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
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Carleigh Shaffer
Answered 2022-05-10 Author has 10 answers
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 = d P d z R 2 r 2 4 μ
where R is the radius of the blood vessel, d P / d z is the pressure gradient driving the flow, and μ is viscosity. The shear stress only has a z r component
τ z r u z r d P d z 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 d P d z R 4 μ .
We can re-express shear stress using flow rate
τ r z ( R ) 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.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-05-17
What happens to pipe length when the pipe diameter changes?
Intuitively, when the diameter of pipe is decreased, there will be more friction loss, more water pressure, and a higher flow rate. Is there a direct relationship/equation derived to see the affect of the pipe lengths?
Deriving from Hagen-Poiseuille's equation, we get:
  Q = Δ P π r 4 8 μ L
where :
Q = flow rate
Δ   P = change in fluid pressure
r = radius of pipe,
μ = dynamic viscosity of fluid,
L = length of pipe
To keep similar flow rates, are we able to use Poiseuille's derived formula to find the new lengths of pipe with a change in r (pipe radius)?
asked 2022-05-10
Pressure changes in Continuity equations and Poiseuille's Law?
Continuity says Q=AV, and we know that velocity and pressure are inversely related. So if we are in a closed system, like vasculature for example, Q is constant and any decrease in vessel radius would be expected to raise velocity, which would result in lower pressure.
If we look at Poiseuille's Law, on the other hand, we see the opposite! If Q is constant, then a decrease in radius/cross sectional area we should expect pressure to be raised!
What's going on?
asked 2022-05-07
Difference between Pressure and Pressure Energy in fluids
For a fluid with viscosity to flow through a pipe that has the same cross-sectional area at both ends, at a constant velocity, there has to be a pressure difference according to Poiseuille's Law. Why exactly is there a change in pressure required to keep the velocity constant?
Is it because according to Bernoulli's principle that Pressure or Pressure-Energy gets converted to Kinetic Energy to speed up the fluid so the mass flow rate at both ends of the pipe stays the same?
So if that's the case in light of Bernoulli's principle, that means the change in pressure in Poiseullie's Law is there so that pressure energy gets converted to Kinetic Energy to fight off the viscosity of the fluid to keep the velocity of the fluid constant?
And
What exactly is Pressure? I know it is the Force divided by the area. I understand that concept, but I've seen the terms Pressure and Pressure Energy used interchangeably when talking about fluids, which creates for some amount of confusion. Aren't Pressure and Pressure Energy different? But when we talk about fluids in light of Bernoulli's principles, it seems as if Pressure and Pressure Energy are the same, which is pretty confusing.
asked 2022-05-20
Why is pressure gradient assumed to be constant with respect to radius in the derivation of Poiseuille's Law?
Poiseuille's Law relies on the fact that velocity is not constant throughout a cross-section of the pipe (it is zero at the boundary due to the no-slip condition and maximum in the center). By Bernoulli's Law, this means that pressure is maximum at the boundary and minimum at the center. But in the book I have it is assumed that the pressure gradient is independent of radius (distance from the center of the pipe), and the pressure gradient is thus extricated from a radius-integral. Can anyone justify this?
asked 2022-05-20
For the electrical resistance of a conductor, we have
R = ρ l A
Noting the structural similarity between the Hagen-Poiseuille law and Ohm's law, we can define a similar quantity for laminar flow through a long cylindrical pipe:
R V = 8 η l A r 2
So there's a structural difference of a factor of r 2 between the two. What's the intuition behind this?
asked 2022-05-17
Why exactly is the resistance of a conductor inversely proportional to the area of its cross-section?
Before I explain my query, I would like to clarify that I am a ninth-grader who got this question while studying the formula R 1 A where A is the area of cross-section.
I have often asked this question to my teachers and they always give me the classic "corridor and field example". They told me that if 5 people walk in a corridor, they will find it harder to get across than if they were to be walking through a field- the same goes for electrons passing through a conductor. My counter-argument would be that if the width of the conductor increases, so will the number of positive ions (my textbook says that positive ions in conductors hinder the flow of current) and hence, more the resistance.
I would really appreciate it if the answer could be explained to me in simple terms as I'm not well versed with the more complex formulae involved in this concept. If not, do let me know of the concepts I should read about (preferably the specific books) to understand the solution better.
asked 2022-05-08
Pneumatic flow formula
I'm looking for a pneumatic formula in order to have the flow. I found many formula, but only for hydraulic :
Δ P = R p Q where R p is the resistance, Q the flow, and Δ P the pressure potential:
R p = 8 η L π R 4
with η = dynamic viscosity of the liquid ; L = Length of the pipe ; R = radius of the pipe
I don't know if I can use them, since I'm in pneumatic ! After doing research on the internet, I found some other variables like : sonic conductance, critical pressure coefficient, but no formula...
I think that I have all the information in order the calculate the flow : Pipe length = 1m ; Pipe diameter = 10mm ; Δ P = 2 bars
Thanks !

New questions