Blood flow

Blood flow

Blood flow is the flow of blood in the cardiovascular system. Mathematically, blood flow is described by Darcy's law (which can be viewed as the fluid equivalent of Ohm's law) and approximately by Hagen-Poiseuille equation.

Blood is a heterogeneous medium consisting mainly of plasma and a suspension of red blood cells. Red cells tend to coagulate when the flow shear rates are low, while increasing shear rates break these formations apart, thus reducing blood viscosity. This results in two non-Newtonian blood properties, shear thinning and yield stress. In healthy large arteries blood can be successfully approximated as a homogeneous, Newtonian fluid since the vessel size is much greater than the size of particles and shear rates are sufficiently high that particle interactions may have a negligible effect on the flow. In smaller vessels, however, non-Newtonian blood behavior should be taken into account.

The flow in healthy vessels is generally laminar, however in diseased (e.g. atherosclerotic) arteries the flow may be transitional or turbulent. The first equation below is Darcy's law, the second is the Hagen-Poiseuille equation:

F = frac{Delta P}{R}

R = (frac{nu L}{r^4})(frac{8}{pi})

where:

F = blood flow
P = pressure
R = resistance
ν = fluid viscosity

L = length of tube
r = radius of tube

In the last equation it is important to note that resistance to flow changes dramatically with respect to the radius of the tube. This is important in angioplasty, as it enables the increase of blood flow with balloon catheter to the deprived organ significantly with only a small increase in radius of a vessel.

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