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Continuity equation - mass

  1. Aug 22, 2010 #1
    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 [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?


    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Aug 22, 2010 #2


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    Actually it is

    [tex]u_i|_{x_i+\Delta x_i}[/tex]

    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.
  4. Aug 22, 2010 #3
    That makes sense. Thank you very much.
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