Is this equation conservative or non-conservative?

In summary, the Homework Statement is discussing the Navier-Stokes equation which is a equations for compressible flow. The equation is written for a control volume, but the first term on the right hand side is absent in the material volume form.
  • #1
humphreybogart
22
1

Homework Statement


This is the Navier-Stokes equation for compressible flow. nj is the unit normal vector to the surface 'j', and ni is the unit normal vector in the 'i' direction. Is this equation written for a control volume or a material volume?

Homework Equations


upload_2016-7-10_19-0-39.png


The Attempt at a Solution


I believe it's for a control volume, since it's in integral form and expressing fluxes out of a cube (taking advantage of conservation of momentum). However, I know that integral forms of non-conservative equations also exist, so I'm not sure.
 
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  • #2
humphreybogart said:

Homework Statement


This is the Navier-Stokes equation for compressible flow. nj is the unit normal vector to the surface 'j', and ni is the unit normal vector in the 'i' direction. Is this equation written for a control volume or a material volume?

Homework Equations


View attachment 103049

The Attempt at a Solution


I believe it's for a control volume, since it's in integral form and expressing fluxes out of a cube (taking advantage of conservation of momentum). However, I know that integral forms of non-conservative equations also exist, so I'm not sure.
Would the first term on the right hand side be present in the material volume form?
 
  • #3
Chestermiller said:
Would the first term on the right hand side be present in the material volume form?
I'm tempted to say 'no', because no fluid enters or leaves a material volume. So the term would disappear. I'd like to see the integral and differential form for conservative, and the integral and differential form for non-conservative.
 
  • #4
humphreybogart said:
I'm tempted to say 'no', because no fluid enters or leaves a material volume. So the term would disappear. I'd like to see the integral and differential form for conservative, and the integral and differential form for non-conservative.
The integral form for material volume is the same as for control volume, except that the first term on the right hand side is absent. The differential forms for both are identical. See this link to see why the integral form of the material volume development reduces to the same differential form as the control volume development: https://en.wikipedia.org/wiki/Reynolds_transport_theorem
 
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  • #5
Chestermiller said:
The integral form for material volume is the same as for control volume, except that the first term on the right hand side is absent. The differential forms for both are identical. See this link to see why the integral form of the material volume development reduces to the same differential form as the control volume development: https://en.wikipedia.org/wiki/Reynolds_transport_theorem
Thank you.
Chestermiller said:
The integral form for material volume is the same as for control volume, except that the first term on the right hand side is absent. The differential forms for both are identical. See this link to see why the integral form of the material volume development reduces to the same differential form as the control volume development: https://en.wikipedia.org/wiki/Reynolds_transport_theorem
Great! I seen in another post a reference to Bird's Transport Phenomena book. Thanks.
 

1. What does it mean for an equation to be conservative or non-conservative?

An equation is considered conservative if the sum of energy or mass remains constant over time. In other words, the equation represents a system where energy or mass can neither be created nor destroyed. On the other hand, an equation is non-conservative if the sum of energy or mass changes over time, indicating that energy or mass is being added or removed from the system.

2. How can I determine if an equation is conservative or non-conservative?

An equation can be determined to be conservative if it satisfies the mathematical property of being path independent. This means that the value of the equation is the same regardless of the path taken to get to a certain point. If the equation does not satisfy this property, then it is considered non-conservative.

3. What are some examples of conservative and non-conservative equations?

An example of a conservative equation is the law of conservation of energy, where the total energy in a system remains constant. An example of a non-conservative equation is the Navier-Stokes equation, which describes fluid flow and takes into account external forces that can change the energy of the system.

4. What are the implications of an equation being conservative or non-conservative?

The implications of an equation being conservative or non-conservative are significant in understanding the behavior of a system. Conservative equations indicate that energy or mass is conserved, and therefore, the system is stable and predictable. Non-conservative equations, on the other hand, suggest that energy or mass is not conserved, and the system may exhibit chaotic or unpredictable behavior.

5. Can an equation be both conservative and non-conservative?

No, an equation can only be either conservative or non-conservative. These terms are mutually exclusive and represent two different types of systems. However, some equations may have both conservative and non-conservative components, but as a whole, they will be classified as either one or the other.

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