What is the profile of the second fluid in laminar flow with two fluids?

[jencam]

Hi

I am working in an application where we pump a fluid in a water -filled tube, where the second fluid has a viscosity of 3-4 times the viscosity of water. We have laminar flow (Re around 20).

If the fluids were 100% compatible (e.g. colored water vs clear water) I have determined that if I pump say 100 µl (V), the edge of the second fluid (B) will be att 2V, and the amount of B at a certain cross section will be linearly increasing so at V there will be 50% of B and at 0 µl I will have 100% fluid 2.

For different viscosity I have found a shorter envelope of B in water. I can imagine why, but I don't seem to have the brains to compute how much.

Is there a way to describe the profile of B in water. Maybe there are more parameters needed like velocity, other material properties, ...

I don't need an exact answer. Rather a simple one if there is. Laminar theory works quite well for water/water and I don't seem to need to account for diffusion even though my tubing is 0.8 mm I.D.

Regards

Jens Cameron

 

[Andy Resnick]

It's difficult for me to get a good picture of what you are doing, but recall that Poiseuille flow (which may be appropriate here) has a parabolic velocity profile. This may account for the appearance of gradients in your tube.

 

[jencam]

Hi Andy. Thanks for your response.

Poiseuilles law is what I use for water-water flow. Works quite well and the paraboloid velocity profile will in effect give a linear decrease in "concentration" (I know the fluids are not mixed in laminar flow but for me the effect is the same).

The actual application is that I wish to sample blood from an animal through a catheter. In the animal I have "pure" blood and the catheter is (in this abstraction) filled with pure saline (0.9 mg/ml NaCl in H2O). Properties of saline is equivalent to water but prevents hemolysis.

In the system, blood is withdrawn from the animal into the catheter, where the two fluids mix (in my vocabulary), and further up in the system is a valve where I can insert air to block the laminar flow. After the valve the blood sample is transported between two air bubbles to a test tube.

I have developed a simple theory using Poiseuilles equations and done some experiments that matches theory quite well when I use colored water/colorless water. With blood/saline I get better results (lower dilution) than with water/water phase and I would like to improve my model to handle flows with different properties.

My guess is that I have a flow of blood that is not purely axial in the tube. This flow i sdue to that the more viscous blood pushes the water away (or say that the water is more mobile than the blood). One extreme is the air I insert at the valve - the edge between water and air is very short/sharp - there is no air left in the tube when water comes flowing (no parabolic profile).

I know the question may be very complex and depend on intra-material properties like surface tension, diffusion (blood is water-soluble), not to mention that blood isn't really a fluid at all but a mix of water, salts, fatty acids, proteins and solid particles (erythrocytes). It is more a matter of if it is possible to describe the velocity profile (or inter-fluid border) with some formula or if numerically solving Navier Stokes equations is the only way to go. This wouldn't be feasible for me since it will be easier to do an experiment to determine dilution.

I hope you get a better picture with this description

Regards

--Jens