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I am trying to derive the mass conservation equation:

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In one of the step we are given the equation to find Fluid mass gained through a small element of the surface 𝑑𝑆 in a time 𝑑𝑡 is:

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I don't understand how the equation is got.

Here is diagram i was given

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Mass is equal to density times volume, in your case: $$ dM = \rho \cdot dV. $$

To compute $dV$, you need the volume of a parallelogram with base area $dS$ and (perpendicular) height $h$ so that $dV = dS \cdot h$. Your base area is $dS = dA$ in your sketch, and the height $h$ is the distance travelled perpendicular to $dS$.
Since $\hat{\mathbf{n}}$ is by definition normal to $dS$, the height is just the projcection of the total distance travelled $\ell$ onto this axis, so $h = \boldsymbol{\ell}\cdot \hat{\mathbf{n}}$.
What is $\ell$? Distance is equal to velocity times time: $\ell = \mathbf{u}\cdot dt$.

The minus sign is probably because mass is flowing out of your surface and hence it's a loss.

Putting it all together:

$dM = -\rho \cdot dV = -\rho\, \mathbf{u}\cdot \hat{\mathbf{n}}\,dS\, dt$

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The given equation $$dm=-\rho\ \mathbf{u}\cdot\hat{n}\ dSdt$$ or $$\frac{dm}{dt}=-\rho \ \mathbf{u}\cdot\hat{n}\ dS$$ On the left-hand side, You have a rate at which the mass is changing in the given volume. On the right-hand side, You have the mass flux that goes out of the volume in time $dt$ multiplied with density. That way it makes sense.


Now How to derive this:

Note that $\hat{n}$ is a unit vector perpendicular to the surface. We emphasize that $\mathbf{u}$ is the velocity of a fluid at a given point $(x,y,z)$ in space and at a given time $t$. The mass of fluid flowing in unit time through an element $d\mathbf{S}$ of the surface bounding this volume is $\rho \mathbf{u}\cdot d\mathbf{S}$; the magnitude of the vector $d\mathbf{S}$ is equal to the area of the surface element, and its direction is along normal. $$\rho \mathbf{u}\cdot d\mathbf{S}=\rho \mathbf{u}\cdot \hat{n}dS$$ For time dt this would give the above relation $$dm=\rho \mathbf{u}\cdot d\mathbf{S} dt$$ The vector $$\mathbf{j}=\rho \mathbf{u}$$ is called the mass flux density.

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