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

Transcribed Image Text:

Q=VA= R? ΔΡ 8μ L Re = #R 8μ ΔΡ L L Qeath «R ΔΡ *[(1 – e) - 8μ L (1-2)2, In(1/e) ΔΟ=Q-leath πR ΔΡ. 8μ L -[1-(1-¹)- (1-²)², In(1/e) ΔΡ L - Qܠܡ = ܪܠ prRA 8μ P L -[1-(1-¹)- (1-2)², In(1/e) Q=VA= R? ΔΡ 8μ L Re = #R 8μ ΔΡ L L Qeath «R ΔΡ *[(1 – e) - 8μ L (1-2)2, In(1/e) ΔΟ=Q-leath πR ΔΡ. 8μ L -[1-(1-¹)- (1-²)², In(1/e) ΔΡ L - Qܠܡ = ܪܠ prRA 8μ P L -[1-(1-¹)- (1-2)², In(1/e)

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From fluid mechanics the below formulae are derived by applyin... View the full answer