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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:

First Law: Particles in Two-dimensional Equilibrium

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration; but it is important to remember that these conditions are relative. For instance, an object may be at rest when viewed from one frame of reference, but that same object would appear to be in motion when viewed by someone moving at a constant velocity.
Newton's first law tells us about the...

Viscosity of Fluid

Viscosity of Fluid

Viscosity measures the resistance a fluid offers to flow and deformation. It results from internal friction between layers of fluid moving relative to one another. Dynamic viscosity, denoted by the Greek letter mu (μ), quantifies the force needed to move one fluid layer over another. For Newtonian fluids like water and air, the relationship between the shearing stress and the rate of shearing strain is linear, meaning their viscosity remains constant regardless of the applied stress.

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. However, the...

First Law: Particles in One-dimensional Equilibrium

First Law: Particles in One-dimensional Equilibrium

Newton's first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a net external force. It also states that there must be a cause for any change in velocity (a change in either magnitude or direction) to occur. This cause is a net external force. For example, consider what happens to an object sliding along a rough horizontal surface. The object quickly grinds to a halt, due to the net force of friction. If we...

Equilibrium Conditions for a Particle

Equilibrium Conditions for a Particle

When an object is in equilibrium, it is either at rest or moving with a constant velocity. There are two types of equilibrium: static and dynamic. Static equilibrium occurs when an object is at rest, while dynamic equilibrium occurs when an object is moving with a constant velocity. In both cases, there must be a balance of forces acting on the object.
To understand the concept of equilibrium, let us first consider the forces acting on an object. When different forces act on an object, they can...

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Related Experiment Video

Updated: Jun 8, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Minimal model for Brownian vortexes.

Bo Sun¹, David G Grier, Alexander Y Grosberg

¹Department of Physics, Center for Soft Matter Research, New York University, New York, 10003, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 28, 2010

Summary

This summary is machine-generated.

Minimal models of Brownian vortexes demonstrate how thermal fluctuations drive work extraction from static forces. These models reveal how temperature changes affect flux direction and thermodynamic efficiency.

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Area of Science:

Statistical mechanics
Non-equilibrium thermodynamics
Stochastic processes

Background:

Brownian vortexes are noise-driven machines utilizing thermal fluctuations to perform work.
Their operation involves probability currents with temperature-dependent topology and direction.
Understanding these systems is key to non-equilibrium statistical mechanics.

Purpose of the Study:

To present and analyze minimal discrete models of Brownian vortexes.
To elucidate the conditions governing flux reversal in these systems.
To investigate thermodynamic efficiency via entropy production.

Main Methods:

Development of discrete three- and four-state models for Brownian vortexes.
Exact solution using master-equation formalism.
Analysis of probability current topology and entropy production rate.

Main Results:

The models accurately capture the behavior of Brownian vortexes.
Conditions for flux reversal were identified and explained.
Insights into thermodynamic efficiency were gained through entropy production analysis.

Conclusions:

Minimal models provide exact solutions for Brownian vortex dynamics.
Temperature is a critical factor controlling flux direction.
These findings advance the understanding of work extraction in non-equilibrium systems.