Laminar flow exits an inclined tube

[arthurchen]

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.

 

[siddharth23]

It comes out from the top end of the inclined tube?

 

[arthurchen]

It comes out from the top end of the inclined tube?

Yes

 

[mfb]

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?

 

[arthurchen]

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.

 

[mfb]

Oh, and with a surface nearby, then things get complicated.

 

[arthurchen]

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 $\alpha$, my guess is that the flux per length behaves like
$$Q=\oint\boldsymbol{n}\cdot\boldsymbol{u}h dl\sim\int_{-\pi}^{\pi}\Gamma(\alpha,\theta)rd\theta,$$
where $r$ is the radius of the tube, and the flux $\Gamma$ satisfies
$$\Gamma(\alpha,-\pi)=\Gamma(\alpha,\pi)=0,\,\Gamma(\alpha,-\theta)=\Gamma(\alpha,\theta),\,\Gamma(0,\theta)=\text{constant}.$$

I do not need to solve the exact trajectory of the fluid, an approximated distribution of $\Gamma$ is enough.

 

[mfb]

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.