Question regarding ideal fluids vs. non-ideal fluids

[Makarov]

I understand that in an ideal fluid, when the cross-sectional area of the pipe is increased, the pressure that the fluid exerts on the walls of the pipe also increases. Also, when the cross-sectional area of the pipe is decreased, the pressure that the fluid exerts on the walls of the pipe decreases.

However, I am a little confused as the how a non-ideal fluid works. Is it the other way around? When the radius of the pipe is decreased, the total peripheral resistance increases, and the pressure on the walls of the pipe rises?

The reason I am asking this question is because I am studying about the blood vessels of the body. Whenever the blood pressure rises, the body causes the vessels to dialate in order to counteract the increase in pressure and keep it constant. This seems to be contrary to what I want to think (increase in radius = increase in pressure). Is my conclusion about non-ideal fluids correct?

Thanks

 

[Dr. Jekyll]

We assume that the particles within the ideal fluid do not interact and that their volume is negligible. It's not the same in reality, but there is not much deviation (of course, there are some limits). So, many non-ideal fluids act almost the same as the ideal model. So, your conclusion is not quite correct.

You have completely different example with the blood vessels. We cannot ignore the volume of the blood cells and their collisions, so it can't be considered as an ideal fluid. Therefore, some fluid mechanics laws cannot be applied here.

That's my assumption. I'm not sure if it's correct.

 

[Andy Resnick]

I'm not sure what you mean. In Poiseuille flow, there's several parameters- the pressure drop along the pipe, the diameter of the pipe, viscosity of the fluid, etc. I would expect that keeping the volume flow constant, increasing the pipe diameter will decrease the pressure exerted against the wall because the fluid velocity goes down.

Blood is a shear-thinning fluid; the larger the shear rate, the less viscous the fluid. Also, blood pressure can change due to changes in blood volume in addition to vasoconstriction/vasodilation.

 

[Daiquiri]

Consider a flow through a pipe having a diameter D.
For a given mass flux, a bigger passage area means smaller velocity.
Let me call G the mass flux (dm/dt), and V the velocity.
So:

G = \rho V S --> V = G / (\rho S)

The pressure drop due to the friction against the pipe walls is given by:

\Deltap = 1/2 f (L/D) (V^2)/g

Now, let me simplify both equations (because I understand that you're more interested in qualitative explanation) by saying that:

V \propto (G / S)

\Deltap \propto (L/D) V^2

S \propto D^2

Combine these and you obtain that ("\propto" means "proportional to"):

\Deltap \propto L (G^2) / (D^5)

or, for a fixed G:

\Deltap \propto L/(D^5)

So, if you want to decrease the pressure drop and mantain the flow rate of the fluid, you have to either decrease the length of the pipe or to increase it's diameter. That's why our body behaves in a way you have described.

Somewhere in the middle of it there are coefficients and constants like gravity "g", friction factor "f" and some other stuff, but the qualitative explanation is just like that.