# Poiseuille’s equation

Poiseuille equation was discovered by Jean Louis Marie Poiseuille (1799-1869). As explained, each fluid can be considered as an ideal fluid. The ideal fluid does not have viscosity. If we assume an ideal fluid flows in a pipe, each part of the fluid moves at the same rate (v). Unlike the ideal fluid, the real fluid we encounter in everyday life has viscosity. Because it has a viscosity, then when flowing in a pipe, for example, the rate of each part of the fluid varies. The fluid layer that is in the middle moves faster (deep v), on the contrary, the fluid layer attached to the pipe does not move (v = 0). So from the middle to the edge of the pipe, every part of the fluid moves at different rates. To facilitate your understanding, observe the picture below.

R = the radius of the pipe/tube

v_{1 }= fluid flow rate in the middle / tube axis

v_{2 }= fluid flow rate that is r_{2} from the edge of the tube

v_{3} = fluid flow rate r_{3 }from the edge of the tube

v_{4 }= fluid flow rate r_{4} from the edge of the tube

r = distance

For the flow rate of each part of the fluid to be the same, there needs to be a pressure difference at either end of the pipe or any tube that the fluid passes through. What is meant by fluid here is real fluid, for example, water or oil that flows through a pipe, blood flowing in a blood vessel, etc? In addition to helping a real fluid to flow smoothly, the pressure difference can also make the fluids flow in pipes of different heights.

Jean Louis Marie Poiseuille, a former French scientist who was interested in physical aspects of human blood circulation, researched to investigate how factors, such as pressure differences, tube cross-area, and tube size affect the real fluid rate. The results obtained by Jean Louis Marie Poiseuille, known as the Poiseuille’s equation.

The Poiseuille’s equation can be derived using the help of a viscosity coefficient equation that has been calculated previously. We use the viscosity equation because the case is similar even though it is not the same. When deriving the viscosity coefficient equation, we review the flow of the real fluid layer between 2 parallel plates, and the fluid can move because of the attraction (F). The difference is that the Poiseuille’s equation that we will derive states the factors that influence the flow of real fluid in the pipe/tube and the fluid flowing due to the pressure difference. Therefore, the viscosity coefficient equation needs to be adjusted again.

Fluid can flow due to a difference in pressure (fluid flows from a place of high pressure to a place where the pressure is low), then we replace F with p_{1 }– p_{2 }(p_{1}> p_{2}).

When deriving the viscosity coefficient equation, we review the flow of the real fluid layer between 2 parallel plates. Each part of the fluid changes its regular speed as far as l. In this case, the fluid flow rate changes regularly from the tube axis to the edge of the tube. The fluid in the tube axis flows at a greater (v) speed. The more edge, the smaller the fluid speed. Tube radius = distance between the axis of the tube and the edge of the tube = R. The distance between each part of the fluid and the edge of the tube = r. Because the amount of each part of the fluid is very large and the distance from the edge of the tube is also different, then we just write like this:

v_{1} = the speed of fluid that is at the distance r_{1 }from the edge of the tube (r_{1 }= R)

v_{2} = the speed of fluid that is at a distance r_{2} from the edge of the tube (r_{2} <r_{1})

v_{3} = the speed of fluid at distance r_{3} from the edge of the tube (r_{3} <r_{2} <r_{1})

v_{4 }= the speed of fluid that is at distance r_{4 }from the edge of the tube (r_{4} <r_{3} <r_{2} <r_{1})

………………………………………..

v_{n }= the rate of fluid at the distance rn from the edge of the tube (rn <…… <r_{4 }<r_{3} <r_{2 }<r_{1})

The amount of each part of the fluid is huge, and we also do not know exactly how many of the amount is actually, so it is enough to be written with the symbol n. Each part of the fluid changes the speed (v) regularly, from the tube axis (r_{1} = R) to the edge of the tube (rn). From the tube axis (r_{1} = R) to the edge of the tube (rn), the rate of each part of the fluid is smaller (v_{1}> v_{2}> v_{3}> v_{4}> ….> v_{n}).

From the explanation above, we can have a figure that from R to rn, the fluid rate is getting smaller. Pipe length = L. Obtained equation:

Because what we are reviewing is the speed of the fluid flow, then equation 2 becomes:

This is the fluid flow speed equation in the distance r from the pipe with a radius of R. If confused while looking at the picture above… Please note that the fluid flows in a tube, so we need to review the flow rate of the fluid volume.

Inside the tube, there is fluid. For example, we divide the fluid into the tiny part, where each part has a unit area of dA, spaced from the tube axis and has a flow speed v. Mathematically can be written as follows:

dA_{1} = fluid part 1, which is a distance of dr_{1} from the tube axis

dA_{2} = fluid part 2, which is a distance of dr_{2} from the tube axis

dA_{3 }= fluid part 3, which is a distance of dr_{3} from the tube axis

…………………………….

dAn = fluid part n, which is a distance of dn from the tube axis

The part of fluid are very many, so it’s just written with the symbol n, so it’s more practical. The volume flow rate of each part of the fluid can be written mathematically as follows:

Each part of fluid is at a distance of r = 0 to r = R (R = tube radius). In other words, the distance of each part of the fluid varies when measured from the tube axis. We will obtain the fluid volume flow rate equation in the tube:

Q = Debit, R = radius in a pipe or tube, η = coefficient of viscosity, P_{1} – p_{2} = Pressure difference between the two ends of the pipe, L = pipe length, p_{1}-p_{2} / L = Pressure gradient (fluid flow is always in the direction of pressure drop)

Based on the Poiseuille equation above, it appears that the flow rate of the fluid volume (Q) is proportional to the tube radius (R^{4}), the pressure gradient (p_{2 }– p_{1} / L) and inversely proportional to the viscosity. If the tube radius is added (viscosity coefficient and fixed pressure gradient), then the fluid flow speed increases by a factor of 16.

The basic concept of pipe design, use this equation. Fluid debit is proportional to R^{4} (R = tube radius). The radius of the syringe or pipe fingers needs to be carefully calculated. For example, if we double the radius in the needle (r x 2), the liquid discharge that is sprayed = increases the thumb compression force 16 times.

The Poiseuille equation also shows that the radius (r^{4}) is inversely proportional to the difference in pressure between the two ends of the pipe. For example, blood first flows in a blood vessel that has an inner radius of r. If there is a narrowing of the arteries (e.g. r/2 = the radius in the blood vessels is reduced two times), then a pressure difference of 16 times is needed to make blood flow as before (so that the flow speed is constant).