Laminar flow exits an inclined tube

  • #1

Main Question or Discussion Point

I am working on a problem in which viscous flow comes out from an inclined tube, forming some kind of a fountain. In the tube the fluid is Newtonian and the flow can be treated as Poiseuille flow. I want to study the movement of the fluid after it leaves the tube. Can someone point me about the existing study and research on this? Thanks.
 
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  • #2
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It comes out from the top end of the inclined tube?
 
  • #3
  • #4
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Is the fountain very high/large compared to the diameter of the pipe? Is there a good reason to expect a large deviation from a ballistic trajectory?
 
  • #5
Is the fountain very high/large compared to the diameter of the pipe? Is there a good reason to expect a large deviation from a ballistic trajectory?
I attached a photo of the experiment I am trying to model. The fluid has significant deformation.
 

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  • #6
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Oh, and with a surface nearby, then things get complicated.
 
  • #7
Oh, and with a surface nearby, then things get complicated.
Yes. Is it possible to get some asymptotic relation of the depth averaged radial exit velocity (parallel to the surface) distribution? I mean, if the surface is horizontal, surely the exit velocity is a constant and depth are the same. When the surface has an inclination [itex]\alpha[/itex], my guess is that the flux per length behaves like
[tex]Q=\oint\boldsymbol{n}\cdot\boldsymbol{u}h dl\sim\int_{-\pi}^{\pi}\Gamma(\alpha,\theta)rd\theta,[/tex]
where [itex]r[/itex] is the radius of the tube, and the flux [itex]\Gamma[/itex] satisfies
[tex]\Gamma(\alpha,-\pi)=\Gamma(\alpha,\pi)=0,\,\Gamma(\alpha,-\theta)=\Gamma(\alpha,\theta),\,\Gamma(0,\theta)=\text{constant}.[/tex]

I do not need to solve the exact trajectory of the fluid, an approximated distribution of [itex]\Gamma[/itex] is enough.
 
  • #8
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Inside the tube and far away from the exit, flux should still follow a parabolic profile. Close to the exit, things get different. For a horizontal surface, the exiting water will fall back on the stream, and then I don't see how you could avoid numerical simulations. Those are a good idea for inclined planes as well, I think.
 

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