What is the differential form of the continuity equation for mass?

[Dafe]

Homework Statement

I am having problems understanding the differential form of the conservation of mass.
Say we have a small box with sides $$\Delta x_1, \Delta x_2, \Delta x_3$$.
The conservation of mass says that the rate of accumulated mass in a control volume equals the rate of mass going in minus the rate of mass going out. I could write that as,

$$\dot{M}_{ac}=\dot{M}_{in} - \dot{M}_{out}$$

Accumulated mass in differential form:

$$\dot{M}_{ac} = \frac{\partial \rho}{\partial t}\Delta V$$

Rate of mass going in ($$u_i$$ is the velocity in direction $$i$$).

$$\rho\Delta x_2\Delta x_3 u_1|_{x_1} + \rho\Delta x_3\Delta x_1 u_2|_{x_2} + \rho\Delta x_1\Delta x_2 u_3|_{x_3}$$

Rate of mass going out:

$$\rho\Delta x_2\Delta x_3 u_1|_{x_1+\Delta x_1} + \rho\Delta x_3\Delta x_1 u_2|_{x_2+\Delta x_2} + \rho\Delta x_1\Delta x_2 u_3|_{x_3+\Delta x_3}$$

This is where I get confused. I do not understand how this expresses rate of mass going out. What does the notation $$|_{x_i+\Delta x_i}$$ mean?

Thanks.

Homework Equations

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The Attempt at a Solution

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[kuruman]

What does the notation $$|_{x_i+\Delta x_i}$$ mean?

Actually it is

$$u_i|_{x_i+\Delta x_i}$$

It stands for the "velocity perpendicular to the face at coordinate x_i+Δx_i".
If you do dimensional analysis, you will see that each term has dimensions of m-dot. The expression that you wrote assumes that mass is going in through three of the sides of the box and comes out the other three.

 

[Dafe]

Actually it is

$$u_i|_{x_i+\Delta x_i}$$

It stands for the "velocity perpendicular to the face at coordinate x_i+Δx_i".
If you do dimensional analysis, you will see that each term has dimensions of m-dot. The expression that you wrote assumes that mass is going in through three of the sides of the box and comes out the other three.

That makes sense. Thank you very much.