First let me point out that there are many applications in continuum mechanics that are not open systems. For example, the oscillation modes of a droplet, where evaporation effects are negligible; the mechanics of elastic bodies etc. But back to your question: It is unfortunate the most classical books on thermodynamics avoid a detail analysis of the problem using a systematic application of control volumes. On exception is the book by Sandler: Chemical, Biochemical and Engineering Thermodynamics, Wiley 2006. In my estimation the proper use of control volumes for analyzing a complex problem ( whether it be in continuum mechanics, thermodynamics, or engineering) is an essential tool for teaching students to correctly set up the problem. The use of macroscopic control volumes is often the first step in analyzing a problem. See our textbook on Material Balances for Chemical Engineers for examples. Depending on the application one might then derive the differential form of the balance from the integral formulation leading to partial differential equations. Students should be well versed in using control volumes and be comfortable working with either the integral formulation of the differential form of a balance law ( mass, momentum, energy) depending on the application.

First let me point out that there are many applications in continuum mechanics that are not open systems. For example, the oscillation modes of a droplet, where evaporation effects are negligible; the mechanics of elastic bodies etc. But back to your question: It is unfortunate the most classical books on thermodynamics avoid a detail analysis of the problem using a systematic application of control volumes. On exception is the book by Sandler: Chemical, Biochemical and Engineering Thermodynamics, Wiley 2006. In my estimation the proper use of control volumes for analyzing a complex problem ( whether it be in continuum mechanics, thermodynamics, or engineering) is an essential tool for teaching students to correctly set up the problem. The use of macroscopic control volumes is often the first step in analyzing a problem. See our textbook on Material Balances for Chemical Engineers for examples. Depending on the application one might then derive the differential form of the balance from the integral formulation leading to partial differential equations. Students should be well versed in using control volumes and be comfortable working with either the integral formulation of the differential form of a balance law ( mass, momentum, energy) depending on the application.

Dear Brian,

thank you for the answer. I will try to find that book and read it.

Open system and the control volume in continuum mechanics are the same

Juan it all depends on the type of control volume you use in your analysis. In most cases what you say is correct. One example where this is not the case is a material control volume, which is constructed such that its surface moves everywhere with the fluid velocity and therefore no fluid crosses the surface of the control volume. It is often used in continuum mechanics to derive the balance laws for a system. Sometimes in a problem it is convenient to construct a moving control volume, where one of its surfaces moves at a specified normal velocity so that no fluid crosses that surface. Some simple examples of such systems can be found in our book on material balances ( see chapter 3). In reacting systems you can construct a control volume that moves with one of the species velocities in the reacting system. It all depends on the type of problem you want to solve.

Juan, I hate to disagree with my colleague Brian Higgins, but I think that control volumes are better described in Chapter 4 (not Chapter 3) of our text on Material Balances for Chemical Engineers. See Eqs. 4-1 through 4-39. To obtain the "material control volume" from Eqs. 4-20 and 4-21, set "double you" equal to zero and note that the control volume will no longer be "arbitrary" but will move with the species A velocity "vee sub ay". To be more precise, this would be called the "species A material control volume" since no species A crosses the surface of the moving, deforming volume.

When fluid no crosses the surface is close system. Close system is different open close system.

just wanted to be back to the question. I think it would help if I mention that the system I am dealing with is a smooth muscle cell. If you consider a cell with ion crossing its membrane, then this is an open system where its volume due to contraction changes. However, as I have understood, the mass of transported ions are not comparable to the mass of the cell (or even a piece of tissue) which removes their effect on the momentum of the whole system and therefore, leads to a close system.

This is what I have come up with to construct the conservation of linear and angular momentum of the system. Now, the question is: Is this an acceptable assumption or not?!

Two things - One ... all conservation laws apply to an imaginary chunk matter, in this case a chunk of continuum given in a Lagrangian frame. Using simple integral identities (Green, Ostrogradskii, actually same thing) we get Eulerian viepoint, which is tantamount the control volume approach. Basically - wherever conservation law is concerned start withLagrangian vierwpoint, use the derivation of a Jacobian formula ... and you are home free. In the specific case of a muscle, you hit on a very crux of the problem, the continuum set was and remains imaginary, because even from a molecular point of view there is no continuum, molecules keep diffusing, or in vanecular 'keep jumping the imaginary limits, humans came up with". In short, Babak, you need a closure law, formulated in terms of a continuum theory to account for the effect an of ion transfer into the control volume. Two, Brian, I am quoting you: the oscillation modes of a droplet, where evaporation effects are negligible. I badly need to get hold of what you are referring to. My droplets keep evaporating, but I try to shake them to bits before they leave any of the fire watermist devices I design.

Jerzy, ha ha! I was referring to container-less processing in the spacelab using a controlled environment ( to minimize evaporation) and activating the drop oscillation modes with acoustic fields. But I like your method:)

Dear Jerzy,

according to you, the muscle cell (tissue) must be considered as a control volume then which rises another question: by considering a control volume problem, the volume of the system is kept constant and any deformation should happen for the material inside the fixed domain. But, in the case of muscle cells, the domain which is their membrane cannot be fixed and therefore, the system cannot be taken as a control volume problem. Is this true?

Babak, you probably you need to propose some model of interaction between the macroscopic object, like the muscle tissue and the ion transfer through the limiting membrane and its impact on the object (contractions, I presume). When it comes to the control volume situation, note that your mathematical formulation has to reflect the observables you are after. One way is to say a blob, occupying some space in 3-D Euclidean space moves and deforms in time, but another - look up Lagrangian versus Eulerian description, up above - is to create a idea of a blob, immutable in some ideal space and map its position in time onto another, quite 'real' Euclidean space of your observations. This sounds crazy or outright Platonic (actually it is Plato), but actually it is a formulation taken directly from the Rational Mechanics (of continua) - started by Truesdell, John Hopkins U. in late '70. There is a fancy mathematics connecting these two Euclidean spaces, the first one is sometimes referred to as the 'pointwise E-space', any mathematician could help you with it. Don't you panic seeing this, just thing of an ice cube, that first gets to be broken in half, then each half is melted separately, and finally upon heating vapors of both caugt in a common pressure vessel. Same with your muscle tissue, you may apply conservation laws to the ideal muscle, neglect the negligible and it will self-simplify to a standard Euclidean description. This is how many I would approach the tissue issue.

Brian. I am looking for all possible ways to excite modes of oscillation of a droplet. Did not have time to delve into this yet, although did some work in nonlinear vibrations way back, as a student. Some 5 years ago I stumbled upon a novel proces of producing fire-fighting equipment. My devices produce fire water mist directly from a two-phase flow. The process is instantaneous, evaporation inside the device is unimportant - besides it i quite humid inside a two-phase flow :-). I am sure I have fairly strong acoustic waves in the packing chamber and am able to tune said waves up to whatever is needed, just tell me how do YOU get water droplets so excited?

See Chapter 5 of Fundamental Principles of Heat Transfer (available on ResearchGate) for a detailed discussion of (1) fixed control volumes, (2) moving control volumes, and (3) material control volumes.

Dear Babak,

You are on a right track, but we all are trying to push you to go a little bit further. Both "Californians' suggest you use species formulation of the material control volume, as it is done in some engineering chemical physics, or fluid dynamics of reacting flows, or gas dynamics of gas mixtures, I propose you set up an ideal space and move to assigning properties to the ideal material and only move the 'earthly reflection' in a 3-D Euclidean space. In any case you have to break the mental barrier, it is no longer a 3-D box and the muscle tissue inside you are so adamant to call 'the control volume'. (You may, though subdivide the tissue into a plethora of little cubes and see what a limit off ininitsimally small cubes would give you.) Advanced modeling of moving and deforming objects is fairly old, historicaly the first was the marker-and-cell method for 2-D model of a breaking water dam. Now you have a multidute of LAGRANGIAN codes in fluid dynamics, whereas fields other than CFD have produced own versions and often entirely new codes. Perhaps lok up FreeFOAM for deforming bodies, they work mostly in fluid dynamics, but very well might already have some in deforming solids. In any case, you need to jump up a notch in your model formulation of the muscle tissue.