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## Homework Statement

Derive a mathematical relationship which encapsulates the principle of continuity in fluid flow.

## Homework Equations

## The Attempt at a Solution

Imagine we have a mass of fluid ## M##, of volume ##V##, bounded by a surface ##S##. If we take a small element of this volume ##dm##, we can form two equivalent expressions, in terms of the fluid density, ##\rho##.

$$ dm = \rho(V) dV $$ (1)

Now, let the velocity of the fluid be ## \vec{v} ##, where ## v = v(x,y,z,t) ## (i.e ## v ## is a function of both position and time).

$$ dm = \rho (\vec{v} \cdot \hat{n}) \ dA = \rho (\vec{v} \cdot \vec{dS}) $$ (2)

Where ## \vec{dS} ## is the infinitesimal surface area element of ## S ## such that ## \vec{dS} = \hat{n} dA ##

So, in integral form:

$$ M = \iiint \rho(V) dV = \iint \rho (\vec{v} \cdot \vec{dS}) $$

Looking at how ## M ## change as a function of time:

$$ \frac{dM}{dt} = \iiint \frac{d \rho(V)}{dt} dV = \iint \rho(\frac{\vec{dv}}{dt} \cdot \vec{dS}) $$

But ## \vec{v} = v(x,y,z,t) ##:

$$ \implies \frac{d\vec{v}}{dt} = \frac{\partial \vec{v} }{\partial t} + \frac{\partial \vec{v} }{\partial x} dx ... $$

$$ \frac{d\vec{v}}{dt}= \frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot (\nabla \cdot \vec{v}) $$

So:

$$ \frac{dM}{dt} = \iiint \frac{d \rho(V)}{dt} dV = \iint \rho \bigg(\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot (\nabla \cdot \vec{v})\bigg) \cdot \vec{dS}$$

However, since the surface ## S ## and the volume ## V ## is arbitrary, it is equally valid to write:

$$ \frac{d \rho(V)}{dt} = \frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot (\nabla \cdot \vec{v})$$

The trouble is that this doesn't seem to be at all the form of the continuity equation I've seen on Wikipedia! I realise you can use the divergence them to express the surface integral directly in terms of the divergence, but is the result I've got here equivalent to that?

Thank you very much!