1.2. Viscous Fluid Flow (Contd.)

Welcome back students. So, this second lecture of viscous fluid flow, so where we are in the end going to derive the Navier-stokes equation but to be able to derive that we had to learn basic properties about thematerial derivative.

In this section, the professor welcomes students back and emphasizes that this lecture will focus on deriving the Navier-Stokes equation, which is fundamental in fluid dynamics. However, before diving into the derivation, students need to grasp some basic properties of fluid flow, including the material derivative, which describes how fluid properties change along the flow trajectory.

Think of the material derivative like tracking the change in temperature as you move along a river. If you float down the river, your temperature may change based on the surrounding water temperature, which varies from place to place.

And also the rotation, what is rotation, how this angle is rotated, so we started from basics and until now we have derived that the rotation per unit time is given in a vector form as this, that was where we concluded in the last lecture. So, we will continue now, so this is, so there is a term called the vorticity. So, the vorticity of the fluid, this is actually the rate of rotation, this above equation, which let us say, I will name it as equation number 4. So, the vorticity of fluid given by omega in vector form is defined as curl of the velocity vector.

This chunk discusses the concept of vorticity, which represents the rotation in fluid flow. The professor explains that vorticity can be mathematically represented as the curl of the velocity vector, symbolized by omega. Understanding how fluid rotates can help in analyzing flow patterns, energy distribution, and how forces act within fluid systems.

Imagine the whirlpool effect when you pull a plug from a filled sink. The water rotates around a central point, and this rotation captures the essence of vorticity in fluid dynamics. The strength of this rotation can be quantified to analyze how the water moves.

Hence, so this is the vorticity definition and this is; so, if you see, our angle of rotation here is half of the vorticity, it is actually ω/2 and vorticity is actually twice of that, so we can simply write 2, so this is another important result, which we have proved that vorticity is twice.

Here, the lecture highlights the relationship between vorticity and the angle of rotation. The angle of rotation experienced by fluid elements is half of the vorticity. This relation helps in providing a clearer understanding of how rotation governs fluid behavior at microscopic levels and assists in learning about fluid momentum and forces.

Consider the Earth spinning on its axis. The rate of rotation can be observed in the time it takes for the planet to complete a full rotation (24 hours). If you measure the angular displacement (the angle rotated), it is comparable to how vorticity signifies rotational behavior in fluids.

Because the rate of rotation we already derived and found out to be 1/2 of the curl, of the velocity vector, so we will go to the next page to continue. So, you know already for irrotational flow, omega is actually 0, because that is in; that is the vorticity vector should be 0.

This piece explains irrotational flow, which occurs when there is no rotation in the fluid; hence, vorticity (omega) equals zero. This type of flow is significant in many engineering applications, as it simplifies the equations governing fluid motion, allowing for easier calculations and predictions in ideal fluid conditions.

Picture a perfectly calm lake on a windless day. When you drop a pebble into the lake, the ripples extend outwards without causing any rotational motion in the water—this represents irrotational flow as there’s no vorticity produced.

Now, the 2 dimensional shear strain is the average decrease of the angle between the sides AB and BC, in this figure.

Shear strain refers to the deformation that occurs in a material when forces are applied tangentially. The section introduces a concept of 2D shear strain by discussing the average decrease in the angle formed by two sides, which is vital for understanding how fluids and materials respond to applied stresses.

Visualize stretching a rubber band: when you do, the angle between your fingers changes. The consequent change in angle represents shear strain; this is similar to how fluids will adjust under force.

So, the dilatation or extensional strain is defined as the ratio of the; see, the dilation or the extensional strain is defined as the rate of the change in length to the original length.

Dilatational strain refers to how much a material stretches or compresses relative to its original length. Mathematical ratios express this change, crucial for understanding how fluids expand or contract when subjected to temperature or pressure changes. It connects closely with fluid dynamics and thermodynamics.

Imagine a balloon. When you blow air into it, the balloon stretches. The change in its length (volume) compared to its starting size demonstrates extensional strain, similar to how some materials behave under fluid dynamic conditions.

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