Confirming the 4th Change in Direction and Some Relief!!

11:05 pm

So, after trying very hard with the method of characteristics used in the original paper, I realized that, the mathematical part is too difficult for a child like me. So, will not be solving the stokes Navier equation using method of characteristics but will be doing it using pdetool in Simulink. This was the same idea that I had posted before. I realize that this means it is a waste of at least a week, but, I’ll call it use,.. since I know, exactly WHY I’m not using the method of characteristics. Have to enlist the reasons.

So, today, before going to sleep, the agenda is, use Simulink to create a module for the Generates the solution to the Stokes-Navier equations. I think the designing part will be easy, but what we need to verify is whether the output we are getting is right or wrong. To do that, we need to connect the entire system together, or give a temporary input that represents the practical input to the module.

1. Think how to calculate mean pressure at any point in the system which I suppose becomes the input to the Stokes-Navier module.

2. Think up if and how this module and the rest of the modules can work together.

3. Form the Stokes-Navier module, using the document mentioned before.

This document will be edited after about half an hour.

So, here I am, trying to design the Stokes – Navier module. Firstly consider this equation given in the document:

Now consider the Stokes-Navier Equation given in one of the ppts that I’m referring to which helps me understand the meaning of the equation:

Consider this Stokes-Navier equation which is given as the prototype in the mathwork manual with good explaination:

Now consider this Stokes-Navier Equation which is given as the prototype in the pdetool with notations which are somewhat different:

Now I have to create an equivalency to match the coefficients. I’m guessing that it’s as follows:

rho = d = density of blood

mu = c = viscosity of blood

The controversy is about a and f. Going by the signs of the terms, the pde toolbox help document and the ppt that I’m referring to, I guess,

p = a = pressure and

f = f = other forces

Now, hoping that these assumptions are correct, I’ve got to substitute the values of these constants. These values are, unfortunately not given in the paper, so I will have to borrow from google, possibly, wikipedia.

Taking, d = 1060 kg/m³, c = (from wikipedia), a will be the time varying mean arterial input pressure, f I think we’ll consider as 1(not sure).

Just figured out how to apply the elasticity to the generated Cross Section of the vessel.

With the Cross section and  space coordinate equations, we can find out a value S of the cross section. Now, as per the formula given in the base paper, we have,

This equation defines elasticity as the steady cross section value, divided by the product of blood density and change in the cross section due to pressure. Now consider the following:

Here, as we know, S becomes the input, we have to have a module for calculation of c².

From here I realise that I have solved the problem of Elasticity application to vessels.

Then, there was this problem about finding the mean pressure.. well just realized how much a of a fool I am to think that pressure in an artery can come from two directions!! Haha.. So, this problem will come only at the placenta. But there too, we have a division as maternal side of the placenta, and the fetal side of the placenta, so, nowhere we will have to encounter a scenario where we will have to calculate the ‘mean’ pressure! HA! Such a fool! Shi.

So now that I have figured out almost the entire deal I am at peace. Aaah! This feels great! There’s a time for everything. Jab jab jo jo hona hai, tab tab so, so, hota hai. Happy!! So, Again! I’ll do all of this tomorrow!!

Now what remains is the thinking up that I have to do regarding the bifurcations.. There is nothing about the bifurcations that is mentioned in the paper! Only the geometrical dimentions are given! What the hell is that all about? I guess, it should be treated in the same way as the arterial segments, but the only difference is that, there are two different values of cross sections at the second end. yoyo! Very happy and relieved! Everything has its own time!

ATB!