 # Blood as Newtonian fluid In some of the literature I read that blood can be considered as Newtonia junoonib89p4 2022-05-09 Answered
Blood as Newtonian fluid
In some of the literature I read that blood can be considered as Newtonian fluid when a larger vesses with high shear stress is considered... How is the shear stress calculated for aorta and how do they claim that shear stress is more for larger vessel when compared to the smaller ones . What is the relationship between the shear stress and the size of the vessel
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welcome. As a general rule, when you read something provide a reference.
Laminar blood flow in cylindrical blood vessels is Poiseuille flow. In Poiseuille flow the velocity profile is parabolic
${u}_{z}=\frac{dP}{dz}\frac{{R}^{2}-{r}^{2}}{4\mu }$
where $R$ is the radius of the blood vessel, $dP/dz$ is the pressure gradient driving the flow, and $\mu$ is viscosity. The shear stress only has a $zr$ component
${\tau }_{zr}\sim \frac{\mathrm{\partial }{u}_{z}}{\mathrm{\partial }r}\sim \frac{dP}{dz}r$
and the maximum stress occurs on the wall, $r=R$. For fixed pressure gradient this increases with radius, but pressure gradient is not fixed in a branching network.
One way to look at this is using the flow rate
$Q\sim \frac{dP}{dz}\frac{{R}^{4}}{\mu }.$
We can re-express shear stress using flow rate
${\tau }_{rz}\left(R\right)\sim \frac{Q}{{R}^{3}}.$
In vascular branching the total flow rate of an incompressible fluid must be conserved (in binary branching each branch has flow $Q/2$). The magic question is then how vessel radius scales in binary branching.
In principle this could follow any relation (as an engineer, you can just decide), but nature presumably tries to optimize things in some way. There is an empirical relation, called Murray's law (together with somewaht hand-waving derivations) that states that the sum of the cubes of $R$ is conserved. This would imply that the shear stress is constant across the network.