11.1.4. Recap of Fluid Kinematics

Before starting solving the blackboard applications, let me just have a recap that what we already discussed in the fluid kinematics is that you know that any velocity field we can define as 3 scalar components. The scalar component can have whether velocity scalar component can have whether positions and the time the independent part that is what the velocity distribution. Similar way the rate of the change of the velocity is accelerations but in terms of local accelerations and convective part we can define the acceleration terms.

In fluid kinematics, a fluid’s velocity field can be understood through three scalar components, which refer to the fluid's speed in different directions (x, y, and z). The changes in this velocity are termed accelerations, which can occur locally (at a specific point in the fluid) and convectively (caused by the movement of the fluid itself). Hence, understanding how velocity changes helps in analyzing fluid motion and flow.

Think of the flow of water in a river. The speed of water at any given point can be thought of as its velocity. If the water moves faster at some places (due to the riverbed shape or obstacles), that's like how we describe accelerations in the fluid kinematics context.

Similar way when you have a motion of the fluid particles can have 4 type of conditions the motion and deformations like translations, rotations, linear strain, and the shear strain. We discussed more detail in the last class.

Fluid motion can be categorized into four primary conditions: translations (movement of the whole fluid body), rotations (spinning movement of the fluid), linear strain (deformation while maintaining volume), and shear strain (deformation through sliding). Each type of motion impacts how the fluid flows and behaves in different scenarios, which is critical in fluid mechanics.

Imagine a dance performance: dancers moving in unison across the stage represent translation; a group spinning in circles illustrates rotation; when they stretch and compress while maintaining their position, it's similar to linear strain; sliding or weaving through each other exemplifies shear strain.

And if you look it if you have the rotations we can write the rotations quantity in terms of the velocity field. Similar way the shear strain components also we can write in terms of the velocity gradients and we have the vorticity measures what we derived very details we can compute the what could be the vorticity in different place.

In fluid dynamics, rotation can be quantified through a measure called vorticity, which is derived from the velocity field. Vorticity helps to understand how 'twisted' the flow is at any point in the fluid, indicating how much rotation occurs due to the fluid's motion. It has important implications in predicting flow patterns and behaviors.

Consider a whirlpool in a water basin. The rapid spin of water forms a vortex, and the measure of how fast that vortex twists is analogous to vorticity in fluid dynamics, providing insights into how water moves in that area.

Look at that when you have the flow is incompressible flow that means when you have a volumetric strain equal to 0. The delta dot product of the v velocity should be equal to 0.

Incompressible flow occurs when the fluid density remains constant despite changes in pressure or velocity throughout the flow. This leads to a volumetric strain (change in volume) of zero. In simpler terms, for certain fluids like liquids, their volume doesn't change under flow, which affects how we apply the equations of fluid motion.

Think about how a solid metal ball retains its volume when you push it through water. Similarly, incompressible flow means that the liquid retains its volume even as it moves, like water flowing through a pipe without changing its density.

--