# Continuity equation - mass

1. Aug 22, 2010

### Dafe

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
N/A

3. The attempt at a solution
N/A

2. Aug 22, 2010

### kuruman

Actually it is

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

It stands for the "velocity perpendicular to the face at coordinate xi+Δxi".
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.

3. Aug 22, 2010

### Dafe

That makes sense. Thank you very much.