Question

Derive an expression of velocity profile fluid flow through a vertical tube.

Answer 1

Table of Contents:
1. Example Problem 2.3-2
2. Solution to Ex. 2.3-2
3. Extra work including heat transfer through circular tube
Example 2.3-2:
A fluid that is very nearly described by the Bingham model is flowing through a vertical tube as the result of a pressure gradient and/or gravitational acceleration.

Buoyancy driven flows of two Bingham fluids in an inclined duct are considered, providing a simplified model for many oil-field cementing processes. The flows studied are near-uniaxial and stratified, with the heavy fluid moving down the incline, displacing the lighter fluid upwards. Flow of two Bingham fluids in a cylindrical duct: heavier fluid (cement) flows downwards in axial direction. Lighter fluid (mud) is below the cement. These calculations were implemented with PLTMG. (References: Ian Frigaard, Otmar Scherzer )
The radius and length of the tube are R and L. respectively. It is desired to obtain a relation between the volume rate of flow Q and the combined pressure and gravity forces acting on the fluid.