This notebook has been written in Mathematica 4.0 by
Mark J. McCready
Professor and Chair of Chemical Engineering
University of Notre Dame
Notre Dame, IN 46556
It is copyrighted to the extent allowed by which ever laws pertain to the World Wide Web and the Internet.
I would hope that as a professional courtesy, this notice remain visible to other users.
There is no charge for copying and dissemination
More recent versions of this notebook may be available at:
This notebook uses the notation of and is intended as a companion to:
An Introduction to Fluid Dynamics by Stanley Middleman, John Wiley, 1999.
This notebook solves the "lubricated" flow in a pipe problem and in doing so demonstrates several major fundamental points in Transport Phenomena.
1. For flows where more than one fluid is flowing together, there is a separate equation for the velocity profile for each fluid. These equations differ just by the fluid properties.
2. To solve these two-fluid problems you need to recognize:
• The if gravity can be ignored.
• The velocity of the two fluids matches at the interface.
• The shear stress in each fluid matches at the interface.
The last two of these are common boundary conditions.
For this specific situation the major results are:
Lubricated transport can cause a flow rate (compared to this fluid flowing alone in the pipe) on the order of the viscosity ratio of two fluids and the maximum enhancement occurs when the less viscous fluid occupies the outer 30% of the radius.
The problem we are considering is the flow of a very viscous liquid (Fluid I) confined to the center of the pipe, with a much less viscous liquid (Fluid II) flowing around Fluid I and contacting the walls. The pipe diameter is D and the thickness of the lubricating layer is h.
1. There is a separate equation for the velocity profile of each fluid, but only one equation for the stress profile.
2. The form of the differential operator (the differential equation) for a circular geometry leads to a logarithmic term in the solution. This term does not give a physically realizable profile for the middle of the pipe and must be neglected. (It is kept for fluids confined away from the middle.
3. These are readily solved and the key is using the boundary conditions to match the constants.
For a two-layer flow you expect:
• Shear stress match at the interface
• Velocity match at the interface
• No slip at the solid surface
4. The enhancement is highest when the outside fluid occupies about 30% of the pipe. Below this value, there is an advantage to increasing the less viscous flow to reduce shear. Above this point, there is not enough flow area available for the desired fluid and the enhancement drops.
5. The maximum enhancement scales as the value of the viscosity ratio and is approximately 50% of this value.
6. We can see explicitly that in contrast to single phase flow (and in contrast to intuition) there is a range where increasing the flowrate of the lubricating fluid will decrease the pressure drop.