It is very important to know the difference between Euler’s method and Lagrange’s. This is key to a deep understanding of the mechanics problem.
Lagrange coordinate system
The Lagrange coordinate system is also called the material coordinate system or reference coordinate system. The Lagrange coordinate system is called a material coordinate system because it is a coordinate system described by following an object. When the time is 0, describe the position using the object’s position as coordinates. For example, if the object A has initial coordinates of (1, 2), it is expressed as (1, 2, 0), and the position at time t is expressed as (x, y, t).
Euler coordinate system
Euler’s coordinate system is also called a spatial coordinate system, but it is a coordinate system that emphasizes position rather than an object. The difference from the Lagrange coordinate system is that (x, y, t) in the Lagrange coordinate system means the coordinates of the object at that point over time. However, from the Euler point of view, it means what coordinates the object has at the time.
Reynolds Transport Theorem
The Reynolds transport theorem is the rate of change of the physical properties of a particular area over time. The area of general interest in fluid mechanics is called the “control volume”. Since the fluid in this control volume varies with time, it must be used as an Euler coordinate system.
Above is the general form of Reynolds Transport Theorem. If the control volume is not stationary but moving, then the speed should be the relative speed of the control volume.
If we make various assumptions from Reynolds’ transport theorem, we can derive several equations. Using the law of conservation of mass, the equation for calculating the flow rate comes out, the force of the fluid can be calculated, and Bernoulli’s law can be derived.