5.4 Linear Mass Equation

Mass is conserved in a "material" perspective which means in "its" material space. Which is to say, mass stays in the space it occupies until physics decides it should move to occupy its next space - conserved in such events is a loose idea of "conserved." A second perspective about mass is how it changes relative to a fixed-in-space (or partially fixed) perspective. The apparent contradiction, "matter, as physics moves it can be observed from observer selected, matter-independent perspective - and with the "mass is conserved" idea remaining intact. In real-time this is not possible. But in the limit of vanishing time, the realm of understanding becomes mathematic, becomes calculus. And, of course, it is helpful in understanding this if you believe in make-believe.

Basic Thermodynamics ~ J. Pohl ©

www.THERMOspokenhere.com (190-E123)

5.4 Linear Mass Equation

Transformation of the "material" perspective a "spatial" perspective.

Abstract:  The objective of this writing is to develop an elementary, least complicated, proof of transformation. The plan is to write a beginning description of fluid motion using mathematics with "material coordinates" and then upon definition of "spatial coordinates," use proper, clear math to change the initial perspective, to transform the mathematical representations, to the second, "spatial coordinates," often called a "control volume," perspective. The development below is mathematically incomplete and perhaps, incorrect.

We consider a simple flow of a fluid. The fluid, ideal. has no velocity
profile (slug-flow). Its bounding "marker" subscript ("m") means material. Here
material means, "imagined-as-attached-to-that-matter" 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 (m,L). (m,R) is the "right" coordinate. These coordinate points move in accord with the physics of the fluid. Assume the fluid ideal; having no velocity profile (slug-flow). Let the selection of system occur at time "t₁".

The, " * ", star 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)(1)

The mass can be expressed as an integral sum with its boundaries as limits:

(2)(2)

We seek a general, rate form equation. Consequently the derivative of equation (2) is taken.

(3)(3)

Equation (3) has the perspective, material, meaning constant identity and amount of system matter. Leibnitz Differentiation can expand (3) to become (4):

(4)(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.

(5)(5)

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.

(6)(6)

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:

(7)(7)

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.