When leukocytes adhere to the endothelium of a blood vessel, they undergo an interesting rolling motion. This is caused by the flow of blood over the leukocyte and the force (F) binding the leukocyte  to the endothelium. We would like to determine this force.  Consider a blood vessel of radius R100 um with a flow of blood Q 0.2 x 10 cm³/s passing  through the vessel. When viewed under a microscope, a leukocyte of radius r = 5 um is seen to be
rolling along the wall at a rotational velocity of 0.05 revolutions/s0.314 rad/s. Thedensity of blood is p= 1.05 g/cm³ and its viscosity is = 0.04 g/cm.s.  

Build a model system that would allow us to determine what the binding force of a  leukocyte to the endothelium might be. We will look at the rolling of a latex microsphere along the  wall of a round tube (of radius R.). This microsphere will be made to adhere to the wall of the roundtube with a known force .

a) Derive a set of dimensionless variables for relating the angular velocity of rolling to the othervariables.

b) We have available microspheres of radius 20 μm, each of which will adhere to ta plastic substratewith a force F 106 dynes. We also have available a fluid with exactly the same density and  viscosity of blood. We plan to run a series of experiments in a plastic tube of radius Rm to  determine how the rolling rate of the microsphere varies with the flow rate (Qm). Determine whatconstraints, if any, need to be put on these experimental variables.
c) In the model experiment, we find that, for the range of flow rates investigated,  varies linearly  with 

 Determine the force (F) that makes the leukocyte adhere to the vessel wall.  

Wm wo + ax Qm 0.001 rev/s 10 rev/cm 2x x 10 rad/s 201 rad/em