# An Overview of the Laminar Flow Equation

### Key Takeaways

• Pressure difference and viscous force significantly influence the attributes of laminar flow.

• Reynolds number not only signifies the flow behavior but also shear stress within the flow system.

• The laminar flow equation facilitates numerical analysis of influencing flow parameters for complex system designs.

The analysis of flow characteristics is extremely important in engineering fluid systems. In CFD, this analysis is based on the governing flow equations and assumptions made depending on the flow motion and their boundary interaction within the flow field. Most engineering systems experience turbulent flow regimes, such as the water distribution system, and are generally thought to be hard to analyze. However, the many parameters—such as viscosity, velocity, and pressure drop in laminar flow—that occur in pipes or cylindrical ducts also need thorough analysis.

Students and professionals alike should thus have a familiarity with the governing laminar flow equations and their significance in driving the CFD flow modeling and analysis process. Here, we will discuss the flow attributes and analyze the equation in detail.

## Discussing Laminar Flow Attributes

Laminar flow can be recognized through its streamlined nature of flow without any eddies or swirls. This flow type is mostly associated with fluids having high viscosity flowing through a pipe or tube. However, the flow attributes can significantly alter due to the change in velocity or viscosity of the flowing fluid. Reynolds number helps system designers predict whether the flow in the system is laminar or not. Numerically, Reynolds number can be expressed as:

Where,

ρ: density of the fluid

V: fluid velocity

D: hydraulic diameter (of pipe, tube, or duct)

μ:  fluid viscosity

For Reynolds numbers up to 2300, the flow is considered to be laminar.

However, there are various factors to take into account in the analysis of laminar flow. These include pressure difference, viscous force, and elastic force. Poiseuille’s equation looks at the pressure drop in a steady, incompressible laminar fluid system flowing through a circular pipe of the constant cross-section. Since the fluid is incompressible, the elastic force can be neglected. For a pipe of radius R, pressure gradient dp/dx, and x as the axial direction, the equation can be expressed as:

Using the Navier-Stokes equation, the axial momentum equation is:

∂p/∂r=0, since the pressure p(x) is only the function of axial coordinate x.

The axial pressure uₓ(r) can be integrated with the above equation to be written as:

Note that µ is the dynamic viscosity and C₁ and C₂ pertain to the boundary conditions. At axis, r=0, C₁=0. Similarly, at the no-slip boundary condition, the velocity at the wall is assumed to be zero. Thus, at r=R, uₓ(R)=0. Therefore,

Thus, for a circular pipe or duct exhibiting laminar flow, the parabolic velocity profile is:

The maximum velocity is observed at r=0, i.e., the center of the axis:

The volumetric flow rate in the pipe is:

Thus, the average velocity of the flow is:

With rearrangement, we can get Poiseuille’s equation for pressure drop calculation as:

Using this equation, pressure drop ‘∆p’ in the laminar regime across ‘l’ length of the pipe can be easily calculated. To summarize the laminar flow equation:

• ➔ The flow rate is directly proportional to the radius of the pipe. Thus, the small increase in pipe diameter can significantly increase the flow rate in the system.
• ➔ The flow rate is inversely proportional to the length of the pipe as well as the coefficient of viscosity of the fluid.

## Shear Stress and Friction Factor

The fluid in motion gives rise to shear stress due to the interaction between fluid particles. From Poiseuille’s equation derivation, we have derived that in laminar flow, velocity is maximum towards the center, indicating shear stress. Thus, the shear stress, in the case of both laminar and turbulent flow, is based on the shear stress in the wall. This can be calculated by determining the friction factor, which according to the Darcy-Weisbach equation is:

In laminar flow, the friction factor is not affected by the pipe roughness, given any resistance is minimized with the viscosity of the fluid.

## Solving Laminar Flow Equations With CFD Solvers

In computational fluid dynamics (CFD) analysis, the governing flow equations can be used to numerically analyze the pressure drop, flow velocities, and shear stress that influence the simulation of a fluid system. However, computation can be fairly difficult in a system where flow is governed by complex geometries and boundaries in comparison to a simple cylindrical pipe system.

CFD solvers allow you to select the correct flow regime and characterize it easily even. Through CFD modeling, the laminar flow equations can be accurately analyzed. CFD simulation platforms, such as Omnis from Cadence, can facilitate the running of simulations in complex systems to achieve the speed and fidelity levels required.