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Hydraulic dynamics question
We know the length of a pipe, say L, and the starting point pressure p1, ending point p2, cross-sectional area A. What else do we need to compute the mean velocity of flow in this pipe?
My full question actually runs as follows. Given the mean cardiac output(volume flux) be 5.5L per min, the radius of the aorta is about 1.1cm. In the systemic circulatory system, the mean radius of a capillary is about 3µm. The mean pressure at the arterial end of the capillary bed (beginning of the capillary system) is estimated to be about 30mmHg, and about 15mmHg at the venous end (end of the capillary system). The length of capillary is 0.75mm.
Calculate the mean velocity in the aorta and in a capillary.
My idea is to use the volume flux 5.5L/min and the radius of aorta, divide the flux by the crossectional area to find the mean velocity in the aorta. However, I have no idea how to deal with the velocity in a capillary. I tried to use Poiseulle's law, but given that we only know the difference of pressure and length of the capillary, it is still unsolvable. Could anyone tell me how to deal with it?

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

Flow through a cylinder with pores
I'm building a model to study the flow of fluid through a cylinder with pores on its surface.
For flow through a cylinder, the velocity of the fluid flow is given by the Hagen-Poiseuille equation.
I would like to ask for suggestions on references from which I can look at derivations for a cylinder with porous walls. By porous, I mean openings on the surface of the cylinder. Example, the pores present in the fenestrated capillary.

Matching Bernoulli equation and Hagen Poiseuille law for viscous fluid motion
I thought that Bernoulli equation could be used only in the case of non viscous fluid. But doing exercises on 2500 Solved Problems In Fluid Mechanics and Hydraulics (Schaum's Solved Problems Series) I found that this procedure is followed.
In the case of viscous laminar flow, Bernoulli equation is written as
$\begin{array}{}\text{(1)}& z_1+\frac{v_1^2}{2g}+\frac{p_1}{\rho g}=z_2+\frac{v_2^2}{2g}+\frac{p_2}{\rho g}+h_L\end{array}$
Where $h_L$ is the head loss (due to viscosity) calculated using Hagen Poiseuille law.
$\begin{array}{}\text{(2)}& h_L=\rho g\frac{8\eta L\overline{v}}{R^2}\end{array}$
Is this a correct way to solve exercises involving viscosity?
Furthermore are there limitation to this use of Bernoulli equation (in case of viscosity)?
In particular
If the flow is not laminar, I cannot use $(2)$, but can I still write $(1)$ in that way?
Is $(1)$ valid only along the singular streamline or between different ones (assuming the fluid irrotational)?

For the electrical resistance of a conductor, we have
$R=\rho \frac{l}{A}$
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:
$R_V=8\eta \frac{l}{A{r}^2}$
So there's a structural difference of a factor of $r^2$ between the two. What's the intuition behind this?

How does the radius of a pipe affect the rate of flow of fluid?
Poiseuille's law states that the rate of flow of water is proportionate to $r^4$ where $r$ is the radius of the pipe. I don't see why.
Intuitively I would expect rate of flow of fluid to vary with $r^2$ as the volume of a cylinder varies with $r^2$ for a constant length. The volume that flows past a point is equal to rate of flow fluid, and thus it should vary with $r^2$
I cannot understand the mathematical proof completely. Is there an intuitive explanation for this?

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?