# An Overview of Poiseuille's Law for Resistance

### Key Takeaways

• Poiseuille’s law is the basic theory describing the fluid flow through a pipe or tube.

• Poiseuille’s law helps in understanding blood pressure, determining the flow rate of intravenous fluids, and predicting vascular resistance.

• The application of Poiseuille's law for resistance is more suitable for uniform liquids or Newtonian liquids with comparatively less turbulence.

Poiseuille’s law helps in determining the viscosity of fluids used in DNA testing and chemical analysis, among many other uses

Pressure difference and fluid viscosity influence the volume flow rate of a smooth laminar flow. In pipe flows, this relationship is governed by Poiseuille’s law. When the dimensions of a pipe are changed, a difference in flow rate is observed. Resistance is experienced by the fluid flowing through the pipe with a change in cross-section. The resistance to flow depends on the viscosity, length, and radius, and can be calculated using Poiseuilles’s law for resistance.

## Fluid Flow Rate and Poiseuille’s Law

Poiseuille’s law is the basic theory describing the fluid flow through a pipe or tube. There are various factors influencing the fluid flow through a pipe: the viscosity of the fluid, pressure difference across the pipe, length of the pipe, and cross-section or diameter of the pipe.

Focusing on the fluid property viscosity, the fluid flow rate decreases with an increase in the viscosity of the fluid. As the fluid viscosity increases, the fluid becomes thicker and sticks to the surface of the pipe or tube. The volume of fluid flowing per second is less for high viscous fluids. In other words, the volume flow rate of a fluid decreases with an increase in viscosity.

The pressure difference is at equilibrium when the fluid is stagnant. For a fluid to flow, there should be a pressure difference between upstream and downstream. The fluid always flows from the high-pressure end to the low-pressure end. The pressure difference induces fluid motion forward, which is opposed by the viscous drag force due to fluid viscosity. When the pressure difference becomes greater than the viscous drag force, the fluid starts to flow.

Poiseuille’s law gives the mathematical expression for the volume fluid flow rate relating the viscosity, pressure gradient, and dimensions of a pipe. According to Poiseuille’s law, the fluid flow rate can be given by the equation:

P is the pressure difference between higher pressure P2 and lower pressure P1, r is the radius of the pipe, l is the length of the pipe, and  is the fluid viscosity.

## Poiseuille’s Law for Resistance

Fluid flow characteristics are different for the same fluids flowing through different pipes and different fluids flowing through the same pipe. In a given pipe, the flow characteristics of fluids of different viscosities are different. When allowing the same fluid to flow through pipes of different dimensions, the flow behavior varies. For fluid flow, there is a dependency on fluid properties as well as pipe dimensions.

Similarly, the resistance experienced by the fluid flow is also different. The resistance to fluid flow can be defined as the ratio of pressure difference to the flow rate. From Poiseuille’s law expression, the resistance to the flow can be derived as:

The resistance to the flow shares a linear relationship with the viscosity and the length of the pipe. As the viscosity of the fluid increases, more fluid particles stick to the pipe surface and block the smooth flow, establishing a higher resistance to flow. In a fluid of low viscosity, there is a reduction in fluid friction compared to a fluid of high viscosity. The decrease in fluid friction results in reduced resistance to flow. The resistance to flow is zero for a fluid with zero viscosity.

The fluid flow resistance is inversely proportional to the fourth power of the radius of the pipe. Consider water flowing through two pipes of the same length with a radius of 1 cm (pipe 1) and 2 cm (pipe 2), respectively, provided all other factors are identical in both pipes. If the resistance offered to the flow by pipe 1 is R, then the resistance to the flow of pipe 2 is R/16. The resistance to flow by pipe 2 is 16 times lower than that given by pipe 1. As the radius doubles, the resistance to flow decreases by 16 times.

## Applications of Poiseuille’s Law

Poiseuille’s law is often applied in the medical field: understanding blood pressure, determining the flow rate of intravenous fluids, predicting vascular resistance, etc. Poiseuille’s law helps determine the viscosity of fluids used in DNA testing, chemical analysis, and engine fuel testing. Some other applications of Poiseuille’s law include:

• The design of space vehicles, configuration of spacecraft, and types of equipment used by astronauts.
• The design of gas turbines and internal combustion engines as well as the analysis of the external flow rate of the motors used in vehicles or large equipment.

The application of Poiseuille's law for resistance is more suitable for uniform liquids or Newtonian liquids with comparatively less turbulence. With CFD simulations, it is easy to determine the flow rate, resistance to flow, dependency on pipe dimensions, and viscosity.