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The "mass equation" is states mathematically the a principle: "Mass is conserved in a physical event." The mass in question is pre-specified; some call "selection of a system." The mass is modeled, of course. For the simplest model, the BODY, the mass equation has its simplext form. Mass of physical reality, modeled as a BODY, as one might expect, has the simplest mass equation.
Although Eqn-1 and Eqn-2 are equivalent, their matematical forms are not. Eqn-1 is said to be non-homogeneous while Eqn-2 is its homogeneous equivalent. More about math distinctions as we go.
The approximation, the model of matter as a BODY material perspective."
Mechanics: The "BODY" (or particle) is the simplest model of physical reality. Beginning topics of mechanics address physical systems that are solid. BODY is a model of reality for which the real physical mass is replaced by a system mass having the actual mass but not distributed in space, that is to reside at a "point." in space. The equation statements "mass of a BODY" (Eq-1) and its "mass is conserved" are quite simple.
Fluid Mechanics: For study of events of a fluid, a "spatial perspective" for system specification is useful. The "spatial perspective" involves mathematic definition of a region of space, a volume of finite interior, that contains the system fluid. The volume is specially defined including all aspects of its surface. The surface of the enclosing volume might be defined to move. Some call this mathematical tool a "control volume," our preference is to "system analysis from a spatial prespective.
So to bring the two ideas together. A BODY has its own coordinates and using a "material perspective" can be studied to predict physical events. For a fluid, "spatial coordinates" describe the fluid as a volume with special surface specifications possible when of use. For both perspectives to be true their mathematics and physics must be equivalent.
A Mathematics ~ Physics Consideration: The goal is to develop a least complicated proof that the "material perspective" and "spatial perspective" are equivalent.
Figure 1 depicts a simple flow of a fluid in one dimension - X. The fluid (taken as incompressible, for now) has bounding coordinates (left and right) marked with subscript ("m") to mean a material designation: the coordinate "goes whever the particles it was first associated with go - for evermore.
It is logical to identify a 1-D system as being that mass of fluid bounded by material coordinates ("m"). The "left" coordinate is notated "m,L" and "m,R" is the "right" coordinate. These coordinate points move in accord with the physics of the fluid. Assume the fluid incompressible and ideal; having no velocity profile (slug-flow). Let the selection of system occur at time "t*1".
The super-script asteric, " * ", notation admits that the system selection was made at a specific ("t*") instance in time. At the end of development, realizing in retrospect, the "instant of time" could have been "any instant." Upon completion of the proof, the notation is changed: (t*) becomes (t).
Specification of material coordinates as boundaries (left/right moving) with fluid particles, assures us that the bounded mass, the material mass is constant:
|(1) Mass within the material boundarys is constant in time.|
The mass can be expressed as an integral sum with its boundaries as limits:
We seek a general, rate form equation. Consequently the derivative of equation (2) is taken.
Equation (3) has the perspective, material, meaning constant identity and amount of system matter. Leibnitz Differentiation can expand (3) to become (4):
It is our choice to manipulate the equations of the "material perspective," however we choose provided math rules are not violated. A natural perspective of persons observing events of fluids is "spatial." The perspective is "instantaneous" and of the same space as the material space. Figure 2.
Next we write an equation that applies superposed to Figure 2. We define velocities of a "spatial perspective" to have the notation: W. Headed toward the transformation, we write an equation of identity.
Equation (5) is valid. It is an identity the use of which can change the perspective of Equation (4). Both equations pertain to the fluid, both are valid, hence they can be added. Equation (4) is rewritten with the "to-be-added" identity below it.
I admitted that this is not a proof - parts are missing. Specifically I'm sure that adding the lower equation to (4) must also have a change of limits of the integral. (Right now I'm tired - something is wrong.) A step further yields:
So there is discretion in specifying W. But not V; it is determined by physics. W is the human-selected, possibly spatial, view. When W is set equal to V everywhere, the terms right of equality become "zero" and (7) collapses (in a proper way), returning it to its parent, (4).
Another case is when W is set identically equal to "0." I believe a constant-volume, "spatial perspective" results. The right side terms become the mass-rates "in" or "out" often used in texts.
This proof is inadequate. I think the limits of the integral must change but right now I am unsure how. Help me if you will.
In closing, this path changes a material perspective of observation, called "closed system" by some, to a spatial perspective or "control volume." Does the volume "control" anything? Spatial is a better descriptor, I think.
One other thing. In the work above (incorrect as it is) there is but physics and math. Wherever I read about this topic, the writing includes many "clues for differentiation." Names of famous people, names of equation terms... Leibnitz, Lagrange, Euler, convective terms, substantial derivative, continuity equation, divergence theorem. My opinion is these terms are not needed.