What are the oscillation modes in low-gravity capillary-dominated flow?

[member 428835]

Hi PF!

I am solving a problem for low-gravity capillary-dominated flow, where liquid rests in a rectangular (2D) channel. The text I'm reading introduces a velocity potential $\hat u = \nabla \phi$, and then states that each oscillation mode is $\partial_n\phi|_\Gamma = \nabla \phi \cdot \hat n|_\Gamma = \vec u \cdot \hat n|_\Gamma$ where $\Gamma$ is the meniscus surface. So oscillation modes are simply $\vec u \cdot \hat n|_\Gamma$? I'm a little confused.

I'm thinking of a vibrating string, where modes are characterized by standing waves, which makes me think position, rather than normal components of velocity. Can someone explain this to me please?

 

[Andy Resnick]

Hi PF!

I am solving a problem for low-gravity capillary-dominated flow, where liquid rests in a rectangular (2D) channel. The text I'm reading introduces a velocity potential $\hat u = \nabla \phi$, and then states that each oscillation mode is $\partial_n\phi|_\Gamma = \nabla \phi \cdot \hat n|_\Gamma = \vec u \cdot \hat n|_\Gamma$ where $\Gamma$ is the meniscus surface. So oscillation modes are simply $\vec u \cdot \hat n|_\Gamma$? I'm a little confused.

What text are you reading? I have a good reference in my office (Slattery's Interfacial Transport Phenomena) that may have some insight.

 

[member 428835]

What text are you reading? I have a good reference in my office (Slattery's Interfacial Transport Phenomena) that may have some insight.

I'm reading Low Gravity Fluid Mechanics by Myshkis et al. 1976.

 

[Andy Resnick]

Hi PF!

I am solving a problem for low-gravity capillary-dominated flow, where liquid rests in a rectangular (2D) channel. The text I'm reading introduces a velocity potential $\hat u = \nabla \phi$, and then states that each oscillation mode is $\partial_n\phi|_\Gamma = \nabla \phi \cdot \hat n|_\Gamma = \vec u \cdot \hat n|_\Gamma$ where $\Gamma$ is the meniscus surface. So oscillation modes are simply $\vec u \cdot \hat n|_\Gamma$? I'm a little confused.

I'm thinking of a vibrating string, where modes are characterized by standing waves, which makes me think position, rather than normal components of velocity. Can someone explain this to me please?

A clue: $\vec u \cdot \hat n|_\Gamma$ is the jump mass balance across a dividing surface; this term is associated with the speed of the dividing surface and there is a corresponding term like $( \vec v - \vec u)(\vec v \cdot \vec n - \vec u \cdot \vec n)$ in the jump momentum balance equation. I couldn't figure out how mode decomposition figures into this, tho.

 

[Andy Resnick]

I'm reading Low Gravity Fluid Mechanics by Myshkis et al. 1976.

Huh... when I was a grad student, I had the good fortune to work with Lev Slobozhanin for a couple of years.

 

[member 428835]

Huh... when I was a grad student, I had the good fortune to work with Lev Slobozhanin for a couple of years.

I'm unfamiliar with him (or her). I google searched them though, and it seems they're pretty on top of capillary phenomena. Thanks for your tip too.

A clue: $\vec u \cdot \hat n|_\Gamma$ is the jump mass balance across a dividing surface

Could you elaborate here? I just thought of it as the velocity component normal to the free surface.

 

[Andy Resnick]

Could you elaborate here? I just thought of it as the velocity component normal to the free surface.

Lev is definitely a 'him'.

That velocity term is part of the total term representing a parcel of moving interfacial fluid- the particle can move, but the interface itself can move as well:

https://books.google.com/books?id=V...AEIKTAA#v=onepage&q=jump mass balance&f=false

When there is surface tension, the jump momentum balance incorporates local interfacial curvature (pressure jump):

http://herve.lemonnier.sci.free.fr/TPF/NE/03-Slides.pdf