3.5. Incompressible flow and Navier–Stokes equations

So we will talk about the difference between the thermodynamic and mechanical pressure. So do you think they both are same, no they are not. So mechanical pressure so the pressure that we derive we find out during the Navier–Stokes equations or any other such equation is the thermodynamic pressure. So the 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. or it is written lambda + 2/3 mu divergence of V and this is equation number 19. If you substitute the value of τ xx, τ yy and τ zz during the previous equation this is what you are going to get okay. So what does this mean? This means that mean pressure in deforming viscous fluid is not = thermodynamic property called pressure.

In fluid mechanics, there are two types of pressure discussed: thermodynamic pressure and mechanical pressure. While thermodynamic pressure is a property associated with the fluid's thermodynamic state, mechanical pressure (denoted as p bar) is derived from the behavior of stresses within the fluid. Specifically, mechanical pressure relates to the average of three normal stresses, which are pressure components acting on the fluid at various orientations. The equation shows this relationship, where p bar is equal to the negative third of the sum of three normal stresses, and further interprets the influence of the fluid's properties like the viscosity (mu) and the divergence of velocity (V). This distinction illustrates that pressure within a deforming viscous fluid isn't purely a thermodynamic property.

Imagine a balloon filled with air. The air pressure inside is thermodynamic pressure, which changes with temperature and the amount of air. When you press the balloon, the internal forces generate mechanical pressure that reacts to the deformation and stress applied. So, while you can measure the air pressure (thermodynamic) in the balloon, the stresses created by your hand pressing on it can be considered mechanical pressure.

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. So let us say =0 actually this is called Stokes Hypothesis all right. The second way is divergence of V can be if it is 0 then also this is possible. This is possible this is more commonly possible because divergence of V 0 for incompressible flow. So that is when we deal with water and hydraulics you see I mean all the time we assume incompressible flow.

For both thermodynamic pressure and mechanical pressure to equate in a fluid, certain conditions must be satisfied. The first condition is captured in the Stokes Hypothesis, which states that the coefficient pertaining to shear stresses (lambda) and the divergence of velocity must equal zero. If the divergence of velocity (often denoted as V) is zero, this implies that the fluid is incompressible, which is a standard assumption in hydraulic applications such as water flow. This understanding is crucial because it helps simplify the Navier–Stokes equations for incompressible flow.

Think of a swimming pool filled with water. When you dive in and move around, the water doesn’t compress; it flows and moves around you without a significant change in density. Here, we consider the water as incompressible, and thus, we can simplify the calculations of flow using conditions like divergence of velocity being zero, which aligns with the idea that there's a consistent flow of water around your movements.

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. So if you look at the Navier–Stokes equations above this will be 0 this the last term divergence of V all right. Therefore, in equation 20 if we assume mu is constant okay as well because in most of the hydraulic purposes we assume mu is constant.

In the context of the Navier–Stokes equations, analyzing the scenario of incompressible flow is essential. Incompressible flow implies that the fluid's density (rho) remains constant. This results in the divergence of velocity (V) equating to zero. By substituting this into the Navier–Stokes equations and also assuming viscosity (mu) is constant—typical in hydraulics—we can simplify the equations significantly, yielding a more manageable form specifically tailored for incompressible flow conditions.

Consider how water flows from a garden hose. The flow is smooth, and the viscosity of water doesn’t change significantly as it moves through the hose. We assume it remains incompressible, allowing us to use simplified equations (like the Navier–Stokes equations for incompressible flow) to predict how the water will behave under different conditions.

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 (()) (13:14) to discuss about mention Euler and the Bernoulli theorem how it has its origin okay. So if we assume that viscous terms are negligible as well in equation 21 then Navier–Stokes NS can be reduced to. This is equation number 22.

When we consider inviscid flow (flow without viscosity), the Navier–Stokes equations can be further simplified. By neglecting the viscous terms, we derive the Euler equation, which is significant in the study of potentials in fluid mechanics. The Euler equation facilitates discussions about concepts like wave mechanics and the origins of the Bernoulli theorem, providing insights into fluid behavior under ideal conditions where viscosity is not a factor.

Imagine a perfectly smooth waterslide with no water resistance (like what happens in inviscid flow). The motion of a person sliding down can be described using simplified rules (Euler equations), ignoring the frictional forces present in real-life scenarios. This allows for a clearer understanding of how fluid moves under ideal conditions.

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