# Determine the Flow Rate Using Poiseuille's Law for Laminar Flow

### Key Takeaways

• Laminar flow is the flow regime in which the infinitesimal parallel layers flow without describing the adjacent layers.

• In laminar flow, the velocity distribution is parabolic in shape at the cross-section of the flow.

• Poiseuille's law for laminar flow states that the flow rate is proportional to the pressure difference between the ends of the pipe and the fourth power of its radius.

Pheromones released by ants take a laminar flow regime and make the ants form a marching army

Have you ever wondered why ants always travel in lines? Ants release a chemical called pheromones. The successive ants smell the fragrance of these pheromones and follow the path taken by their predecessors. The pheromones released by ants take a laminar flow regime, causing the ants to form a marching army.

From the release of pheromones to large fluid systems, the laminar flow regime creates patterned flow characteristics. There are several applications where fluid movements have a laminar fluid flow. When dealing with streamlined fluid flow systems, a knowledge of Poiseuille’s law for laminar flow is vital, as it governs the flow rate.

## Fluid Flow Regimes

The flow regime of a fluid is significant for the design of any fluid system. There are various factors related to flow that need to be considered to maintain certain flow characteristics in a system. For example, fluid friction is a critical parameter to be considered to determine the amount of energy required to maintain a specified flow pattern.

Fluid flow regimes can be broadly classified into two types: turbulent flow and laminar flow.

1. Turbulent flow
Turbulent flow deals with the irregular motion of fluid particles. There is no definite frequency associated with turbulent flow. Turbulent flow lacks definite layers and observable patterns.
2. Laminar flow
Laminar flow is characterized by the opposite features of turbulent flow in terms of order and patterns. The fluid flow regime of interest in this article is laminar flow.

## Laminar Fluid Flow

Laminar flow is the flow regime in which the infinitesimal parallel layers flow without describing the adjacent layers. There is no mixing between the layers, which are flowing one over the other at different speeds. In laminar flow, the fluid particles flow in streamlines, which can be defined as flow in observable paths and with definite patterns.

Let’s look at some of the laminar fluid flow parameters.

### Viscosity

In laminar flow, viscosity is a significant part and the flow is also referred to as viscous flow. In the case of water, it is an inviscid fluid that does not have much influence on viscosity except at the boundary layers. However, there are other fluids where viscosity influences the laminar flow regime; for example, lubricating oil.

### Flow Velocity

In laminar flow, the velocity distribution is parabolic in shape at the cross-section of the flow. The maximum velocity of the laminar flow equals about two times the average velocity in the pipe flow.

### Average Flow

Average velocity measurements in fluid flow represent the velocity of the entire fluid contained in the pipe at that point. In laminar flow, as the velocity profile is parabolic, the average velocity is obtained by means of integral calculus.

### Reynolds Number

Reynolds number is the metric that gives the ratio of the viscous to the non-linear initial forces acting in the fluid. The Reynolds number is the parameter that is used to distinguish between laminar flow and turbulent flow. The fluid flow is laminar when the Reynolds number is less than 2300.

## Determining Flow Rate Using Poiseuille’s Law for Laminar Flow

When the fluid flow regime through a pipe is laminar, the volume rate or flow rate of the fluid depends on factors such as the pressure difference and resistance offered by viscosity. The resistance due to viscosity is influenced by the length of the pipe, the nature of viscosity, and the fourth power of the radius of the pipe cross-section.

Poiseuille's law for laminar flow states that the flow rate is proportional to the pressure difference between the ends of the pipe and the fourth power of its radius r. The law is commonly known as the Hagen-Poiseuille equation or law in fluid dynamics. The flow rate, Q, can be given by the following equation:

ΔPis the pressure difference, l is the length of the pipe μ, is the dynamic viscosity, and r is the radius of the pipe.

To study the laminar flow through pipes and other fluid systems, the Omnis CFD platform from Cadence can be utilized. The workflow in the Omnis platform helps in the design of complex fluid flow systems. Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you.