3. Viscous Fluid Flow (Contd.)

Welcome back students to the final lecture of this module viscous fluid flow where we are deriving the Navier–Stokes equations. In the last lecture we wrote the general deformation law for Newtonian Viscous Fluid okay this equation we call it equation number 18.

In this section, the professor is introducing the topic of viscous fluid flow and its relation to the Navier-Stokes equations. The deformation law for Newtonian viscous fluids is fundamental, as it describes how these fluids respond to applied forces. It's essential to distinguish this law, referred to as equation number 18, as it will lead to the derivation of the Navier-Stokes equations in this lecture.

Imagine kneading dough. When you apply pressure (force) to the dough, you can see it stretch and change shape. This represents how a viscous fluid deforms under stress, much like how the dough conforms to the forces applied to it.

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. 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.

The professor is highlighting the distinction between thermodynamic and mechanical pressure. While they might seem similar, in the context of fluid flow and equations like the Navier-Stokes, they refer to different concepts. Thermodynamic pressure is associated with the state of the fluid (like temperature and phase), while mechanical pressure results from the fluid's environment and deformation under stress.

Think of a balloon. The air pressure inside is thermodynamic, as it relates to the gas laws and the state of the air. However, when you squeeze the balloon, the pressure you feel on the outside is mechanical pressure, resulting from the physical deformation of the balloon's material.

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.

This portion introduces the calculation of mechanical pressure in a viscous fluid. The professor explains that the mechanical pressure can be derived as a function of the normal stresses acting on the fluid in three orthogonal directions (x, y, and z). Specifically, it states that the mechanical pressure is equivalent to negative one-third of the sum of these three normal stresses, providing a formula for easy calculation.

Imagine sitting on a soft sofa. The pressure you exert on the sofa can be thought of as the normal stresses. If you sit in three different positions (flat on your back, on your side, and leaning forward), the total pressure felt by the sofa can be viewed as the sum of these pressures. The mechanical pressure of the sofa 'feeling' your weight is similar to how fluids experience internal stresses.

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 is called Stokes Hypothesis.

The Stokes Hypothesis provides conditions under which mechanical pressure and thermodynamic pressure can be considered equal. If either the parameters lambda and mu fulfill the specific requirement or if the divergence of velocity is zero, these two pressures can equate, typically observed in incompressible flows. Understanding this hypothesis is crucial in fluid mechanics as it simplifies analysis under certain flow conditions.

Think about a water bottle under pressure. If you keep filling it without allowing any air to escape (zero divergence), the pressure inside (mechanical) can align with the thermodynamic pressure if no heat is exchanged with the environment, representing the conditions outlined in the Stokes Hypothesis.

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.

The professor emphasizes the goal of the lecture: to derive the Navier-Stokes equations for Newtonian viscous fluids by applying the deformation law discussed previously. This is a significant step in fluid dynamics, as these equations govern the motion of fluid substances under various forces and conditions.

Consider cooking spaghetti. Following a recipe (like Newton's Law) is akin to using the deformation law, then combining all the ingredients methodically to reach the finished dish (Navier-Stokes equations). Just as in cooking, precise steps lead to understanding and mastering fluid dynamics.

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 part, the concept of incompressible flow is introduced. The professor mentions that incompressible flow implies that the fluid density does not change (rho is constant), which leads to the divergence of velocity being zero. This concept is crucial because many fluids, like water, are treated as incompressible in many engineering applications.

Think of a full balloon. When you squeeze it, the volume remains relatively constant, and the shapes change, while the internal pressure adjusts. This is like incompressible flow, where the density stays constant even when shapes and conditions around it change.

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.

Here, the professor indicates that for incompressible flow, if viscosity (mu) is assumed constant, it simplifies the Navier-Stokes equations further. This simplification allows engineers to analyze most hydraulic systems effectively, as many practical fluid scenarios involve constant viscosity under typical conditions.

Think about oil flowing through pipes. If the oil temperature doesn't change much (constant viscosity), the calculations needed to ensure smooth transport are easier. Just like using a consistent method to successfully bake a cake avoids complications, using constant viscosity simplifies fluid calculations.

And now moving to the last part again that is called Inviscid flow.

This transition marks the introduction to the concept of inviscid flow, where the effects of viscosity are negligible. The discussion will later tie into wave mechanics and the Euler equations for inviscid flows, which are vital in defining how fluids behave when viscosity does not significantly affect their motion.

Picture a smooth sheet of ice. When sliding over it, air moves effortlessly with little resistance, akin to inviscid flow where viscosity is minimal. This concept is essential in fluid dynamics as it often simplifies equations governing fluid behavior, especially in high-speed or low-viscosity scenarios.

So if we assume that viscous terms are negligible as well in equation 21 then Navier–Stokes NS can be reduced to…

This discussion leads to the Euler equation for inviscid flow. The professor mentions how the Navier-Stokes equations can be simplified drastically when viscous effects are ignored. This simplification is crucial for understanding inviscid flow behavior, such as in high-speed aerodynamics.

Think of a high-speed train slicing through the air. The air behaves almost as if it has no viscosity due to the train's speed, allowing engineers to simplify the fluid dynamics equations. This likens to observing how a stone thrown into still water creates ripples, while a fast-moving object creates fewer disturbances.

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