Fluid stream on a Curved Surface

[roldy]

So the other day I was thinking about this problem that arose when I was watching water spiral down a bowl shaped drain.

If you can imagine a spherical shell and on the inside there is a fluid stream that is tangent to the interior curvature of the wall. The entrance of the fluid stream is at some point (r,$$\theta$$, $$\phi$$) where r is the inside radius of the sphere.
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Question: What nozzle pressure would you need in order for the fluid to remain attached to the curved surface? Another question that follows is what nozzle pressure would you need in order for the fluid to travel fully around the circumference of the sphere? To make things simpler, I'm assuming the fluid to be an inviscid, Newtonian fluid with laminar flow. Now since I'm assuming an inviscid flow, the only other thing that would affect the travel of the fluid stream would be gravity. With inviscid flow, the friction between the wall and fluid stream would cause the fluid stream velocity to decrease as it goes around the circumference of the sphere, hence prohibiting the fluid from going all the way around the sphere.

I've tried to think of a way to model this with an equation using the Navier Stokes equations. I would like to solve for the pressure in terms of some diameter. Eventually I would like to perform a fluid simulation but first would like to perform a theoretical study. How should I approach this problem?

 

[boneh3ad]

So the other day I was thinking about this problem that arose when I was watching water spiral down a bowl shaped drain.

If you can imagine a spherical shell and on the inside there is a fluid stream that is tangent to the interior curvature of the wall. The entrance of the fluid stream is at some point (r,$$\theta$$, $$\phi$$) where r is the inside radius of the sphere.

Question: What nozzle pressure would you need in order for the fluid to remain attached to the curved surface?

The flow will always remained attached, as there is a pressure gradient induced by the curvature holding the fluid to the wall. The only risk of separation is if you were looking at a flow around the outside of the tube.

To make things simpler, I'm assuming the fluid to be an inviscid, Newtonian fluid with laminar flow.

This doesn't make sense. Laminar/turbulent flow and Newtonian fluids have no meaning when talking about an inviscid fluid.

Now since I'm assuming an inviscid flow, the only other thing that would affect the travel of the fluid stream would be gravity. With inviscid flow, the friction between the wall and fluid stream would cause the fluid stream velocity to decrease as it goes around the circumference of the sphere, hence prohibiting the fluid from going all the way around the sphere.

Another question that follows is what nozzle pressure would you need in order for the fluid to travel fully around the circumference of the sphere?

You contradicted yourself, but I get what you are saying. It really is just a questions of if the flow has enough velocity to make it around the circumference before hitting the ground. Since it is inviscid, there will be no circumferential deceleration as it falls, so it is only a matter of gravity pulling it down; a fact to which you already alluded.

I've tried to think of a way to model this with an equation using the Navier Stokes equations. I would like to solve for the pressure in terms of some diameter. Eventually I would like to perform a fluid simulation but first would like to perform a theoretical study. How should I approach this problem?

Honestly you only need Bernoulli. Just use the pressure drop from the nozzle to get a velocity until you have a high enough velocity to make it around the tube before the flow hits the ground. It is essentially just a kinematics problem with a fluid spin on it.

 

[roldy]

I'll try and model the problem in some sort of 3d picture with a better explanation of the problem. I think I worded it wrong in my attempt to explain a 3 dimensional problem.

 

[boneh3ad]

I don't think you did. I think you are just making the problem harder than it needs to be.

 

[pmacias]

I find your question and curiosity about fluid dynamics on a curved surface to be very interesting. It is a complex problem that requires a deep understanding of fluid mechanics and mathematical modeling. In order to solve this problem, you will need to consider various factors such as the fluid properties, geometry of the curved surface, and external forces like gravity.

To approach this problem, I would recommend starting with simplifying assumptions, such as assuming the fluid to be inviscid and Newtonian with laminar flow. This will help in developing a theoretical model and equations to solve for the pressure needed for the fluid to remain attached to the curved surface and to travel around the circumference of the sphere.

Next, you can use the Navier-Stokes equations, which describe the motion of fluid particles, to develop your theoretical model. You will need to consider the boundary conditions, such as the initial velocity and pressure at the entrance of the fluid stream, as well as the curvature of the surface, to solve for the pressure required.

Once you have developed your theoretical model, you can then perform fluid simulations to validate your results and make any necessary adjustments. This will also give you a better understanding of the fluid behavior on a curved surface and how different factors, such as viscosity and gravity, affect the flow.

In conclusion, your approach to this problem is a good one and I encourage you to continue your theoretical study and perform fluid simulations to gain a deeper understanding of this complex problem. Good luck!