When to use which dimensionless number

[Chestermiller]

1. I didn't use continuity in differential form (at least not $\nabla \cdot \vec u = 0$). I used a control volume approach. Should I not have?
2. I was using cylindrical coordinates (notice my arguments for the Laplacian). But I ended up using just the $z$ direction.
3. Is the volume constant? I suppose we could consider a spreading droplet of water. I was imagining a set up where we could drain fluid at some downstream point, but perhaps we can talk more on this later?
4. Yea, I definitely was assuming creeping flow, and neglected flow in $r$ and $\theta$.
5. I'd like to learn this! Ok, so what's next? Shall I give it another go or would you like to take over?

So you are saying you have a trough with a very viscous fluid flowing down the trough? And no new fluid is being introduced on the high end?

 

[member 428835]

So you are saying you have a trough with a very viscous fluid flowing down the trough? And no new fluid is being introduced on the high end?

I was assuming the flow was very viscous. Did I enforce that no new fluid is being introduced? If so, this was not my intent (although we could do that too).

So in your analysis, how would you do things?

 

[Chestermiller]

I was assuming the flow was very viscous. Did I enforce that no new fluid is being introduced? If so, this was not my intent (although we could do that too).

So in your analysis, how would you do things?

Have you solved this without surface tension yet? If not, what do you think that solution looks like?

 

[member 428835]

Have you solved this without surface tension yet? If not, what do you think that solution looks like?

By "this" are you referring to the post in 103? If so, well the governing equation looks identical to that of Poiseuille flow, but obviously with a different geometry and BC, and the pressure I don't think would be constant here, though without surface tension then yes.

The way I understand the problem, flow is either driven by capillary forces or gravity (or perhaps both). The significance of each would depend on scaling. What do you think?

 

[Chestermiller]

By "this" are you referring to the post in 103? If so, well the governing equation looks identical to that of Poiseuille flow, but obviously with a different geometry and BC, and the pressure I don't think would be constant here, though without surface tension then yes.

The way I understand the problem, flow is either driven by capillary forces or gravity (or perhaps both). The significance of each would depend on scaling. What do you think?

By "this," I meant the overall problem. I have trouble seeing how capillary forces are going to be important unless the wedge angle is very small (and the height of water in the trough is small) or, at the very leading edge of the fluid as it advances in the trough. I'm going to try to set up this problem without surface tension an see where it takes me. I'll assume that the height of fluid in the trough is changing very gradually with axial position along the trough. Is that the kind of approximation you have been using?

 

[member 428835]

By "this," I meant the overall problem. I have trouble seeing how capillary forces are going to be important unless the wedge angle is very small (and the height of water in the trough is small) or, at the very leading edge of the fluid as it advances in the trough. I'm going to try to set up this problem without surface tension an see where it takes me. I'll assume that the height of fluid in the trough is changing very gradually with axial position along the trough. Is that the kind of approximation you have been using?

I think this thread was started considering capillary-driven flows, where gravity is ignored. There's no induced pressure gradient from external forces and no gravity, so isn't a capillary pressure force the only mechanism for driving flow?

This being said, I like the idea of including inertia! Since gravity is absent, surface tension (and capillarity) is relevant, right? I definitely used length much larger than height. If it's okay with you, can we proceed with surface tension?

 

[Chestermiller]

I think this thread was started considering capillary-driven flows, where gravity is ignored. There's no induced pressure gradient from external forces and no gravity, so isn't a capillary pressure force the only mechanism for driving flow?

This being said, I like the idea of including inertia! Since gravity is absent, surface tension (and capillarity) is relevant, right? I definitely used length much larger than height. If it's okay with you, can we proceed with surface tension?

Sorry Josh. I'm totally confused by this. Apparently, I just don't understand the essence of the problem.

 

[member 428835]

Sorry Josh. I'm totally confused by this. Apparently, I just don't understand the essence of the problem.

We can continue with your interpretation since I enjoy learning from you.

What I was imagining is a low-gravity situation, where the pressure gradient drives flow in a wedge but the pressure gradient is interpreted in the Young-Laplace equation since surface tension is relevant.

 

[member 428835]

Were you still interested in doing this Chet?

 

[Chestermiller]

Were you still interested in doing this Chet?

My energy for doing this has waned. But, here is the game plan I would use.

Assume that the liquid is very viscous and flowing very slowly. Assume that the surface of the liquid is horizontal in the direction perpendicular to the trough axis, so the surface elevation h is a function only of z. Use this to determine the local pressure gradient in the z direction. The shear stress on the fluid flowing down the trough is zero at the free surface, and the cross section of fluid at each axial location is basically half a rhombus. So the local pressure gradient would be the same as if you had a full rhombic closed channel with twice the volumetric flow rate. So, look up the relationship for fully developed laminar flow in a duct of rhombic cross section. Do a differential mass balance on the section of trough between z and z plus delta z to determine the rate of change of the liquid surface height with time as a function of z.