Name: fluid flow and area changes
Uploaded: 2026-01-30T03:08:26.000Z
Duration: 1 min 28 s
Channel: JoVE

The continuity equation derives from the principle of mass conservation, stating that the mass of fluid entering and leaving a system remains constant.
For incompressible fluids, the product of flow area and velocity remains consistent throughout the flow path, adjusting to changes in cross-sections.
As the cross-sectional area decreases, the fluid velocity increases to maintain a constant flow rate.
This relationship is crucial for analyzing systems such as water pipelines, stormwater management, and open channels, ensuring stability and efficiency in various fluid flow applications.
In water supply systems, the continuity equation helps maintain steady distribution by predicting velocity changes when pipes narrow, which is essential for avoiding pressure drops.
It helps design detention basins for stormwater management. When runoff enters the basin during a storm, the outflow rate must match the inflow rate, adjusted to prevent flooding downstream.
For open channels like rivers or canals, the continuity equation helps understand how variations in width or depth affect water levels and flow speeds. This understanding is vital for designing effective flood control and navigation systems.

continuïteitsvergelijking

De continuïteitsvergelijking stelt dat de massastroom constant moet blijven voor een constante stroom van een onsamendrukbare vloeistof in een beperkt systeem. Dit principe is van toepassing op systemen waarin vloeistof door verschillende dwarsdoorsneden stroomt, zoals sproeiers, spuiten en leidingen. De massastroom wordt uitgedrukt als: <img alt="Equation 1" src="/files/ftp_upload/18038/18038_Equation_1.svg" style="text-align: center; height: 12px;"> Voor onsamendrukbare stroming, waarbij de dichtheid constant blijft, wordt de continuïteitsvergelijking vereenvoudigd tot: <img alt="Equation 2" src="/files/ftp_upload/18038/18038_Equation_2.svg" style="text-align: center; height: 12px;"> Deze vergelijking illustreert dat de snelheid toeneemt wanneer de dwarsdoorsnede afneemt en vice versa, waardoor massabehoud wordt gegarandeerd. Een voorbeeld is de stroming door een spuit. Wanneer de zuiger de vloeistof door een sproeier comprimeert, laat de continuïteitsvergelijking zien dat als het uitlaatoppervlak de helft is van het inlaatoppervlak, de uitlaatsnelheid twee keer de inlaatsnelheid moet zijn. Een ander voorbeeld is de stroming in een door zwaartekracht aangedreven tank. Door zowel de vergelijking van Bernoulli als de continuïteitsvergelijking toe te passen op punten langs de stroming, kunnen we de uitlaatsnelheid en het volumetrische debiet bepalen: <img alt="Equation 3" src="/files/ftp_upload/18038/18038_Equation_3a.svg" style="text-align: center; height: 75px;"> <img alt="Equation 3" src="/files/ftp_upload/18038/18038_Equation_3b.svg" style="text-align: center; height: 20px;"> waarbij h de hoogte van de vloeistof is, D de diameter van de tank en d de uitlaatdiameter. Deze relatie laat zien dat het debiet convergeert naar een stabiele waarde naarmate de uitlaatdiameter kleiner wordt ten opzichte van de tankdiameter. Deze afleidingen illustreren het nut van de continuïteitsvergelijking bij het voorspellen van vloeistofgedrag onder verschillende beperkingen, waardoor massabehoud in dynamische systemen wordt gewaarborgd.

Continuïteitsvergelijking

The continuity equation derives from the principle of mass conservation, stating that the mass of fluid entering and leaving a system remains constant.
...

Education

JoVE Core

Civil Engineering

Civiele techniek

Unit 1

Fluid Dynamics

Stromingsleer

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

Continuity Equation

fluid flow and area changes

Fluid flow and area changes are linked by the continuity equation. For a steady flow of an incompressible fluid in a confined system, the mass flow rate stays constant. This idea is useful in nozzles, syringes, and pipes where the cross-sectional area changes.

The mass flow rate is written as an equation that connects density, area, and velocity. For incompressible flow, the density stays the same, so the equation becomes simpler. When the area gets smaller, the fluid speed must increase. When the area gets larger, the speed must decrease. This keeps mass conserved as the fluid moves.

A syringe shows this relationship clearly. When the plunger pushes fluid through a nozzle, the outlet speed rises if the outlet area is smaller than the inlet area. If the outlet area is half the inlet area, the outlet velocity is twice the inlet velocity.

A gravity-driven tank gives another example. Here, Bernoulli's equation and the continuity equation can be used together at points along the flow. They help find the outlet velocity and the volumetric flow rate. In this setup, h is the fluid height, D is the tank diameter, and d is the outlet diameter. The flow rate approaches a stable value as the outlet diameter becomes small compared with the tank diameter.

Fluid Flow and Area Changes

bernoulli's equation for flow along a streamline

bernoulli-s-equation-for-flow-along-a-streamline

Pressure and Speed in Bernoulli's Equation

Bernoulli's equation links pressure, speed, and height in a moving fluid along a streamline. It is used for incompressible, inviscid fluids under steady flow. The idea comes from applying Newton's second law to a small fluid element. That element feels forces from pressure differences, gravity, and changes in velocity. The resulting equation shows how fluid energy is shared. In the equation, P is pressure, ρ is fluid density, v is velocity, g is the acceleration due to gravity, and h is...

bernoulli's equation for flow normal to a streamline

bernoulli-s-equation-for-flow-normal-to-a-streamline

Pressure Changes Across Curved Streamlines

Bernoulli's equation for flow normal to a streamline describes how pressure changes across curved streamlines. When a fluid follows a curved path, the outer side of the curve experiences centrifugal force, which pushes the fluid outward. The pressure is therefore higher on the inner side, near the center of curvature, and lower farther out. The size of this pressure difference depends on the fluid's velocity and the radius of curvature. In flows with nearly straight streamlines, the pressure...

bernoulli's equation: problem solving

Venturi Meter Flow Rate from Pressure

A Venturi meter measures fluid flow rate in a pipeline by using pressure changes and Bernoulli's equation. In a sewage system, it can be used to determine wastewater flow rate by reading the pressure difference across the meter. The narrowed throat makes the fluid speed up, and that change in speed is linked to pressure drop. The first step is to find the cross-sectional areas of the pipe and the Venturi throat. These areas are needed to interpret the pressure difference shown on the pressure...

static, stagnation, dynamic and total pressure

static-stagnation-dynamic-and-total-pressure

Pitot-Static Measurement in Bernoulli Flow

Static pressure, dynamic pressure, stagnation pressure, and total pressure are key ideas in fluid dynamics. They are often explained with Bernoulli's equation. In that equation, p is static pressure, 1/2ρV^2 is dynamic pressure, γz is hydrostatic pressure or potential energy pressure, and pT is total pressure. For incompressible flow, the total pressure stays constant along a streamline. Static pressure is the pressure a fluid exerts when it is at rest, or when flowing fluid is brought to rest...

free jet

Liquid Speed from Reservoir Outlets

Free jets show how liquid moves after leaving a reservoir through an opening and flowing into the air without resistance. The jet speed is found with Bernoulli’s principle, and it depends on the height of liquid above the opening. This relationship assumes atmospheric pressure at both the reservoir surface and the jet exit. The liquid column’s height controls the speed at the nozzle because the fluid’s potential energy changes into kinetic energy during flow. As the liquid leaves the...

continuity equation

Fluid flow and area changes are linked by the continuity equation. For a steady flow of an incompressible fluid in a confined system, the mass flow rate stays constant. This idea is useful in nozzles, syringes, and pipes where the cross-sectional area changes. The mass flow rate is written as an equation that connects density, area, and velocity. For incompressible flow, the density stays the same, so the equation becomes simpler. When the area gets smaller, the fluid speed must increase. When...

energy line and hydraulic gradient line

Fluid Head Profiles in Pipes and Channels

Fluid head profiles in pipes and channels show how pressure, speed, and height change in a flowing fluid. These profiles come from Bernoulli's equation for steady, incompressible, inviscid flow. In that equation, total head includes pressure head, velocity head, and elevation head. The energy line (EL) shows the total head available in the flow. It sits above the datum and includes all three energy parts. Under ideal conditions, the EL stays horizontal because the total energy does not change...

design example: designing water slide

Water Slide Speed and Bernoulli's Principle

Water slide speed is controlled by Bernoulli's principle and the height of the slide. Engineers use these ideas to keep the ride exciting while making sure the water stays safe for riders. As water moves downhill, gravity makes it speed up. The water at the bottom depends on the height it starts from. A taller slide gives the water more potential energy at the top. That energy changes into kinetic energy as the water descends, so the flow becomes faster. Bernoulli's principle helps describe...

key terms and concepts

Learn how the continuity equation links area, velocity, and mass conservation in steady fluid flow, with examples from syringes and gravity-driven tanks.

Fluid Flow and Area Changes | JoVE K12

JoVE Core Civil Engineering provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

civil engineering

Fluid Flow and Area Changes

Description

Transcript

Tags