4.2.3. Divergence of Velocity

Let us start on the continuity equations. The basically mass conservation equations, mass conservations for infinitely small control volumes as we discuss it that means the control volumes what we consider it, its the dimensions dx dy dz they are very very close to the 0. So when you have the control volumes infinitely small in that case we are looking at how we can have a differential equations format for mass conservation equation.

In fluid mechanics, we often use 'control volumes' to analyze fluid flows. A control volume is a specific region in space where we examine the mass and momentum of the fluid. The dimensions of this control volume (dx, dy, dz) are incredibly small, approaching zero. This means that we are interested in very tiny sections of the fluid where we can apply mathematical equations to understand conservation laws, like the conservation of mass.

Think of a control volume like a small balloon filled with air. If you imagine slicing this balloon into infinitesimally thin layers, each layer could be analyzed to see how much air is entering or exiting, allowing us to apply principles of physics to understand how air behaves as it flows.

As we derive it the basically when you have a basic components like we are looking for rho is a density field which is a scalar quantities functions of positions x, y, z in a Cartesian coordinates, then you have the time component. Then we have the time component, v is the velocity field which is having a scalar component of smaller u, v and w which will be the functions of positions x, y, z, t.

In our analysis, we need to define two crucial properties: density (denoted as ρ) and velocity (denoted as v). Density is a measure of how much mass is contained in a given volume and is represented as a function of position (x, y, z). Similarly, the velocity of the fluid is not just a single number; it has three components (u, v, w) that correspond to movement in the x, y, and z directions and can also change over time.

Imagine a flowing river. The density of the water can vary depending on factors like temperature or pollution, while its velocity might be different at various points across the river. For instance, water near the banks may flow slower than water in the middle where it’s deeper and faster.

This is the components indicating for us change of the mass storage within the control volume. That is what will be equal to the divergence of the max flux within the control volume. This is what derived from the Gauss theorems for a control volumes.

The change in mass storage within our control volume can be understood as how much mass is building up or depleting over time. This change is related to the 'divergence of mass flux', which essentially measures how much mass is flowing out of the control volume compared to how much is flowing in. Using Gauss's theorem enables us to relate these mass flow rates to the integral forms over the control volume surfaces.

Consider a balloon again. If you blow air into it, the mass inside the balloon increases (mass storage increases) while if you let some air out, the mass decreases. The rate of air flowing in or out can be statistically represented as 'flux,' helping us understand how the mass inside the balloon changes. Similarly, in our control volume, we want to understand how fluid mass is flowing in and out.

Using the Taylor series expansions we can approximate it. So basically as I consider at the centroid of this box the velocity is equal to u v and w what will be the velocity at the different faces that is what we are going to look it there is a front face, the rear face, top, bottom, left side and the right sides.

To understand how velocity changes at different points on our control volume, we use Taylor series expansion, a mathematical tool that helps approximate values of functions around a point. By examining the velocity at the centroid (the center) of our control volume, we can compute the velocities at the various faces of that volume. This helps us visualize how velocity varies spatially within that small region.

Imagine a spinning top. While the top spins, the speed of its tip varies slightly based on its position. By observing the tip's speed at the center and then using approximations, we can forecast how fast it’s moving at different points along its path. Similarly, in our control volume, we apply Taylor series to predict velocity variations.

If you look at all the terms okay this is what mass flux as any surface if I consider it okay let me I consider this is the x direction surface the surface area will be dy dz the surface area. The velocity mass flux is a perpendicular to that.

When we analyze fluid motion, we calculate the mass flow through surfaces of our control volume. For example, on a surface within our control volume aligned along the x-direction, the area is dy dz. The mass flux that crosses this surface is influenced by the fluid's density and velocity in that direction. By systematically considering these contributions from all surfaces, we can relate them to fundamental equations of fluid motion.

Think about a water pipe. The water flowing through has a certain density and velocity. If we want to figure out how much water is leaving the pipe at one end, we can multiply the cross-sectional area of the pipe by the flow speed to get the mass flow rate. Similarly, in fluid analysis, we examine how much fluid crosses various boundaries in our control volumes.

Physically you can understand it that we are looking for change of the mass storage within the control volume with respect to time.

The mass conservation equations highlight the principle that in a closed system (like our control volume), the mass cannot be created or destroyed, only transferred or transformed. Analyzing these equations helps us understand how mass storage fluctuates over time, considering all inputs and outputs across the surfaces of the control volume.

This is similar to a bathtub filled with water. If you add water and drain some away at the same time, you'll need to account for both actions to see if the water level rises, falls, or stays the same. The water level represents mass storage, and understanding the conservation of mass concept helps explain how that's adjusted with time.

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