Deriving Velocity Profiles for Flow of Oil Down Glass Rod

[CanoIsDbo]

I am having a huge problem with understanding the derivation of velocity profiles for these type of problems. The one i am having the most difficulty is the flow of oil going down vertically along the surface of a glass rod. Where p=900 kg/m^3 and u=120 mPa's. How do I start this problem?

 

[fluidman]

here is the key

Start by writing Navier-stokes equations in cylindrical coordinates.And make reasonable assumptions to simplify the equations. Like, 1 The flow is steady and two dimensional in r-z plane. Here z is the coordinate pointing downwards.You can also chose it to be pointing upwards. In both cases you should pay attention to the signs in the equations. 2 The flow is driven by gravational forces. 3 The shear forces on interface of oil and air are negligible. 4 The flow is in only z direction. Thus the r and θ velocities component go to zero.
Now you need to do some integrations. I am leaving the rest to you...
Good luck!

 

[piercas]

Hello there,

Thank you for reaching out about your difficulty in understanding the derivation of velocity profiles for the flow of oil down a glass rod. I am happy to help you with this problem.

Firstly, it is important to understand the concept of velocity profiles. Velocity profiles describe how the velocity of a fluid changes at different points in a flow. In the case of flow down a glass rod, we can use the Navier-Stokes equations to derive the velocity profile. These equations describe the conservation of mass and momentum in a fluid flow.

To start this problem, we need to first define the variables given in the problem. In this case, p represents the density of the oil (900 kg/m^3) and u represents the dynamic viscosity (120 mPa's). We can also define the geometry of the problem, which in this case is a vertical glass rod.

Next, we can use the Navier-Stokes equations to derive the velocity profile. These equations involve partial derivatives, so we need to use calculus to solve them. The equations can be simplified for the case of a vertical flow, and we can also make some assumptions such as the flow being steady and incompressible.

Once we have the simplified equations, we can solve for the velocity profile by using boundary conditions. In this case, the boundary conditions would be the velocity at the surface of the glass rod and the velocity at the center of the rod. By solving these equations, we can obtain an equation that describes how the velocity of the oil changes as it flows down the glass rod.

I understand that this may seem like a complex problem, but with a solid understanding of the Navier-Stokes equations and some calculus, you should be able to derive the velocity profile for this specific scenario. I hope this helps guide you in solving this problem. Good luck!