3.1.7. Conservation of Mass

Very good morning all of you. Today we are going to discuss on differential analysis of fluid flow which is very interesting chapters in the book of Senjal Chembala and also the F. M. White book which is the foundations of the computational fluid dynamics and So, considering that let us start the differential analysis of the fluid flow.

In this introduction, we are greeted and the topic of discussion is set to the differential analysis of fluid flow. This method is crucial for developing algorithms in computational fluid dynamics (CFD). It signifies an advanced approach beyond integral methods, focusing on analyzing fluid behavior at infinitesimally small points rather than large control volumes.

Think of this as examining a flowing river. Instead of looking at the river as a whole (integral approach), we choose to look at tiny sections of the river, measuring properties like speed and pressure at specific points (differential analysis). This close inspection helps us understand flow dynamics more accurately.

We use as a control volume considers there is a dicks is mountain over a decks and we have the control volumes to just to estimate how much of force is acting on these dicks. So, if you look at these problems, we consider a control volumes and we try to look at the velocity components, what could be expected inflow, outflow, also the outflow here and what could be the velocity vectors and based on that we apply the mass considerations and the momentum equations to estimate the force.

This chunk explains the concept of control volumes in fluid mechanics. A control volume is an arbitrary space through which fluid flows. By observing the inflows and outflows and their associated force, we can derive important factors like force and mass conservation. The differential approach, however, looks at each point within the flow, seeking deeper insights into velocity, pressure, and density.

Imagine you’re checking the water level in a bathtub (control volume). You can measure the inflow from the tap and the outflow through the drain. However, to understand how water moves and behaves in different areas of the tub, you might need to measure the water at multiple points within the tub (differential approach).

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.

Here, the discussion focuses on the significance of reducing the dimensions of control volumes to infinitesimally small values. When the dimensions (dx, dy, dz) approach zero, the control volume is effectively a point, allowing for a precise mathematical analysis through differential equations. This step is crucial in deriving equations for mass conservation and momentum in fluid dynamics.

Imagine zooming in on a beach. At first, you see the entire beach (large control volume), and then you start looking at smaller sections until you are examining just a grain of sand (infinitesimally small). This allows you to analyze very specific effects like how a wave moves just one grain of sand.

So, if I write this partial differential equations for mass and linear momentum, the basically I will get the four basic equations. 4 basic partial differential equations. One is mass conservation equations and the 3 equations which is a vector forms of linear momentum.

In this section, the lecturer discusses the culmination of the differential approach, leading to the four fundamental partial differential equations. Among these, one equation pertains specifically to mass conservation, while the other three describe the linear momentum in three Cartesian coordinates. This forms the foundation for solving fluid dynamics problems effectively.

Think of this as creating a recipe for a complicated dish. You first identify the ingredients (mass conservation equation) and then the cooking methods (momentum equations). Together, these components will allow you to recreate the dish perfectly, similar to how these equations help predict fluid behavior.

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.

The Reynolds Transport Theorem is essential for linking system properties to control volumes. When applying this theorem to an infinitesimally small control volume, we relate changes in mass and flow through surface integrals to obtain a simplified mass conservation equation. The key variables here are density (rho) and flow velocity, both integrated over the volume.

Consider a balloon being filled with water. As you add water (mass input), some water spills out (mass output). By applying the Reynolds Transport Theorem, you can express the relationship between the water being added and spilled, similar to how engineers relate mass flow in fluid systems.

If I have a there is a part of the control surface mass inflow is coming into this control volume and it is going out. See if I follow that so this surface integral part we can simplified it and to write as very simple way that the control volumes d v is equal to sum of mass flux in minus sum of mass flux out.

This part emphasizes mass flux in and out of a control volume. Essentially, the rate of change of mass within the control volume must equal the mass inflow minus the mass outflow. This brings together the concepts of inflow and outflow, establishing a clear equation for mass conservation.

Think of a reservoir being filled with rainwater. The water flowing into the reservoir (inflow) and the water spilling over the sides (outflow) can be represented mathematically. By knowing how fast it rains (inflow) and how much is overflowing (outflow), you can assess the change in water level in the reservoir (change in mass within control volume).

The first part let us come for deriving mass conservation equations for infinitely small control volumes. So, looking at infinite small control volumes, so we can derive this mass conservation equations two ways. One is the divergence theorems and to apply the divergence theorems.

In this segment, we explore Gauss's Theorem and how it allows for transforming volume integrals of a divergence of vectors to surface integrals. This relationship is crucial for deriving compact forms of the mass conservation equations, which can be applied universally to fluid flow, whether compressible or incompressible.

Visualize pouring water into an irregularly shaped bucket. Gauss's Theorem is like a method that helps us figure out how much water has moved at various points inside the bucket by inspecting the edges, rather than measuring every spot inside.

--