8. Newton's Second Law

So, similar way if I am looking at Newton's second law also see that force is equal to mass into acceleration, force is vector component, acceleration is the vector component, okay and both are the parallels okay, so force and the vector, at the particle levels like in solid mechanics, the force we can put is mass into acceleration.

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). In this relationship, both force and acceleration are vector quantities, meaning they have both magnitude and direction. When discussing the law at a particle level, we treat each particle's mass and acceleration to understand how forces affect them.

Think about pushing a shopping cart. The more force you apply (like pushing harder), the faster (or more accelerated) the cart moves. If the cart is heavier (more mass), you need to push harder to get it to accelerate. This illustrates how force, mass, and acceleration are related.

Now, let us look it at the particle levels, if I had to find out what is the acceleration of the fluid particles? Is nothing else is a time derivative of velocity of the particles, as you know it from class 10, 11th and 12th I am just doing this time derivative of the velocity; particle velocities with respect to time and that is what represents the accelerations, at the particles levels you will have these.

To find the acceleration of fluid particles, we take the derivative of their velocity with respect to time. This means we measure how the velocity of the fluid particles changes over time. Acceleration can be thought of as a change in speed or direction for these particles, and it's calculated directly from their velocity.

Imagine a car speeding up on a road. If you know how fast the car is going at different points and how that speed changes, you can determine the car's acceleration. For fluid particles, it's similar; we track how their speeds change over time to understand their acceleration.

Now, if you look at this the velocities has a variability in a positions and the time because of that though when you define a; derivative with respect to time you will have a local component okay, you will have a with a x particle directions, y particle directions and z particle direction.

Accelerations can be broken down into local components along different spatial coordinates (x, y, and z). This means that the total acceleration of a fluid particle can vary depending on its position in space. By analyzing these local components, we can gain a more comprehensive understanding of how particles in the fluid are behaving in three-dimensional space.

If you're riding a bicycle on a hilly path, your acceleration isn't the same throughout. When you're going uphill, you slow down (negative acceleration), and when going downhill, you speed up (positive acceleration). Each direction you move in affects your total acceleration, just like how the fluid particles' acceleration changes based on their position.

This is nothing else if you are considering is a 2 variables like I just discussed you the Taylor series, if you remember it defining for the 2 variables in this case, I have a Taylor series of 4 variables the x, y, z and the t. If you expand it and take it only these 4 terms, you will get this component nothing else, we are near to the same Taylor series concept what we discussed in a single variables, independent variables, how the Taylor series expansion, the 2 independent variables how the Taylor series expansion.

The Taylor series is a mathematical concept used to approximate functions. In the context of fluid dynamics, we can extend this idea to functions of multiple variables (like x, y, z, and time) to understand how the acceleration of fluid particles can be expressed and approximated mathematically. This approach helps simplify complex dependencies based on the position and time of the particles.

Consider making a cake: if you change the amount of flour (x), sugar (y), and baking time (z), each adjustment affects the final cake differently. Similarly, using a Taylor series allows us to see how changes in position and time affect the motion of fluid particles.

If you look at that this is what is called the local accelerations, the velocity changes as with respect to the time only, so that means at the probe equations because of only the velocity changes at that locations will give us the local accelerations. Velocity is also changes because of change of the velocity field that is the velocity variances x gradient, y gradient, z gradient and the u, v, w component.

Local acceleration refers to changes in velocity over time at a specific location in the fluid. In contrast, convective acceleration accounts for changes in velocity due to the movement of the fluid through varying velocities in space. Understanding the distinction between these allows us to analyze fluid behavior more effectively.

Think of a river: a fish at a specific spot (local acceleration) experiences changes in flow speed due to the water speed at that exact location. However, as it swims downstream (convective acceleration), it moves to sections where the water flows differently, experiencing new speeds. Both types of acceleration are important for understanding the fish's movement.

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