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

连续性方程源于质量守恒原理，该原理指出进入和离开系统的流体质量保持不变。
对于不可压缩流体，流道各处的流动截面积与流速的乘积保持恒定，以适应横截面的变化。
当横截面积减小时，流体速度会增加，以保持恒定的流速。
这种关系对于分析水管道、雨水管理以及明渠等系统至关重要，可确保各类流体流动应用中的稳定性与效率。
在供水系统中，连续性方程通过预测管道变窄时流速的变化来帮助维持稳定的输配水，这对于避免压力下降至关重要。
它有助于为雨水管理设计调蓄池。当径流在暴雨期间进入调蓄池时，出流速率必须与入流速率相匹配，并进行调整，以防止下游发生洪水。
对于河流或运河等开放性水道，连续性方程有助于理解宽度或深度的变化如何影响水位和流速。这种理解对于设计高效的防洪和航运系统至关重要。

连续性方程

连续性方程表示，对于密闭系统内不可压缩流体的稳定流动，质量流速必须保持恒定。此原理适用于流体通过不同横截面积的系统，例如喷嘴、注射器和管道。 质量流速表示为： <img alt="Equation 1" src="/files/ftp_upload/18038/18038_Equation_1.svg" style="text-align: center; height: 12px;"> 对于密度保持不变的不可压缩流，连续性方程简化为： <img alt="Equation 2" src="/files/ftp_upload/18038/18038_Equation_2.svg" style="text-align: center; height: 12px;"> 该方程说明，当横截面积减小时，速度会增加，反之亦然，从而确保质量守恒。一个例子是通过注射器的流动。当柱塞压缩流体通过喷嘴时，连续性方程表明，如果出口面积是入口面积的一半，则出口速度必须是入口速度的两倍。 另一个例子是重力驱动罐中的流动。将伯努利方程和连续性方程应用于流动方向上的点有助于确定出口速度和体积流量： <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;"> 其中 h 是流体的高度，D 是罐的直径，d 是出口直径。该关系表明，当出口直径相对于罐直径变小时，流量会收敛到一个稳定值。这些推导证明了确保动态系统中质量守恒的连续性方程在预测各种约束条件下流体行为的实用性。

连续性方程源于质量守恒原理，表明流入和流出系统的流体质量保持不变。

Education

教育

JoVE Core

Civil Engineering

土木工程

Unit 1

第一单元

Fluid Dynamics

流体力学

KEY TERMS AND CONCEPTS

关键术语与概念

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

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