9.4.2. Fluid Element Characteristics

Now, let us come back to our basic fluid mechanics okay, so as in the last class I talked about 2 types of descriptions; one is Eulerian frame of descriptions, another is Lagrangian frame of description and that the descriptions we try to visualize as form of virtual fluid balls okay.

Fluid elements in mechanics are typically represented as volumes that contain many particles of fluid. These elements help us understand how fluids behave both at small (molecular) and larger (macroscopic) scales. The Eulerian approach focuses on specific locations in the space through which fluid flows, while the Lagrangian approach follows individual fluid particles over time. In our context, we can visualize these fluid particles as 'virtual fluid balls' that can exhibit various motions.

Imagine a river where you are standing on the bank observing the water flow. You can think of the river's water as many small balls (fluid elements) moving past a fixed point (Eulerian). Alternatively, you might drop a ball into the river and follow its journey downstream (Lagrangian). The concept of 'virtual fluid balls' helps visualize how these particles interact with one another as they flow.

So, when you have a balls then it can have go through any type of motions and the differences like it can be translations like variations that you have a ball, it can be translated...

Fluid elements can move through space due to the velocity of the surrounding fluid. The displacement of a fluid particle is determined by its velocity at a given time and how long it moves. Mathematically, this is expressed as displacement equal to velocity multiplied by time.

Think of a soccer ball on a field. When a player kicks it, the ball moves in the direction of the kick. The distance traveled by the ball can be calculated by knowing how hard the player kicked it (velocity) and for how long the ball rolls across the ground (time). Just as the ball's path can be predicted using these factors, fluid elements likewise translate through their surroundings.

So, the fluid element can go for a rotation, so that means we can compute what could be this angular velocity, if it is going through a rotation...

In addition to translating through space, fluid elements can also rotate. This rotation is described by angular velocity, which measures how quickly an object spins. The rate of rotation for fluid elements can depend on the velocity differences at various points within the fluid, leading to tiny particles rotating around each other.

Consider the way leaves swirl in a whirlpool or a drain. The water spins around as it flows down, and each leaf rotates around the center of that spin. Similarly, fluid elements in fluid mechanics can rotate around points in the fluid, influenced by the surrounding fluid currents.

the fluid can go through 2 types of motions; translations and rotations also, it can have a deformations of linear strains and the shear strain.

Fluid elements can undergo deformations which refer to changes in their shape or size due to external forces or internal velocity gradients. Linear strain corresponds to stretching or compressing, while shear strain relates to how the shape of the fluid element changes without changing its volume.

Imagine pulling on a piece of taffy. When you stretch it, you see it elongate (linear strain). If you twist it, the shape changes without changing its volume (shear strain). Fluid elements can behave similarly under flow conditions, depending on how they are pushed or pulled by the surrounding fluid.

Now, coming to the rate of rotations, which is called angular velocity okay, what could be the angular velocity of the fluid point...

Angular velocity measures how fast something spins around an axis. In fluid mechanics, we calculate the angular velocity of a fluid point using the differences in velocity across four corners of a small fluid element. This helps us understand how the velocity field influences the rotations within the fluid.

Think of an ice skater performing a spin. As they pull their arms in, they spin faster. In fluid dynamics, if certain fluid particles speed up compared to those around them, this creates a rotational effect. Each small 'element' of fluid can spin based on its surrounding velocities just like the skater's movements influence their spin.

Now, coming to shear strain rate, what is this? The shear strain rate is a half rate of decrease of angle between 2 initially perpendicular lines...

The shear strain rate assesses how the shape of an element changes when it experiences shear stress. This change can be calculated using the change in angle between two lines that used to be perpendicular. The half of this change reflects how much deformation occurs due to the shear forces present in the fluid.

Think about how a deck of cards can bend when a force is applied. If you push the ends of the deck toward each other, the angles between the cards change from being perpendicular to slanted. This is akin to how fluid elements experience shear strain when forces act on them, altering their shape without changing their volume.

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