# Laminar flow exits an inclined tube

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1. Jan 28, 2015

### 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.

Last edited: Jan 28, 2015
2. Jan 30, 2015

### siddharth23

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

3. Jan 30, 2015

### arthurchen

Yes

4. Jan 30, 2015

### Staff: Mentor

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. Jan 30, 2015

### arthurchen

I attached a photo of the experiment I am trying to model. The fluid has significant deformation.

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6. Jan 30, 2015

### Staff: Mentor

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

7. Jan 30, 2015

### arthurchen

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.

8. Jan 30, 2015

### Staff: Mentor

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.