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Internal flow -- the HP Equation

Page history last edited by Joe Redish 8 years, 11 months ago

Working content >MacroModels > Fluids

 

Prerequisites

 

As liquid flows through a pipe in a steady state, there is a tricky balance of forces. For simplicity, let's consider a liquid flowing through a cylindrical pipe at a steady rate. Because of adhesion between the liquid and the walls, the fluid sticks to the walls and doesn't move at the wall. But since it's moving in the middle, there is a variation in the velocity as a function of radius. This means that cylindrical shells of fluid are flowing over each other. This flow will be resisted by the liquid's internal viscosity. As a result, there has to be another force acting in the direction of flow to keep the fluid flowing.

 

Since there is a varying velocity, figuring out the balance of forces and the flow is quite tricky and involves some serious calculus. If you want to see the full detail, visit the page Internal flow -- the HP equation (advanced). For now, we'll work with a less accurate but simpler model that still gives the correct result.

 

Instead of worrying about the variation of flow with radius, let's assume a model in which the flow in the tube is uniform and doesn't vary in the tube. We know there is a velocity dependent drag, so let's include in our model a friction force between the wall of the tube and the fluid that's proportional to the velocity. We know that's not the actual source of resistance here, but since we know that a balance of forces is necessary and that the true resistive force is proportional to velocity, this should work. We'll just have to adjust the constants in the formula to match the more complex analysis, and this model will help us think about what's happening.

 

So consider a pipe with a liquid moving down it. Isolate a small cylinder of liquid (shown in blue in the figure below). We assume that it's moving at a constant speed throughout and that there is a resistive force with the wall proportional to the velocity. Since the cylinder of liquid is moving at a constant speed, the resistive force must be balanced by some force pushing it down the pipe. The only thing touching the cylinder is the rest of the liquid on either side. Each side exerts a pressure force on the disk of liquid. Since there has to be a force balancing the resistive force, it must arise from a pressure difference. We conclude that if the fluid keeps moving at a constant speed, the pressure must drop as we go downstream. Here's the picture.

The resistive force is shown in the blue cylinder of fluid with arrows pointing to the left. The fluid upstream from the cylinder exerts a pressure force on it to the right and the downstream fluid exerts a pressure force on the cylinder pushing to the left.

 

We can now derive a relation between the amount of fluid flowing (Q) and the pressure drop if we make the following simple assumptions:

  • The liquid is moving at a constant velocity, v.
  • The liquid has a constant uniform density, so we can talk about the volumetric flow, Q = Av.
  • The resistive force on the cylinder is proportional to its velocity and its length, L. We'll choose the proportionality constant so as to match the more sophisticated analysis:

 

Fresistive = 8πμLv

          where μ is the viscosity of the liquid. (The 8π comes from doing integrals.)

With these modeling assumptions, here's what the free-body diagram for the cylinder of fluid looks like.

 

 

where the AL and AR mean the areas on the left and the right. We take their directions from seeing which way the external fluid pushes on the stuff inside. They both have the same area, A. The upstream pressure must be greater than the downstream to keep the cylinder going. We'll define Pupstream - Pdownstream = ΔP.

 

Since these forces have to balance to maintain a constant velocity we get

But since we know that the volumetric flow can be given by Q = Av, we can replace v by Q/A and get a relationship between the amount of flow and the pressure drop across the cylinder.

 

In the last line, we have put in the fact that we have a circular pipe so the area A = πR2. The factors in parentheses are all constants that have to do with the size of the pipe so we can write is as the resistance of the pipe, Z. In this form the equation relates the volumetric flow to the pressure drop:

This is known as the Hagen-Poiseuille equation and it tells us that flow rate increases with pressure difference, and decreases with pipe length or fluid viscosity.  These make sense.  The more pressure that is applied to the fluid at the pipe opening, the faster the fluid should flow.  The longer the pipe or the more sticky the fluid, the harder it is to get the fluid to flow.  One surprising result of the Hagen-Poiseuille equation is that flow rate increases with the fourth power of the tube radius.  Therefore, pipe diameter has a huge effect on how easily fluid flows down a pipe. This has important biological implications 

 

There is a trade off in designing circulatory systems.  There are two primary costs.  The first is making the blood vessels and the blood.  The smaller the total volume of the system, the less materials required and the lower the costs.  However, the other cost in the system is making the pump to drive the flow through the blood vessels.  The pump must be large enough to drive a given volumetric flow through the blood vessels.  If you look at the Hagen-Poiseuille equation, as the blood vessels get smaller, the pump must work harder (create a bigger pressure) to get the same volume of fluid to flow.  Apparently, this is a significant effort as we use about 10% of our resting metabolic rate to power the circulatory system (Steven Vogel, Comparative Biomechanics: Life's Physical World (Princeton U. Press, 2003) pg. 209).

 

Joe Redish and Karen Carleton 10/26/11

 

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