Boundary Divergence and Potential Formulation in Fluid Mechanics

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Discussion Overview

The discussion revolves around the concepts of boundary divergence and potential formulation in fluid mechanics, particularly in the context of a tank draining under gravitational potential. Participants explore theoretical aspects, boundary conditions, and the application of divergence in fluid dynamics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that there is no unbalanced flux in the system, noting that the outward flux at the upper surface is negative while at the lower surface it is positive, suggesting a cancellation.
  • Free surface conditions are discussed, with one participant mentioning dynamic boundary conditions of pressure continuity and kinematic boundary conditions related to fluid velocity.
  • One participant raises a point about the divergence being a local concept, indicating that while a vector field may be divergenceless overall, it can exhibit non-zero divergence at specific points on the free surfaces.
  • Another participant expresses confusion about how to write boundary conditions and seeks resources for further reading on the topic.
  • Concerns are raised about reconciling divergence on fluid boundaries with Euler's formulation of fluid mechanics and the appropriateness of potential formulation for the problem at hand.

Areas of Agreement / Disagreement

Participants express differing views on the specifics of the problem, particularly regarding boundary conditions and the application of potential formulation. There is no consensus on how to address the issues raised, and multiple competing perspectives remain.

Contextual Notes

Participants note difficulties in understanding the relationship between divergence and boundary conditions, as well as the applicability of potential formulation in this scenario. There are unresolved questions regarding the definitions and implications of these concepts.

tankFan86
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Does anyone have any ideas about this?
 

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tankFan86 said:
Does anyone have any ideas about this?
Your going to have to be slightly more specific than that. What specifically are you having problems with?

P.S. Is this homework?
 
Which part of the attached pdf (above) is not specific enough? Assume the tank is draining in a gravitational potential.
 
1. There is no unbalanced flux.
At the upper surface, at every moment, the outward flux is negative. At the lower surface, the outward flux is positive, and these cancel each other out.

2. The free surface conditions will be
a) the dynamic boundary conditions of continuity of pressure (as long as surface tension is neglected.
b) The kinematic boundary conditions that the normal component of the fluid velocity there equals the surface velocity.
 
tankFan86 said:
Which part of the attached pdf (above) is not specific enough? Assume the tank is draining in a gravitational potential.

Hoot isn't asking about the specifics of the problem, but rather the specifics of where YOU are having trouble with it. In other words, what have you attempted and where did you get stuck.

Since you didn't answer on whether this is a HW/Coursework problem or not, we are forced to assume that it is. Please post questions like this only in the HW/coursework forum from now on.

Zz.
 
1. The idea of the divergence is a local concept. When we say a vector field is divergenceless, we mean that at every point, the divergence is zero. However, if you pick a point on either of the free surfaces the divergence of the velocity field will be non-zero at that point.

2. These are ideas that I have heard of but still do not understand. How do you write these conditions? Where would I read more about them?
 
ZapperZ said:
Hoot isn't asking about the specifics of the problem, but rather the specifics of where YOU are having trouble with it. In other words, what have you attempted and where did you get stuck.

Since you didn't answer on whether this is a HW/Coursework problem or not, we are forced to assume that it is. Please post questions like this only in the HW/coursework forum from now on.

Zz.
I hope this is the correct forum. The homework help forum did not seem appropriate, as this is not part of a problem set or any particular class/assignment. I cannot find anything in the literature that addresses these issues.

I suppose the two greatest difficulties so far are reconciling the divergence on the boundary of the fluid with Euler's formulation of fluid mechanics, and realizing why the potential formulation is inappropriate for this problem (or how to apply it if it is).

Thank you for your input.
 

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