Catheters that are placed in arteries and veins will alter the flow through the vessel. Consider a
Question:
Catheters that are placed in arteries and veins will alter the flow through the vessel. Consider a catheter of radius cR (see fig 3.19) that is placed in a blood vessel of radius R. Assume that the catheter I concentric with the centerline of the blood vessel, and determine the reduction in flow rate relative to a vessel of the same radius without a catheter. Assume steady flow and assume that the pressure drip is the same with the fluid without the catheter; treat the blood as a Newtonian fluid.
Shetch of the velocity profile for steady flow around a catheter
From fluid mechanics the below formulae are derived by applying the following conservation laws on an elemental control volume.
1. Continuity equation
Conservation of momentum The boundary conditions are:
1. No slip at blood vessel wall, i.e Vvw=0
2. No slip at catheter wall, i.e Vcathw=0
With the above the flow rate in a cylindrical blood vessel without catheter with laminar flow
Where m = dynamic viscosity Pa.s
?P / L = Pressure drop per unit length, Pa/m
With catheter inside
Where radius of catheter r = eR
Hence reduction in volume flow rate
Here it is assumed that is same for with and without catheter.
Reduction in mass flow rate
Mathematical Applications for the Management Life and Social Sciences
ISBN: 978-1305108042
11th edition
Authors: Ronald J. Harshbarger, James J. Reynolds