3. Fluid Mechanics

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 introductory segment, the speaker sets the stage for the discussion on fluid flow analysis. They mention the relevance of two important texts on the subject, indicating that the focus will be on the differential analysis as opposed to the integral analysis introduced previously. The key concept here is that understanding fluid dynamics will benefit from another perspective, specifically how we can analyze fluid flow at very small scales.

Think of analyzing the flow of water through a garden hose. Rather than focusing on the entire hose at once, the differential approach allows us to examine small sections along the hose to see how the flow rate, pressure, and velocity change from point to point.

Looking that let us as we discuss about integral approach in the previous classes which we generally use as a control volume. 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.

This chunk highlights the integral approach previously discussed, utilizing a control volume concept to estimate forces. The idea is that a control volume encompasses a section of fluid, allowing analysis of force interactions within that volume without needing to delve into the specifics of the fluid properties inside it. The speaker emphasizes that the integral method views a larger section, whereas differential analysis will zoom in to individual points within that section.

Consider a large swimming pool. Using the integral approach is like measuring the overall temperature of the pool water without checking each specific corner. The differential approach, however, would allow you to measure the temperature at different spots within the pool to see where it is warmer or cooler.

If I consider that the basic equations what I will get it that is very the conservation of mass for infinitely small control volume.

In this part, the content introduces the concept of using 'infinitesimally small' control volumes. As sizes become smaller (approaching zero), we can analyze fluid behavior at a specific point, enabling us to derive equations that govern fluid flow. This marks the transition from merely observing entire volumes to understanding fluid dynamics at a micro-level, which is crucial for deriving differential equations.

Imagine zooming in on a grain of sand on a beach. If you look at the whole beach, you might see waves crashing and retreating. However, by focusing in on a single grain of sand, you can better understand the interaction of tiny water droplets and how each contributes to the beach's erosion or formation.

So, if I write this partial differential equations for mass and linear momentum, the basically I will get the four basic equations.

Here, the speaker reveals that by applying the differential approach, one can derive four fundamental partial differential equations governing fluid flow: one for mass conservation and three for linear momentum. This is fundamental to understanding how different fluid variables, like density and velocity, interact. The equations are coupled; changes in one affect the others, which reflects the complex nature of fluid dynamics.

Consider a car driving through different terrains. The way the car’s speed, engine power, and tire traction interact can be understood like these equations: changing one factor (like speed) affects other factors (like momentum and stability).

So if you look at these equations, which are very simple equations, that means we can get the mass flux. If I know the discharge, if I know the density, I can know this mass flux is coming in or going out.

This section emphasizes the concept of mass flux, defined as the mass per unit time flowing through a surface. The relationship between mass flux, density, and volumetric flow rate is critical for solving fluid dynamics problems. By understanding how different variables interplay, one can assess how mass is conserved as it flows into and out of control volumes.

Think of a water fountain: the amount of water being pumped up (discharge) is directly related to how heavy and dense the water is. If you know either the water’s density or the flow rate, you can predict how much water will come out over time, illustrating the conservation of mass at play.

Let us come for the mass conservation equations for a control volume which is infinitely small. 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 this segment, the speaker applies the Reynolds transport theorem to derive the conservation of mass for small control volumes. This theorem establishes how changes within a control volume dictate the net mass flowing in and out. By substituting the defined extensive properties (mass in this case), we can mathematically express how the mass is conserved across those volumes.

Consider a container with a lid: if water is added (inflow) or taken out (outflow), the amount of water inside directly reflects what’s been added or removed. The Reynolds transport theorem helps us quantify that relationship mathematically.

If I look at these equations, equations 2 and this is what the equations 1, which is from the Reynolds transport theorems. Now, interestingly, what I am looking at, I am not looking for the change this control volume part, which is, anyway, I don't have much problems.

The divergence theorem is introduced to relate volume integrals to surface integrals, emphasizing how mass flow can be calculated as the fluid passes through a defined control volume. This theorem transforms the calculation of mass changes into more manageable forms by considering the flow just at the surface boundaries of the volume in question.

Picture a large balloon—if we want to count the number of air particles inside, rather than poking the balloon and measuring each one, we can simply observe the air particles escaping from the surface. This illustrates how analyzing the surface is sometimes enough to comprehend what's happening inside.

So, we can write it these ones in very compact forms so if I look at this part see if some things if your integrals that is what is becomes a zero okay if i doing a volume integrals of something is equal to zero that means this component is supposed to be zero so that means if i can write it the finally the mass conservation secretions from Gauss diversals theorem it comes out to be very compact form is del this is the del operators v is equal to 0.

Finally, the mass conservation equations are distilled into a compact mathematical form using the divergence theorem. Here, the operator signifies that in a suitable mathematical environment, mass is conserved where the divergence of the velocity field remains zero. This compact form can be applied across various fluid dynamics problems, highlighting the elegance of the mathematics behind fluid mechanics.

Think of pouring a perfect cup of coffee. If the rate of coffee going into the cup equals the rate of the coffee already in the cup exiting through a hole at the bottom, it remains full. This is a real-world manifestation of the principle of mass conservation being presented in a simple form.

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