Is this equation conservative or non-conservative?

[humphreybogart]

Homework Statement

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

Homework Equations

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.

 

[Chestermiller]

Homework Statement

This is the Navier-Stokes equation for compressible flow. n_j is the unit normal vector to the surface 'j', and n_i 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?

 

[humphreybogart]

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.

 

[Chestermiller]

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

 

[humphreybogart]

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.

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.