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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:

Typical Model Studies

Typical Model Studies

Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.

The Buckingham Pi Theorem

The Buckingham Pi Theorem

The Buckingham Pi theorem provides a structured method to simplify fluid dynamics problems by reducing complex systems of variables to dimensionless terms.

Navier–Stokes Equations

Navier–Stokes Equations

For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...

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 Between Parallel Plates

Steady, Laminar Flow Between Parallel Plates

Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.

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Related Experiment Video

Updated: Jul 15, 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 Simulation of Three-Dimensional Free Surface Flows Using the K-BKZ-PSM Integral Constitutive Equation.

Juliana Bertoco¹, Antonio Castelo², Luís L Ferrás^3,4

¹Center for Mathematics, Computing and Cognition - CMCC, Federal University of ABC - UFABC, Santo André 09210-580, Brazil.

|September 28, 2023

Summary

This summary is machine-generated.

A new numerical method accurately simulates complex viscoelastic fluid flows with free surfaces. This method effectively handles confined flows and extrudate swell, providing insights into fluid behavior.

Keywords:

Boger fluids K–BKZ PSM finite difference free surface

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Last Updated: Jul 15, 2025

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Area of Science:

* Rheology and Computational Fluid Dynamics (CFD).
* Polymer processing and fluid mechanics.

Background:

* Simulating viscoelastic fluid dynamics, especially free surface flows, presents significant computational challenges.
* Integral constitutive equations like the K-BKZ-PSM model are crucial for describing complex fluid viscoelasticity.

Purpose of the Study:

* To develop and validate a novel numerical method for three-dimensional unsteady free surface flows of viscoelastic fluids.
* To accurately model the free surface evolution and solve integral viscoelastic constitutive equations.

Main Methods:

* A second-order finite difference approach combined with the deformation fields method for integral constitutive equations.
* The marker-and-cell (MAC) method for precise free surface tracking.
* Derivation of a semi-analytical solution for fully developed pipe flows.

Main Results:

* The numerical method effectively simulates complex scenarios, including confined flows and extrudate swell of Boger fluids.
* Validation of the method's accuracy in capturing free surface dynamics.
* A new semi-analytical solution for specific K-BKZ-PSM fluid flow conditions.

Conclusions:

* The developed numerical technique offers a robust tool for simulating viscoelastic free surface flows.
* The findings advance the understanding and simulation capabilities for polymer processing and related fluid dynamics.
* The semi-analytical solution provides a benchmark for validating numerical models in specific flow regimes.