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

In summary, the conversation discusses the differential form of the conservation of mass and how it relates to a small box with sides Δx1, Δx2, and Δx3. The conservation of mass states that the rate of accumulated mass in a control volume is equal to the rate of mass going in minus the rate of mass going out. The conversation also explains the notation |_{x_i+\Delta x_i}, which stands for the velocity perpendicular to the face at coordinate xi+Δxi.
  • #1
Dafe
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0

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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  • #2
Dafe said:
What does the notation [tex]|_{x_i+\Delta x_i}[/tex] mean?

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.
 
  • #3
kuruman said:
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.

That makes sense. Thank you very much.
 

1. What is the continuity equation for mass?

The continuity equation for mass is a mathematical expression that describes the conservation of mass in a fluid flow. It states that the rate of change of mass in a fluid is equal to the negative of the divergence of the mass flux density.

2. How is the continuity equation derived?

The continuity equation is derived from the principle of conservation of mass, which states that mass can neither be created nor destroyed, only conserved. It is derived using the fundamental laws of fluid mechanics, namely the continuity equation, the Navier-Stokes equations, and the conservation of momentum equation.

3. What is the significance of the continuity equation in fluid dynamics?

The continuity equation is a fundamental equation in fluid dynamics as it describes the behavior of fluids in terms of mass conservation. It is used to understand and predict the flow of fluids in various applications, such as in engineering, meteorology, and oceanography.

4. How is the continuity equation applied in real-world situations?

The continuity equation is applied in various real-world situations, such as in the design of pipes and channels for fluid transportation, in the study of weather patterns, and in the analysis of blood flow in the human body. It is also used in the development of models and simulations for fluid flow in different systems.

5. What are the limitations of the continuity equation?

While the continuity equation is a fundamental principle in fluid dynamics, it does have some limitations. It assumes that the fluid is incompressible, meaning that its density remains constant. This is not always the case, as some fluids, such as gases, can compress and expand. Additionally, the continuity equation does not account for external forces, such as gravity, which can affect the flow of fluids.

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