dominate. Euler's equation provides insights into the behavior of ideal, inviscid fluids, making it easier to analyze phenomena such as shock waves or potential flow theory. Recognizing that some systems can be idealized without viscosity allows engineers to model complex behaviors with less computational power while still obtaining valuable information about flow characteristics.

Think about a perfectly smooth water slide. If a person slides down with minimal resistance (akin to an inviscid flow condition), one can predict the speed and motion easily using Euler’s equation. This analogy emphasizes how ignoring factors like friction in certain scenarios can simplify analysis, just as engineers utilize these simplifications to optimize fluid systems in practice.

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