- #1

Dafe

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## Homework Statement

I am having problems understanding the differential form of the conservation of mass.

Say we have a small box with sides [tex]\Delta x_1, \Delta x_2, \Delta x_3[/tex].

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,

[tex] \dot{M}_{ac}=\dot{M}_{in} - \dot{M}_{out} [/tex]

Accumulated mass in differential form:

[tex] \dot{M}_{ac} = \frac{\partial \rho}{\partial t}\Delta V[/tex]

Rate of mass going in ([tex]u_i[/tex] is the velocity in direction [tex]i[/tex]).

[tex] \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} [/tex]

Rate of mass going out:

[tex] \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} [/tex]

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

Thanks.

## Homework Equations

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## The Attempt at a Solution

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