Please explain these steps to me, the Continuity equation?

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The" 2.2.1.1.2.1 CONTINUITY EQUATION" please explain the mathematics steps. I need to understand what is happening. Please explain the mathematics steps.

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  • 7 years ago
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    I'll start by commenting on Equation 2-87.

    This author is using "div v" in an unusual way; he means it as

    i du/dx + j dv/dy + k dw/dz,

    where the derivatives are partials, and the u,v,w are the eastward, northward, and upward components of the flow field. You probably know that "div v" USUALLY means

    du/dx + dv/dy + dw/dz (a scalar).

    Let's just dwell on the fact that this definition of "divergence" makes sense! If dv/dy is a positive number, it means that a northward-flowing parcel goes faster as it moves farther north. Hence, for a box in space, more stuff flows out the north wall than enters through the south wall. This is possible because either (a) more stuff enters through the top/bottom and E/W walls than is going out through those walls, or (b) the density of the material in the box decreases (that would mean d(rho)/dt < 0).

    If the fluid were incompressible, the "rho" would not be needed. One often treats water, or even air, as incompressible, and writes the equation simply as "div v = 0". But you can get that "div v" may not be zero if the material in the box is changing density. "Rho" can be important where gas is forced through ducts and nozzles, or in very large-scale gas-flow problems on planets or stars.

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    Moving to Equation 2-88, this says for a box in space, any change in the mass of the box (middle expression) must equal the mass that flowed in or out (right-hand expression). On the right, the integral with the circle means "integral over a closed surface", and the vector "dA" means the box surface is a set of little squares each replaced by its OUTWARD oriented unit vector. Dotting the "rho v" with the "dA" gives you the volume that flowed out, and that's why a minus sign is needed -- because the middle expression in this equation is the positive change in the mass of the box.

    There may be a typo in this equation (2-88), because you can't "dot" a dot product with another vector. The first time you "dot", you get a scalar for an answer. The right-hand integrand should be "rho v dot dA" rather than "rho dot v dot dA".

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    What's to get in Equation 2-89 is that the leftmost expression is the change, through time, of the whole mass in the whole box; the middle expression represents the sum of all the mass changes, through time, from all the teeny tiny boxes into which the whole box has been divided; the rightmost expression looks at each teeny tiny box and asks only whether the density of that box has changed through time (since its volume hasn't), and then adds up the changes from that viewpoint.

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    Equation 2-90 is the Gauss Divergence Theorem. The idea is again that the stuff going out through a macroscopic boundary (left side of equation) must equal the sum of the "divergence" over all the teeny boxes inside; see above what "divergence" really means. If you need a more detailed understanding of the theorem, see Wikipedia article entitled "divergence theorem."

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    Eqn 2-91 is a transformation of Eqn 2-88, using 2-89 to replace the LHS and using 2.90 to replace the RHS of 2-88. Eqn 2-92 seems a natural result of 2-91, and there's something very odd about the "because" that follows it. My guess is that the "because" was supposed to start a new idea, leading to Eqn 2-94.

    The Eqn 2-93 is meant to stand on its own mathematically, but I don't like "rho dot div v", because "rho" was never a vector to begin with. I'm looking at Gerard Neumann's 1968 "Ocean Currents", where the term that your author is writing as "rho dot div v" is instead "rho div v" with the customary use of "div v" as a scalar. Apparently your author ALSO wants that term really to mean "rho times the sum of the partial derivatves du/dx and dv/dy and dw/dz". I suspect your author of cut/pasting!

    Anyway, you must know that "grad rho" means

    i d(rho)/dx + j d(rho)/dy + k d(rho)/dz.

    The physical meaning of 2-93 is that the local loss of mass from a teeny box must equal the sum of (a) the change that is due to a steady flow of fluid whose density is not spatially homogeneous; and (b) the change that is due to spatially-varying flow of fluid whose density IS homogeneous. My (a) and (b) correspond to the first and second terms on the RHS of 2-93.

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    If you have really followed what I have said about 2-87 through 2-93, you will know what I might have said about 2-94 through 2-96...a slight complaint about notation, and vector-scalar confounding, but the physical meaning is clear enough.

    Good luck going forward!

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