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

Рассмотрим небольшой прямоугольный элемент жидкости с длинами ребер, которые предполагается взять из любого произвольного места в пределах исследуемой массы жидкости.
На этот элемент действуют два типа сил: поверхностные силы, вызванные давлением, и сила тела, равная весу элемента.
Если давление обозначено в центре элемента как p, то среднее давление на различных гранях может быть описано в терминах p и его производных.
Чтобы понять влияние сил на элемент жидкости, можно выразить результирующую поверхностную силу в направлении y, и аналогично для поверхностей x и z.
Результирующая поверхностная сила, действующая на элемент, теперь может быть представлена в векторной форме и в дальнейшем выражена в виде результирующей поверхностной силы на единицу объема.
Вес элемента может быть выражен с помощью удельного веса жидкости.
Второй закон Ньютона для жидкого элемента может быть выражен с помощью результирующей поверхностной силы на единицу объема и веса элемента.
С помощью этого можно получить уравнение поля давления.

Основное уравнение для поля давления

Basic Equation for Pressure Field

Основное уравнение для поля давления в механике жидкости отражает баланс сил в любом сегменте жидкости, обеспечивая основополагающее понимание того, как давление изменяется в жидкостях под действием различных сил. Как правило, на любую часть жидкости действуют два основных типа сил: поверхностные силы и объемные силы. Поверхностные силы возникают из-за разницы давления в точках внутри жидкости, что приводит к итоговым силам, которые могут меняться в зависимости от локального градиента давления. Объемные силы, с другой стороны, распределены по всей жидкости и в основном возникают из-за силы тяжести, действующей на массу жидкости.

 Уравнение поля давления выводится путем анализа этих сил в сочетании со вторым законом Ньютона, который связывает силу с ускорением массы жидкости. Это уравнение предлагает упрощенное представление о том, как силы взаимодействуют, создавая изменения давления в жидкости. Когда давление в жидкости увеличивается или уменьшается, это часто приводит к движению или ускорению, обусловленному этим балансом сил.

 Уравнение поля давления необходимо для понимания поведения жидкостей и газов, от течения в реках и трубах до изменений атмосферного давления и погодных условий. Понимая, как распределяется давление в жидкости, можно прогнозировать ее движение и проектировать системы для управления потоком в различных областях применения: от гидравлического оборудования до сетей распределения воды и даже аэродинамики.

Рассмотрим небольшой прямоугольный элемент жидкости с длинами ребер, которые предполагается взять из любого произвольного места в массе жидкости ...

Education

JoVE Core

Civil Engineering

Unit 1

Раздел 1

Fluid Statics

Гидростатика

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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