2.1. Lecture - 53

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.

In fluid mechanics, there are two types of pressures referred to: thermodynamic pressure and mechanical pressure. Thermodynamic pressure is the pressure related to the state of the fluid (temperature, volume, etc.), whereas mechanical pressure arises from the stresses within the fluid. In the context of Navier–Stokes equations, the mechanical pressure calculated is often different from the usual thermodynamic pressure that we might expect.

Think of a tire. The air pressure inside a tire (mechanical pressure) might be different from what you consider when you look at the temperature effects on that air's ability to hold pressure (thermodynamic pressure). They are interrelated but not the same.

The mechanical pressure p bar is negative one-third of sum of three normal stresses. 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.

The mechanical pressure (represented as p bar) in a fluid can be calculated using the average of three normal stress components acting in different directions: τ xx, τ yy, and τ zz. The formula indicates that mechanical pressure is considered as a negative one-third of the sum of these stress components, reflecting the internal forces distributed in the fluid.

Imagine a sponge submerged in water. The sponge absorbs stress from all sides—top, bottom, and sides. The average stress it experiences, similar to the average pressure exerted on the sponge from the water, can be thought of in this manner.

...if you look at the above equation p bar will be = p either lambda + 2/3 mu = 0 or divergence of V = 0.

For mechanical pressure to be equal to thermodynamic pressure, certain conditions must be met. This can occur if either the term involving lambda and mu equals zero or if the divergence of velocity (divergence of V) is zero. This condition is often valid in cases of incompressible flow where certain simplifications can be made.

Think of water flowing through a hose. If there's no change in the water's flow (velocity remains constant), the pressures can be thought of as balanced. In this state, the mechanical pressure can equal the thermodynamic pressure, similar to how fluids behave in pipelines.

Now coming back to this Stokes Hypothesis it says even in case of compressible fluid is lambda + 2/3 mu = 0 then we can have.

The Stokes Hypothesis suggests that the relationship between the viscous coefficient (lambda) and the shear viscosity (mu) can determine whether both mechanical and thermodynamic pressures can be equal in compressible fluids. However, experiments often show that this condition is rarely satisfied, as lambda values are generally positive in practical scenarios.

Think of a filled balloon that might change its shape depending on the air pressure inside (lambda) and how flexible the balloon's material is (mu). If the conditions aren’t just right, the pressures won’t balance out as we expect.

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

The Navier–Stokes equations describe how fluids move by relating the velocity, pressure, density, and viscosity of the fluid. These equations are derived from the fundamental laws of motion (Newton’s laws) and take into account how the fluid deforms under various forces.

Consider how water flows in rivers versus how it behaves when poured from a bottle. The Navier-Stokes equations help scientists understand the different flow behaviors, like turbulence in rivers, as opposed to the smooth flow of water from a bottle.

When it comes to Navier–Stokes equations for practical purposes this is the equation that we are going to use.

In the case of incompressible flow, where the fluid density remains constant and the divergence of velocity is zero, the Navier–Stokes equations simplify. This simplified form is often used in hydraulic applications, making it more practical for engineers.

Think of how water, which we assume is incompressible, flows through various systems like pipes and channels. The simplicity of the Navier-Stokes equations in these situations helps engineers design systems efficiently without complicating factors like compressibility.

So actually we are going to study wave mechanics as part of Inviscid flow because there we will have potential theory and other things.

Inviscid flow refers to flow where viscous forces are negligible. This allows us to simplify the fluid equations significantly. The study of inviscid flow leads to understanding wave dynamics and phenomena that occur in ideal fluids, which lack internal friction.

Imagine a smooth, frictionless slide in a playground. If a child were to slide down, they would move with relatively little resistance, analogous to how fluids behave in inviscid flow situations.

This equation is first order right in velocity and pressure. Thus, it is simpler than another change is in viscous fluid flow.

When discussing inviscid flow, we can derive the Euler equation, which governs the flow behavior of fluids without considering viscosity. This leads to the Bernoulli equation, an important principle in fluid dynamics that relates pressure, velocity, and height in a flowing fluid, especially in applications involving energy conservation.

Consider the Aerodynamics of a plane. As air flows over the wings, the pressure differs above and below the wings, allowing the plane to lift off the ground. Bernoulli's equation helps explain this phenomenon clearly, linking speed with pressure in the flow.

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