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Conservation of mass Let $\mathbf{v}(t, x, y, z)$ be a continuously differentiable vector field over the region $D$ in space and let $p(t, x, y, z)$ be a continuously differentiable scalar function. The variable $t$ represents the time domain. The Law of Conservation of Mass asserts that$$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=-\iint_{S} p \mathbf{v} \cdot \mathbf{n} d \sigma$$ where $S$ is the surface enclosing $D$a. Give a physical interpretation of the conservation of mass law if $\mathbf{v}$ is a velocity flow field and $p$ represents the density of the fluid at point $(x, y, z)$ at time $t$b. Use the Divergence Theorem and Leibniz's Rule,$$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=\iiint_{D} \frac{\partial p}{\partial t} d V$$to show that the Law of Conservation of Mass is equivalent to the continuity equation,$$\nabla \cdot p \mathbf{v}+\frac{\partial p}{\partial t}=0$$(In the first term $\nabla \cdot p \mathbf{v},$ the variable $t$ is held fixed, and in the second term $\partial p / \partial t,$ it is assumed that the point $(x, y, z)$ in $D$ is held fixed.)

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Vector Functions

Missouri State University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

11:33

Conservation of mass Let $…

03:25

Suppose the density $\rho$…

So you have a d, some bounded region region. It is smoke service. We have a row density. There is the conservation of mass observation of mass equation equation that it states that or if you integrate over b, the density, the den some x y c under perhaps the time. So this is the total mass or to mass on de. So you want us to get to mass on b, you need to integrate over d. This should be. It is sin that the change with respect to time of this is equal to minus the integral the surface of rho. V. Where we see is the velocity velocity to the velocity has direction so vector pointing where there in the x y has some direction. So all the interpretation should be that or if you have some some close region, you have some mass over there all the way that this whole mass can change over here is, if some little particle escapes, and so will this n is measuring n. That b is measuring in which amount amount the or rho times, and the achamothe differential mass. The piece of mass leaves lives d, because, if you have a like origin, is like this see that you're in a box if the velocity is completely parallel to the surface, while in that case, if that is b- and this is m her in that case, that B is equal to 0, but that is no problem, because in this small region, where the velocity is or thrown out to the surface, there is no change in mass where, when changes mass is happening, is when the velocity has component in the normal direction to the Surface so that it is a living and since since the end is out out of d, then b and being positive, that means that the velocity is out because it has the same sign as a sin. So that is why we need to put a minus here, because for this this, this change being negative means that some mass is living, but mass living means that this. This part here is positive. This this interval here being positive, means that mass is living is living withing their domain. So that's why you have to put a minus sign, so that is the ego interpretation on this equation and secondly, we have that or if we assume that this is valid to do so, if you will differentiate this row, not x y z and then t, if You assume that differentiating this is the same as just differentiating piece by piece inside v, so the partial taking the the partial rates inside then the then the conservation conservation of mass equation, which is which is this equation. This is the concentration of mass. This equation is equivalent equivalent to have a dram row. This change in the density plus divergence of rho v is equal to 0, so these both being equivalent, then what would say that this isn't our way to say conservation of mass. So all the reason why that is true is if we assume this assume that this hole, and then we also have a by this equation. We have a lie: equation of constipation of mass. We have this row v equal to minus the integral over b forensia volume is equal to the integral over the surface of a row times b dotes, since this is equal to that. We can say that we can move it to the other side to the equation, and then we change a sign and we should still get this. Is this the should be distant? This is not as so integrating well, it is valued also to take the partial libative with respect to time inside. So we, by moving these to that side, you have that this has to be true, and then you can apply the divergence theorem to see. Ratithis is so vector field rho times b, is a vector field so that what are the efficiency of this? That a bactrafield that an over the surface is equal to the divergence of f over the volume, and so you apply that so you get that d row b b, so this integral over the volume plus the integral or over derdthey, both have to be equal to 0, so well, this is this. The b is precisely that so that you have that the re integrated is equal to 0, so that that is equal to 0 or because we have the continuity, because we have this this equation or do not lots.

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