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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 sealar function. The variable $t$ represents the time domain. The Law of Conservation of Mass asserts that $$ \frac{d}{d t} \iiint_D p(t, x, y, z) d V=-\iint_S p \mathrm{v} \cdot \mathrm{n} d \sigma, $$ 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 $\nabla \cdot p v$, 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.)

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 sealar function. The variable $t$ represents the time domain. The Law of Conservation of Mass asserts that
$$
\frac{d}{d t} \iiint_D p(t, x, y, z) d V=-\iint_S p \mathrm{v} \cdot \mathrm{n} d \sigma,
$$
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 $\nabla \cdot p v$, 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.)

Key Concepts

Leibniz's Rule

Leibniz's Rule is a method for differentiating an integral with respect to a parameter, allowing the interchange of differentiation and integration operations under certain conditions. In the context of the continuity equation, it provides a way to differentiate an integral over a time-dependent function, facilitating the derivation of the local form of the conservation of mass from its global integral form.

Divergence Theorem

The Divergence Theorem, also known as Gauss's theorem, is a powerful tool in vector calculus that relates the flux of a vector field through the boundary surface of a region to the divergence of the field within the volume bounded by the surface. It is critical in converting integral conservation laws (over a volume) into differential forms, and vice versa, making it easier to analyze local conservation properties.

Conservation of Mass

This concept states that mass cannot be created or destroyed within an isolated system. In the context of fluid dynamics, it means that the total mass of a fluid within a defined region can only change by the amount of mass entering or leaving the region. The law provides a fundamental basis for analyzing how mass is distributed and how it flows through a space over time.

Continuity Equation

The continuity equation is the local formulation of mass conservation in fluid dynamics. It mathematically expresses that the rate at which mass accumulates in a given volume is balanced by the net rate at which mass is transported into or out of the volume. In its differential form, the equation relates the time derivative of the density and the divergence of the mass flux, ensuring that mass is conserved at every point in the fluid.

Transcript

00:01 So we have a d, some bounded region, with a s smooth surface, and we have a row density.

00:29 There is the conservation of mass.

00:35 Serbation of mass equation equation that it states that are well if you integrate over b the density then the depends on x, y c on the perhaps the time the b so this is the total mass well over a total mass on d so well you want us to get the total mass on the you need to integrate over d.

01:23 This should be d.

01:28 It is saying that the change with respect to time of this is equal to minus the integral where the surface of row b where v is a...

01:57 Is the velocity velocity has direction it's a vector it's pointing here there or in the x y it has some direction so well the interpretation should be that or if you have some enclosed region on d we have some mass over there all the way that this this whole mass can change over here is if some little particle escapes and so while this n is measuring n that b is measuring in which amount the row times n that b in which amount the the differential mass the piece of mass leaves the d because if you have a if like the region is like this say that you're in a box and if the velocity is completely parallel to the surface well in that case if that is b and this is n well in that case n that b is equal to zero but that is no problem because in this small region where the velocity is orthogonal to the surface there is no changing mass where when changes mass is happening is when the velocity has component in the normal direction to the surface so that it is living and since since the n is out of d then b that n being positive that means that the velocity is out because it has the same sign as an so that is why we need to put a minus here because this this change being negative means that so mass is living.

05:07 Mass living means that this, this part here is positive...

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