Name: pressure balance in a fluid element
Uploaded: 2025-02-13T15:43:11.000Z
Duration: 1 min 13 s
Channel: JoVE

エッジの長さが対象の流体質量内の任意の位置から取得されると仮定された小さな長方形の流体要素について考えます。
この要素には、圧力によって生じる表面力と、要素の重量に等しい物体力の 2 種類の力が作用します。
圧力が要素の中心でpとして表される場合、異なる面の平均圧力はpとその導関数で表すことができます。
流体要素に対する力の影響を理解するために、y方向の合力サーフェス力を表わすことができ、同様に、xサーフェスとzサーフェスについても表すことができます。
要素に作用する合力の表面力をベクトル形式で表し、さらに単位体積あたりの合力の表面力として表すことができます。
要素の重量は、流体の比重を使用して表すことができます。
流体要素に関するニュートンの第2法則は、単位体積あたりの結果の表面力と要素の重量を使用して表すことができます。
これにより、圧力場の方程式を求めることができます。

圧力場の基本方程式

流体力学における圧力場の基本方程式は、流体の任意のセグメント内の力のバランスをとらえ、さまざまな力の下で流体内の圧力がどのように変化するかについての基礎的な理解を提供します。一般に、流体の任意の部分には、表面力と体積力という 2 つの主な種類の力が作用します。表面力は、流体内における点間の圧力差から生じ、その結果、局所的な圧力勾配に応じて変化する正味の力が生じます。一方、体積力は流体全体に分散され、主に流体の質量に作用する重力から生じます。

 圧力場方程式は、これらの力をニュートンの第 2 法則と組み合わせて分析することで導き出されます。ニュートンの第 2 法則は、力を流体質量の加速に結び付けます。この方程式は、力がどのように相互作用して流体内の圧力変化を生み出すかを簡略化して示しています。流体全体の圧力が増加または減少すると、多くの場合、この力のバランスによって動きや加速が起こります。

 圧力場方程式は、川や配管の流れから大気圧の変化や気象パターンまで、流体の挙動を理解するために不可欠です。流体内で圧力がどのように分散されるかを理解することで、流体の動きを予測し、油圧機械から給水網、さらには空気力学に至るまでのさまざまなアプリケーションで流れを管理するシステムを設計できます。

小さな長方形の流体要素を考えてみて、エッジの長さは流体質量内の任意の位置から取得されると仮定します...

Education

JoVE Core

Civil Engineering

土木工学

Unit 1

ユニット1

Fluid Statics

流体静力学

KEY TERMS AND CONCEPTS

主要用語と概念

Basic Equation for Pressure Field

pressure balance in a fluid element

The pressure field equation describes how forces stay balanced inside a small fluid element. In fluid mechanics, this balance helps explain how pressure changes from one place to another within a fluid.

Two main force types act on a fluid segment: surface forces and body forces. Surface forces come from pressure differences across nearby points in the fluid, so they depend on the local pressure gradient, which is the rate of pressure change with position. Body forces are spread through the fluid, and gravity is the main example because it acts on the fluid’s mass.

The pressure field equation is derived by combining these forces with Newton’s second law, which connects force and acceleration. The result gives a simplified way to see how pressure variations form in a fluid. When pressure increases or decreases across the fluid, the force balance can lead to motion or acceleration.

This idea is useful for understanding fluid behavior in rivers and pipes, as well as atmospheric pressure changes and weather patterns. It also helps predict fluid movement and support the design of systems such as hydraulic machinery, water distribution networks, and aerodynamics applications.

Pressure Balance in a Fluid Element

concept of pressure at a point

Fluid Pressure at One Point

Fluid pressure at one point is the focus of this lesson. It shows that pressure in a fluid is the same in every direction at a single location. This happens because fluid molecules move randomly and collide with nearby molecules in a balanced way. At a given point, the pressure depends on the surrounding fluid molecules. It is affected by depth and density, not by the shape of the container or the way the fluid is oriented. In a fluid at rest, the pressure acts equally in all directions...

basic equation for pressure field

The pressure field equation describes how forces stay balanced inside a small fluid element. In fluid mechanics, this balance helps explain how pressure changes from one place to another within a fluid. Two main force types act on a fluid segment: surface forces and body forces. Surface forces come from pressure differences across nearby points in the fluid, so they depend on the local pressure gradient, which is the rate of pressure change with position. Body forces are spread through the...

pressure variation in a fluid at rest

Hydrostatic Pressure and Fluid Depth

Hydrostatic pressure describes how pressure changes with depth in a fluid at rest. In a stationary fluid, the fluid acceleration is zero, so the forces inside the fluid balance. That means pressure at a point below the surface depends on depth, not on the container’s shape or size. Only vertical forces affect the pressure change in this situation. These forces come from the weight of the fluid above a point. When pressure is compared at two different levels, the difference in pressure equals...

measurement of fluid pressure

Manometers and Gauges for Fluid Pressure

Fluid pressure is measured with manometers and related gauges. These tools use a liquid column to show pressure differences. The height of the column gives a simple reading of the force the fluid exerts. A basic manometer is the piezometer. It is a vertical tube that is open at the top and filled with the same fluid being measured. When the tube connects to a container, the fluid height in the tube matches the pressure inside that container. For more detailed measurements, the U-tube...

hydrostatic pressure force on a plane surface

hydrostatic-pressure-force-on-a-plane-surface

Hydrostatic Pressure on Submerged Surfaces

Hydrostatic pressure creates force on submerged plane surfaces. The force depends on the fluid pressure and the area of the surface. The way that force acts changes with the surface position. For a horizontal surface, the depth stays the same across the whole area. That means the pressure is uniform. The resultant force equals the pressure at that depth multiplied by the surface area, and it acts through the centroid of the surface. For a vertical surface, pressure increases as depth...

hydrostatic pressure force on a curved surface

hydrostatic-pressure-force-on-a-curved-surface

Finding the Force on Curved Fluid Surfaces

Hydrostatic pressure on curved surfaces is an important idea in fluid mechanics. It helps explain how water pushes on curved structures in civil engineering. Examples include reservoirs, dams, and storage tanks. The pressure changes in both size and direction along the curve. To find the total hydrostatic force on a curved surface, engineers isolate the fluid volume next to that surface. They then study the forces on imaginary horizontal and vertical planes that cut through the volume. The...

buoyancy and stability for submerged and floating bodies

buoyancy-and-stability-for-submerged-and-floating-bodies

How Buoyancy Keeps Bodies Stable in Water

Buoyancy and stability explain how submerged and floating bodies behave in a fluid. When a body is fully or partly under water and stays still, the fluid pushes upward on it. That upward push is called the buoyant force. The buoyant force acts through the center of buoyancy. This is the center of the displaced fluid volume. Archimedes’ principle says the buoyant force equals the weight of the fluid displaced by the body. Stability describes whether a body can return to its original position...

design example: application of archimedes' principle

design-example-application-of-archimedes-principle

Floating Block Buoyancy and Center of Buoyancy

Archimedes' principle is used to analyze the buoyant force on a wooden block that floats in seawater. The block has a rectangular section, and its dimensions, specific gravity, and the specific weight of seawater are used in the calculation. This setup shows how floating bodies stay in equilibrium. The first step is to find the weight of the block from its volume and specific gravity. Then Archimedes' principle is applied. The weight of the displaced water must equal the weight of the block.

key terms and concepts

Learn how surface forces, body forces, and Newton’s second law combine to explain pressure changes in fluids, from pipes to weather patterns.

Pressure Balance in a Fluid Element | 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

Pressure Balance in a Fluid Element

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