Introduction to Navier–Stokes equations

Today, we're diving into the Navier-Stokes equations. Let's start by discussing the kinds of pressure in fluids. Can anyone tell me the difference between thermodynamic and mechanical pressure?

I think thermodynamic pressure relates to the state of a fluid, while mechanical pressure has to do with the stresses acting on it?

Good way to summarize that! Mechanical pressure is derived from the sum of normal stresses, whereas thermodynamic pressure is a property of the fluid's state. Remember: they are not always the same.

How do we quantify mechanical pressure?

Excellent question! Mechanical pressure can be expressed as p̄ = -1/3(σ_xx + σ_yy + σ_zz), where σ represents the stress components.

Does this mean that if we are in a deformed state of flow, they're different?

Exactly! In deforming fluids, these pressures diverge, but under certain conditions, they can align, such as in incompressible flow. Remember the term for flexing pressure: it's 'mechanical pressure'.

So the conditions for them to be the same are important?

Yes! Conditions like the Stokes Hypothesis play a key role. Let’s summarize this session: we discussed how mechanical and thermodynamic pressures differ, their significance in fluid dynamics, and how they relate to the Navier-Stokes equations.

Next, let’s delve into the Stokes Hypothesis. What does it propose regarding fluid behaviors?

Doesn’t it say that under incompressible flow conditions, the properties of the fluid don't change?

Right! Particularly, it suggests that if the divergence of velocity is zero, it supports the idea of incompressibility, which eases the analysis of flow.

What happens to the Navier-Stokes equations under this assumption?

Good inquiry! When we assume incompressible flow, we can simplify the equations significantly. This leads directly to the practical forms you'll regularly use in hydraulics.

Is it common for fluids to satisfy the Stokes Hypothesis?

For Newtonian fluids in many applications, yes! However, it’s less so for compressible flows. Let’s summarize: we explored the Stokes Hypothesis and the implications for flow analysis, particularly regarding incompressibility.

Now that we understand the pressure concepts and Stokes Hypothesis, let's derive the Navier-Stokes equations. Can anyone tell me the basis of these equations?

They are derived from Newton's second law applied to fluid motion, right?

Exactly! By applying Newton's law within the framework of fluid mechanics, we derive the Navier-Stokes equations that govern fluid motion. These equations relate to forces, pressure, and motion of viscous fluids.

What happens if we make some assumptions, like constant viscosity?

When we assume that viscosity is constant, we simplify the equations further. For incompressible flow, terms in the Navier-Stokes equations become uncoupled from temperature variations.

So these equations have really practical implications in engineering?

Absolutely! They allow for predictions of flow behavior in hydraulic applications, leading to designs and solutions in engineering practice. In summary, we derived the Navier-Stokes equations, emphasized the assumptions we can make, and discussed their practical significance.

Lastly, let's connect the Navier-Stokes equations to the Euler equations and Bernoulli's principle. How do they relate?

I think we can get the Euler equations by neglecting viscous terms?

Correct! Neglecting viscous terms in the Navier-Stokes equations allows us to derive the Euler equations, which apply to inviscid flow.

And the Bernoulli equation comes from integrating the Euler equation along a streamline, right?

Exactly! This is a perfect example of how fundamental equations interconnect in fluid mechanics. To summarize, we connected the Navier-Stokes equations to Euler’s equations and derived Bernoulli’s equation as a special case.