2. Lectures

So proceeding forward so we will talk a little bit before writing the Navier–Stokes equations we will talk about the difference between the thermodynamic and mechanical pressure. So do you think they both are same, no they are not.

In this chunk, the lecturer introduces the distinction between thermodynamic pressure and mechanical pressure. Thermodynamic pressure is typically related to the properties of the fluid at a microscopic level, while mechanical pressure is derived from the fluid's behavior observed through equations like the Navier–Stokes equations. Understanding this difference is crucial for further discussions about fluid behavior.

Imagine a balloon filled with air. The pressure you feel when pinching it is mechanical pressure, but the pressure that accounts for the energy within the air molecules is thermodynamic pressure. These two pressures interact but are not the same.

So mechanical pressure p bar is negative one-third of sum of three normal stresses all right. What are the three normal stress τ xx, τ yy and τ zz. So p bar can be written as – of 1/3 τ xx + τ yy + τ zz.

Here, the lecturer provides a mathematical representation of mechanical pressure, showing that it can be derived from the stresses in three different directions of a fluid element. Each of these stresses corresponds to how the fluid resists deformation in those particular directions. Understanding this formula allows for the calculation of mechanical pressure in a given flow.

Think of a sponge being squeezed: the pressure exerted on it comes from the stresses produced by squeezing it in multiple directions. Each direction represents one of the stresses τ xx, τ yy, and τ zz.

However, if we want both of them to be same two different ways right. One is if you look at the above equation p bar will be = p either lambda + 2/3 mu = 0 or divergence of V = 0.

This chunk discusses Stokes Hypothesis, which provides the conditions under which mechanical and thermodynamic pressures can be considered equal. These conditions are significant in fluid dynamics, especially for incompressible flow. The divergence of velocity being zero implies that the fluid's density is constant, simplifying the calculations and assumptions in fluid dynamics.

Think about a calm pond: when water isn't moving (incompressible flow), the pressure at any point is simply due to the weight of the water above it. Here, mechanical and thermodynamic pressures align because the flow is steady and undisturbed.

So now going back to the objective of this module the Navier–Stokes equations okay. So desired momentum equation for general Newtonian viscous fluid is obtained by equation 18 that is the deformation law in rewritten Newton’s Law.

This part shifts the focus back to the main objective: deriving the Navier-Stokes equations, which describe how fluids behave under various forces. The Navier-Stokes equations are foundational in fluid mechanics, providing a comprehensive framework to analyze fluid flow. The derivation combines Newton’s laws and the principles of fluid deformation.

Think of the Navier-Stokes equations as the rules of a game of basketball. Just as the rules dictate how players can move and interact on the court, these equations define how fluid particles move and interact based on forces acting upon them.

So to do that we go to a fresh page so incompressible flow what is that if rho is that means incompressible flow that means divergence of V = 0.

In this chunk, the lecturer defines incompressible flow, emphasizing that for certain types of fluids (like water), the density remains constant, thus simplifying the Navier-Stokes equations. The condition of divergence of velocity being zero is critical, as it helps in practice to achieve simpler forms of these equations.

Consider water flowing through a pipe. As long as the flow is steady and we assume it's incompressible, the density of water doesn't change, much like a steady stream of cars passing through a toll where each car's size remains constant, leading to easier calculations of traffic flow.

So actually we are going to study wave mechanics as part of Inviscid flow because there we will have potential theory and other things, but here in Inviscid the Euler it to discuss about mention Euler and the Bernoulli theorem how it has its origin.

Here, the discussion shifts towards inviscid flow, which assumes there are no viscous forces acting on the fluid (like air or superheated gas). The Euler equations describe the motion of such flows and lead to foundations like the Bernoulli theorem, crucial for understanding energy conservation in flowing fluids.

Think about a smooth ball rolling down a hill without friction – the forces acting are only due to gravity and not any viscous drag. This scenario is similar to how inviscid flow works, where viscosity is negligible and we can study pure energy dynamics.

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