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Bernoulli's Equation for Flow Along a Streamline

Bernoulli's Equation for Flow Along a Streamline

Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:

Free Jet

Free Jet

Free jets describe the flow of liquid exiting a reservoir through an opening into the atmosphere without resistance. The velocity (v) of the liquid jet is derived using Bernoulli's principle and expressed as:

Bernoulli's Equation for Flow Normal to a Streamline

Bernoulli's Equation for Flow Normal to a Streamline

Bernoulli's equation for flow normal to a streamline explains how pressure varies across curved streamlines due to the outward centrifugal forces induced by the fluid's curvature. The pressure is higher on the inner side of the curve, near the center of curvature, and decreases outward to balance these centrifugal forces.
The pressure difference depends on the fluid's velocity and radius of curvature. The pressure variation is minimal in flows with nearly straight streamlines.

Steady, Laminar Flow in Circular Tubes

Steady, Laminar Flow in Circular Tubes

Hagen-Poiseuille flow describes a viscous fluid's steady, incompressible flow through a cylindrical tube with a constant radius R. This flow profile is often applied to understand fluid transport in narrow channels, such as capillaries. It serves as a foundational example of laminar flow. In this model, cylindrical coordinates (r,θ,z) are used to describe the radial (r), angular (θ), and axial (z) dimensions within the tube. For Hagen-Poiseuille flow, the velocity profile is...

Bernoulli's Equation

Bernoulli's Equation

In the middle of the nineteenth century, it was observed that two trains passing each other at a high relative speed get pulled towards each other. The same occurs when two cars pass each other at a high relative speed. The reason is that the fluid pressure drops in the region where the fluid speeds up. As the air between the trains or the cars increases in speed, its pressure reduces. The pressure on the outer parts of the vehicles is still the atmospheric pressure, while the resultant...

Dimensional Analysis

Dimensional Analysis

Dimensional analysis is a valuable technique in fluid mechanics for simplifying complex problems by reducing them into dimensionless groups. These groups capture the essential relationships between the variables involved, allowing researchers and engineers to analyze fluid flow without dealing with each variable individually. This approach reduces the number of independent variables, allowing for easier analysis and better understanding of physical phenomena.
In fluid mechanics, dimensional...

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Related Experiment Video

Updated: Aug 22, 2025

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Published on: February 27, 2016

Numerical Analysis of Fiber/Air-Coupling Field for Annular Jet.

Yudong Wang^1,2,3, Hongzhi Wei¹, Yumei Chen¹

¹College of Biological and Chemical Engineering, Guangxi University of Science & Technology, Liuzhou 545006, China.

|November 11, 2022

Summary

This study numerically investigates melt-blowing fiber and airflow interactions. Results reveal airflow velocity fluctuations near the spinning line can cause fiber whiplash, impacting micro-nanofiber material production.

Keywords:

annular die coupled field melt-blowing numeral calculations

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Last Updated: Aug 22, 2025

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Area of Science:

Materials Science
Fluid Dynamics
Chemical Engineering

Background:

Melt-blowing is crucial for micro-nanofiber production, involving polymer melts and high-velocity airflow.
The complex interaction between melt-blowing fibers and airflow is often simplified or ignored in numerical models.
Understanding this coupling is essential for optimizing fiber formation and material properties.

Purpose of the Study:

To numerically investigate the coupled flow field of flexible fibers and airflow in an annular melt-blowing die.
To analyze the influence of fiber presence on airflow dynamics within the die.
To identify key flow characteristics that affect fiber behavior during the melt-blowing process.

Main Methods:

Utilized the Volume of Fluid (VOF) method for numerical simulation.
Employed computational fluid dynamics (CFD) software to model the flexible fiber/air-coupling flow.
Focused on an annular melt-blowing die geometry.

Main Results:

Pressure distribution was consistent across central symmetry planes of the ring die.
Velocity distribution exhibited variations, with dynamic changes along the spinning line.
Minimal pressure fluctuations were observed on the spinning line.
Significant velocity fluctuations around the spinning line were identified as a cause of fiber whiplash.

Conclusions:

The presence of melt-blowing fibers significantly influences the airflow field.
Velocity fluctuations near the spinning line are a critical factor leading to fiber whiplash.
This research provides insights into optimizing melt-blowing processes for improved micro-nanofiber production.