Eulerian and Lagrangian Descriptions
Concepts Explained:
The experiment that has been done in the given film about the Eulerian and Lagrangian descriptions has used the concepts of the Dynamics of flow. In this regard, the services of Kinematics have been used in order to explain the motion of the particles in free space. In this way, through the explanation of the motion of particle in free space, we become able to explain the Eulerian and Lagrangian descriptions. Also, in this experiment, the mathematical relationship between these two descriptions is also developed.
The description of the motion of the material particle in terms of displacement, velocity and acceleration by an open arrow in for a specific time is known as the Lagrangian Description. In this kind of description, the velocity in any arbitrary point in free space is a function of the initial position and the change in time. This kind of description can be viewed by the usage of the computer simulation. In this way, a clear description comes before us.
Another concept that is told by the video is about the Eulerian Description. In this description, there has been considered a fixed arbitrary point through which the material point passes. The indication of displacement, velocity and acceleration at that fixed point in terms of time is known as the Eulerian Description. The direction of all these three vectors is indicated by the solid arrow.
So this report is going to explain the mathematical relationship between these two descriptions through the concepts of Fluid Mechanics.
Experiment Done:
Consider a nuclear plant is located at the corner of a river. From this plant, the radioactive material is included into the water of the river that is so dangerous for the creature of water in that river. We are interested here that to determine the intensity of the radioactivity in the water of the river.
Consider a sample of the water of river is taken into the lab. The radioactive tracer is placed in the sample of water. If we look with the help of microscope, we will see the water as collection of many particles. Hence, in order to see the particle in more detail, we apply the computer simulation for the sample of the water.
In this simulation of the water sample, we will be able to see the particles as the small holes moving from left to right corner of the readioactive tracer. In order to know and measure the intensity of the radioactivity, we apply the pressure gauge in the form of a small probe that will act as the detector of the motion of the material particle.
When the material particle will collide with this probe, it rotates. In this way, at the position of the probe, we consider the fix point. As at the two places, this probe is placed, so there will be to fixed points inside the readioactive tracer; one at the left corner and one at the right corner.
At first, we suppose that the radioactive tracer is placed in a uniform position. So the equal number of material particles will pass through the two fixed points. So the intensity of the radioactivity will be same for the two points. But in actual case, this does not happen.
In the radioactive tracer placed in the water sample, there will be unequal distribution of the material particles at the left and right points. So this will create the intensity difference between the two points.
Hence, the intensity difference between the two fixed points and between the material particle and left point will give us the required relationship between the two descriptions. In this way, we will get the required results for this experiment and those will be our findings for it.
Findings:
So after the experiment has done, we obtained the following findings given as under.
The difference between the Lagrangian counter and the Eulerian counter is given by the following relation.
∆t (∂R/∂t) ___________ (1)
And the difference between the two Eulerian counters at different time intervals is given as under,
∆x (∂R/∂x) ___________ (2)
But the change in distance (∆x) can be written as, ∆x = ∆tU. Hence, equation (2) becomes as,
∆tU (∂R/∂x) ___________ (3)
Hence, the total change will gain the following form,
∆t [(∂R/∂t) + U (∂R/∂x)] _______ (4)
Hence in vector notation,
DR/Dt = [(∂R/∂t) + U (∂R/∂x) _________ (5)
= ∂R/∂t + (U---. ∆) R
In the arbitrary co-ordinates,
∆t [(∂U----/∂t) + (U---.∆) U---] _________ (6)
Hence, the Lagrangian acceleration is expressed in terms of Eulerian notation as follows.
DU---/ Dt = (∂U----/∂t) + (U---.∆) U--- _________ (7)
Conclusion:
From the help of the given video and knowledge and concepts of the Fluid Mechanics and Dynamics, we became able to develop the different relationships between the Eulerian and Lagrangian Descriptions. This was the actual goal of making this report. So we have achieved the required purpose in the form of the foundations.
From this experiment, we became able to understand the basic relationship between these two descriptions. After developing the relations, we have found that there is very vital application of these two descriptions in the field of the fluid mechanics. We gained a lot of knowledge from this report.
You must be logged in to post a comment.