The Biggest Myth in Fluid Mechanics | Bernoulli’s Principle Explained #engineering #fluidmechanics

追加: 2026-05-31

[00:00:00]If you've ever heard that faster fluid means  lower pressure, you've heard Berni's principle.

[00:00:05]But that explanation is incomplete and in many  situations, it's actually misleading. In this video, we're going to break down Berni's  equation properly, where it comes from, what it really means, where engineers use it, and  just as importantly, where it completely fails.

[00:00:22]On engineering, we're diving into everything  you need to know about Boni's principle.

[00:00:28]What is Berni's principle? The principle is named  after the Swiss mathematician and physicist Daniel Boni who published it in his book hydrodnamica in  1738. Boni's principle is often explained by using the following statement. As the velocity of a  fluid increases, its pressure decreases. However, this statement on its own is incomplete and  often misleading. The correct statement is that for steady incompressible non-iscous flow along a  streamline, the total mechanical energy per unit volume remains constant. Bonoli's equation is  therefore fundamentally an energy conservation statement applied to fluid motion. Let's imagine  forcing a liquid through a narrow opening in a pipe. As the pipe constricts, continuity requires  the velocity of the fluid to increase if density remains constant. If no energy is added or  removed from the system, Boni's equation predicts that this increase in kinetic energy must  be accompanied by a decrease in static pressure.

[00:01:33]As the pipe constricts, this creates an area of  lowest pressure at the narrowest section of the pipe known as the venturi throat which entrains  or pulls in the surrounding atmospheric air.

[00:01:45]This is known as the venturi effect and is  something we will discuss later on in the video.

[00:01:51]Boni's equation helps relate pressure, velocity,  and elevation in a flowing fluid. But it does not independently explain all fluid motion. In  many real systems, conservation of momentum must also be considered. As it flows, three things  change. Velocity energy, the kinetic energy of the fluid. Pressure energy, the pressures the  fluid experiences. And the potential energy, the height or elevation of the flowing fluid. Due  to conservation of energy, the sum of pressure, velocity energy, and height energy remains  constant along a streamline. So if velocity increases and height is unchanged, pressure  must decrease. Boni expressed this relationship in what's now known as the Boni equation, showing  how velocity, pressure, and height energy balance along a streamline. P + 1/2 row V ^ 2 plus row  GH equ= constant where P equals static pressure, row equals fluid density, V equals fluid velocity,  G equals gravity, H equals height above reference.

[00:02:59]This is the BI equation, the foundation of fluid  mechanics. Each term represents mechanical energy per unit volume of the flowing fluid. Think of the  equation as three energy buckets. The first term in the equation is the static pressure of the  fluid P. This is the force the fluid exerts on the walls or surfaces. The second term represents  the fluid kinetic energy per volume and is known as the dynamic pressure. As velocity increases,  the kinetic energy term increases. Under Boni's assumptions, this must be balanced by a reduction  in static pressure or potential energy. The final term in the equation represents the potential  energy or height energy and is known as the hydrostatic pressure. As you can see by the term  G, this is the pressure exerted by the fluid due to gravitational forces. G is the gravitational  acceleration which is 9.81 m/s squared on Earth.

[00:03:59]And the H on the equation is the height or  elevation of the fluid from a reference point.

[00:04:04]The higher up the fluid, the more gravitational  energy it has. Before applying the equation, it's important to understand where it comes from.  Deriving Berni's equation. The Bernoli equation for incompressible fluids can be derived by either  integrating Newton's second law of motion or by applying the law of conservation of energy via  the continuity equation. Assumptions of Berni's.

[00:04:31]When deriving Berni's the following assumptions  must be made. The flow is steady. Nothing changes with time. The fluid is incompressible. It has a  constant density. There are no friction losses. So no energy is lost as heat. The first derivation we  are going to look at is when Berni's equation is derived by integrating the uler momentum equation.  Row dv / dt equals negative delta p + row g where row equals fluid density v equals velocity vector  d / dt equals material derivative acceleration of the fluid parcel negative delta p equals pressure  gradient force row g equals body force due to gravity which itself comes directly from Newton's  second law applied to a moving fluid. Force per volume equals mass per volume time acceleration.  Imagine a small element of fluid moving along a streamline. Three forces act on this fluid  element. Pressure forces pushing the fluid forward and backward. Gravity acting on the fluid mass.  The inertia of the moving fluid itself. Applying Newton's second law along the streamline gives the  uler equation. When this equation is integrated between two points in the flow, we obtain half V  ^2 + P over row + GH equals constant. Multiplying through by density or row produces the familiar  Boli equation. Half row V ^ 2 + P + row G H equals constant. We can also derive bonies by  applying the law of conservation of energy via the continuity equation for steady incompressible  flow in a pipe. We can show flow as the following equation a1 v1 = a2 v2 equals q. This tells us  that as the pipe widens the velocity decreases.

[00:06:26]We can rearrange this equation to find the  velocity on the second part of the pipe.

[00:06:30]We can then substitute v_sub_1 and v_sub_2 back  into the boni's equation. The continuity equation comes from the conservation of mass principle. It  states that mass cannot be created or destroyed within a control volume. The conservation of mass  principle is denoted by the following equation where the m with the dot above denotes the mass  flow rate. This can be converted to show the mass flow rate of the fluid as the sum of its density  multiplied by the cross-sectional area multiplied by the velocity of the fluid. Row 1 A1 V1 equals  row 2 A2 V2. Now let's imagine a pipe with water flowing through it. The [music] pipe has a change  in diameter and elevation. As the water moves from 0.1 to 2, three things change. the pressure, the  speed of the flow, and the height of the pipe.

[00:07:23]Because energy is conserved, any gain in one of  these must come from a loss in one or both of the others. At 0.1, pressure P1 pushes the water  forward into the next section of pipe. At 2, pressure P2 pushes back against the flow. So,  the net pressure force acting on the fluid is shown by the following equation. P1 minus P2.  This difference in pressure is what we call the static pressure term in the Boni's equation. The  work done by this pressure difference supplies energy to the fluid. The pipe widens, the velocity  changes according to the continuity equation.

[00:07:59]The kinetic energy per unit volume of the fluid is  denoted by the following part of Boni's equation.

[00:08:05]A half row v^ squ. So the change in kinetic energy  from 0.1 to 2 is shown like this. Because 2 is higher than 0.1, the fluid must gain gravitational  potential energy. Potential energy per unit volume is shown by the following equation. Row GH. So  the change in potential energy is shown here.

[00:08:27]Row G multiplied by the difference between  H2 and H1. Now we apply the energy balance.

[00:08:35]Pressure work equals change in kinetic energy  plus change in potential energy. So P1 minus P2 =/ row V 2^ 2 minus a half row V1^ 2 + row G  multiplied by the difference between H2 and H1.

[00:08:55]Rearranging to collect terms at each point. P1 +  half row V1 squared plus row GH1 equals P2 + half row V2 plus row GH2. And since this applies  between any two points along a streamline, we write the general form P + a half row V² plus  row GH equals constant, which is Berni's formula.

[00:09:24]Forms of Boni's equation. Boni's equation doesn't  come in just one form. Engineers use different versions depending on what they're trying to  calculate. The following table breaks them down.

[00:09:38]If you're enjoying this deep dive into Boni's  principle and fluid mechanics, make sure to hit subscribe and give the video a thumbs up.  We've got plenty more engineering foundations and realworld applications coming your way. You can  also check out our online store for engineering themed gear and tools that help support the  channel. Every purchase genuinely helps us keep making highquality content. And if you want to  take things a step further, consider becoming an Engineering Nest member. You'll get early access  to new videos, behindthe-scenes content showing how we animate and build these breakdowns,  and access to our membersonly chat rooms where we talk about all things engineering. Boni's  principle shows up in many realworld applications.

[00:10:23]Some of the main areas we can see this is when  plane wings generate lift. Fluid flows over the wings faster due to the wings design than the  fluid flowing beneath the wing. This high velocity flow above the wing causes an area of low pressure  and an area of high pressure below the wing where the fluid velocity is slower in line with the  principles of Bernoli's equation. Lift arises from the pressure field created by circulation  around the air foil and the downward deflection of air flow which involves conservation of momentum.  Boni helps relate velocity and pressure locally, but it is not a complete explanation of lift  by itself. When a fluid flows through a pipe that narrows, the continuity equation tells us  the velocity must increase. And Boni tells us that when velocity goes up, pressure must drop.  This pressure drop in the narrowed section is known as the venturi effect. Now let's say that we  create a small hole in the pipe work that is open to atmospheric pressure. The atmospheric pressure  which is higher than the internal pressure within the pipe will cause the air to become entrained  within the pipe work. It is used everywhere in engineering. Venturi flow meters measure flow  rate by comparing the pressures in the wide and narrow sections. It's how aspirators and inductors  draw in chemicals or gases without using a pump.

[00:11:51]Limitations of Boni's equation. Although  Berni's equation is extremely useful, it is important to understand where it breaks  down and where it has limits. If viscosity is significant, mechanical energy is dissipated  as heat and friction losses must be included.

[00:12:09]In turbulent flows, additional energy dissipation  occurs through velocity fluctuations and eddies.

[00:12:17]In compressible high-speed gas flows, density  changes become significant and the incompressible form of Bernoli's equation is no longer  valid. Across shock waves, entropy increases and mechanical energy is not conserved,  meaning Bernoli cannot be applied directly.

[00:12:35]Understanding these limitations is essential when  applying Berni's equation in real engineering systems. Boni's equation is one of the most  important relationships in fluid mechanics.

[00:12:46]It connects pressure, velocity, and elevation  through conservation of mechanical energy.

[00:12:52]But as we've seen, it only works when its  assumptions are satisfied. Used correctly, it becomes an incredibly powerful  engineering tool. Used incorrectly, it can lead to completely wrong conclusions. If  you enjoyed this deep dive, hit subscribe and give the video a thumbs up. Check out our online  store to support the channel or become a member for early access and behindthe-scenes animation  content. Thanks for watching and stay curious.