1.3. Basic Properties of Material Derivative

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 the material derivative.

The material derivative is a crucial concept in fluid dynamics and describes how a physical quantity changes for an object moving with the fluid. It's essentially a combination of the local derivative (change at a point in time) and the convective derivative (change due to motion through space). In this section, we learn its properties, which will help us derive the Navier-Stokes equations later.

Think of a leaf floating on a river. As the river flows, both the leaf and the water's properties, like speed or temperature, change. The material derivative helps measure how these changes affect the leaf, taking into account both its local environment and its movement with the river.

So, there is a term called the vorticity. So, the vorticity of the fluid, this is actually the rate of rotation. The vorticity of fluid given by omega in vector form is defined as curl of the velocity vector.

Vorticity is a measure of the local rotation in a fluid flow. It is represented by the symbol 'omega' and calculated as the curl of the velocity vector. Essentially, if there's any rotation in the fluid particles, the vorticity quantifies it. This understanding is crucial for studying fluid behaviors, especially under rotational movements.

Imagine swirling a cup of coffee with a spoon. The coffee particles near the spoon rotate around it. This swirling action is a result of vorticity; the more intense the swirl, the higher the vorticity. Visualizing this can help understand how fluids behave during rotation.

So, you know already for irrotational flow, omega is actually 0, because the vorticity vector should be 0.

In an irrotational flow, there is no local rotation of fluid elements, meaning vorticity is zero. This situation is important in simplifying fluid dynamics problems, as it indicates the flow is smooth and consistent without turbulence. Recognizing irrotational flows aids in applying various equations in fluid mechanics effectively.

Consider a calm lake on a windless day. If you drop a pebble in, the water ripples outward evenly without swirling, representing an irrotational flow. The water doesn't have any vorticity because there's no local spinning in the fluid elements.

The 2-dimensional shear strain is the average decrease of the angle between the sides AB and BC. So, we can define this as the average decrease of the angle.

Shear strain measures how much a material deforms when subjected to shear stress. In two dimensions, it's defined as the average decrease of angle between lines that are being sheared. Recognizing these strain behaviors helps in understanding how materials respond under various forces and is pivotal in engineering applications like structural analysis.

Imagine two stacked cards being pushed in opposite directions. The angle between them decreases as they shear. The amount of tilt or slide due to this force is the shear strain, illustrating how materials deform under stress, much like bending a piece of cardboard.

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.

Dilatation or extensional strain measures how much a material expands or contracts in response to stress. It represents the change in length relative to the original length of the material. Understanding this concept allows engineers to predict how materials will behave under stretching or compressing forces.

Think of a rubber band when you stretch it. The rubber band gets longer—this change in length relative to its original length is a form of extensional strain. The greater the stretch, the more significant the strain, which is essential to consider for any elastic material's design.

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