3.1.6. Reynolds Transport Theorems

The basic idea comes here is that my control volume which is have the dimensions of like if it is the x this is the y and this is the z coordinate. If this is the x y and z coordinates and this is the dimensions let we consider it is dx dy and dz that is my very simplified a parallel fight control volumes having a dimensions of dx dy dz.

In fluid mechanics, we often analyze fluid flow through a specific region called a control volume. The dimensions of this control volume are defined by small changes in three directions: x, y, and z, which we denote as dx, dy, and dz. This setup allows us to simplify our calculations by treating the control volume as a small box where we can easily apply fundamental principles like conservation of mass and momentum.

Think of a water tank where you're trying to measure how much water enters and exits a specific section. By dividing this tank into smaller sections or layers (represented by dx, dy, dz), we can better understand how water moves through the tank and apply conservation principles to find answers.

If my dx is tending towards 0, the dy tending towards 0, and the dz tending towards 0. That means what I am looking at the my control volume tendings towards a infinitely small the control volumes converging towards a point value.

When we shrink the dimensions of our control volume (dx, dy, and dz) to be infinitesimally small, we approach the idea of analyzing fluid flow at a single point in space. This allows us to write differential equations that describe how properties like pressure, velocity, and density change instantaneously at that point, rather than over a larger, more generalized area.

Imagine zooming in on a specific point in a river to observe the water's speed and pressure at that exact spot. By ignoring everything else around it, you can get a clear understanding of the dynamics at play at that point, much like reducing our control volume to an infinitely small size.

So if you do that, that if you reduce your control volumes to a smaller contour volume as smaller as that it is close to a points. So, in that case we get a set of a partial differential equations that is what we will going to derive it set of the partial differential equations that is the partial differential equations differential equations for mass and linear momentum.

By applying the concept of infinitely small control volumes, we can derive a set of partial differential equations that govern fluid behavior, particularly regarding mass and momentum. These mathematical expressions are crucial as they describe how mass is conserved and how forces act in fluid systems, which is fundamental for fluid mechanics and applications in engineering.

Consider how local weather systems are modeled. Meteorologists use differential equations to predict changes in temperature, wind, and moisture at specific locations, helping them forecast weather by understanding fluid dynamics in the atmosphere.

If I write the partial differential equations for mass and linear momentum, the basically I will get four basic equations: one is mass conservation equations and the 3 equations which is a vector forms of linear momentum. These equations are coupled equations.

The derivation ultimately leads us to four fundamental equations in fluid mechanics: one for mass conservation and three for the conservation of momentum in Cartesian coordinates (x, y, z). These equations are interdependent and need to be solved together, as they share common variables like density and velocity. This coupling highlights the complexity of fluid behavior and the need for simultaneous consideration when analyzing fluid flows.

Think of a busy highway where the movement of cars (momentum) depends on the traffic flow (mass). Each car's speed may affect others, and vice versa. Similarly, in fluid dynamics, the conservation equations influence each other, requiring us to look at the bigger picture to understand how the whole flow system behaves.

Considering that if I put it for a control volumes which is a infinitely small, okay which is infinitely small and for that case if I substitutes the extensive properties b is equal to m and the beta is equal to 1 that is what we derived it in previous classes.

In applying Reynolds Transport Theorems to our analysis, particularly for an infinitely small control volume, we can define extensive properties such as mass in a manner that helps us derive how mass changes over time and how it flows into and out of our control volume. This forms the basis of our mass conservation principle.

Imagine a sponge absorbing water. If we think of the sponge as our control volume, the water's flow in and out of it reflects the changes in mass we are studying. By focusing on how much water can enter the sponge over time, we can understand the broader concept of fluid dynamics.

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