# What is Bernoulli's equation: Definition and 190 Discussions

In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler who derived Bernoulli's equation in its usual form in 1752. The principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. heat radiation) are small and can be neglected.
Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).
Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy, potential energy and internal energy remains constant. Thus an increase in the speed of the fluid – implying an increase in its kinetic energy (dynamic pressure) – occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρ g h) is the same everywhere.Bernoulli's principle can also be derived directly from Isaac Newton's Second Law of Motion. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

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1. ### B Applying Bernoulli's equation to problems involving a perfect gas

I would like to know the opinions of experienced forum users regarding an issue that seems to happen often in problems: namely, applying Bernoulli's equation to perfect gas. Is it permissible to do so, even if only to find reasonable estimates? Two examples I found out might be: - The problem...
2. ### Can you please explain Bernoulli's equation?

Can you please explain why is there work done by F2(on photo of textbook explanation of Bernoully equation (photo below)). I can understand that W2 is caused by F2 which is gravitational force(screenshot photo from YT). But for the explanation in textbook pipe is straight, no height...
3. ### Calculating Pressure in a Flowing Liquid

According to what I get, the problem clearly says that ##p_2= 500 ## kPa. I can also see that ##p_1 > p_2## since higher velocity means lower pressure in a flowing liquid and we can say that ##v_1<v_2##.
4. ### I About Bernoulli's equation for fluid flow

Hello, I am currently studiying Bernoulli's equation and I have trubble understanding something , say we have a horizontal hose (no change in altitude of pressure ) Bernoulli's equation state that an ideal fluid can flow thought the hose with the same velocity , does an ideal fluid need a...
5. ### Determining Reverse Mass Flow Rate Formula for Multiple Diameter Tubes

Alright, this is more of a conceptual question than a HW question that will nonetheless help me design something. In the attached image, air is flowing through a large area tube and then a small area tube from right to left. A collaborator previously figured out the equation for measuring the...
6. ### I Bernoulli's equation and the work energy theorem

Hello physics researchers, teachers and enthusiasts. I notice one little thing is confusing me in the derivation of Bernoulli's equation in the article, they write:$$dW = dK + dU$$where dW is the work done to the fluid, dK is the change in kinetic energy of the fluid, and dU is the change in...
7. ### [Aerodynamics] Bernoulli's equation and Pressure Coefficient

We know that the definition of the pressure coefficient is $$C_p=\frac{p-p_\infty}{q_\infty}$$, where ##p## is the pressure at a point, ##p_\infty## is the ambient pressure (free-stream), and ##q_\infty## is the free-stream dynamic pressure. We also know that the Bernoulli's equation is...
8. ### Velocity at a reservoir (Bernoulli's equation)

Hi there, I am solving a problem which requires me to use Bernoulli's equation and I have across a hint that says that "velocity at the reservoir is negligible because the reservoir is large" and I do not understand what that means exactly. How is velocity at a reservoir negligible?
9. ### I Fluids: Bernoulli's Equation Derivation Question

I figure that either the Force F2 is applied in the opposite direction because of some kind of resistance, but I'm not sure. Thanks!
10. ### Can I apply Bernoulli's equation to this situation?

Can Bernoulli's equation be applied between points 1 and 2, ignoring the another tank ?
11. ### Bernoulli's equation from an elemental fixed streamtube control volume

Elemental fixed streamtube control volume from Professor White’s textbook “Fuid Mechanics”: I was unable to develop the intermediate steps for the following approximations: (continuity equation according to the book ) Where and (Momentum equation according to the book) In...
12. ### Bernoulli's Equation With Losses

Hello community I have been trying to get my head around Bernoulli's equation when factoring in energy loss due to friction. I am trying to understand the concepts and i was hoping someone could remove some doubt from my mind by confirming the following:- 1) Would the following statement be...
13. ### Water flowing out of a tank through a hole

I assume the water to start flowing from rest at position 1, hence ##v_1 = 0##. Applying the continuity equation, ##A_1 v_1 = A_2 v_2##, we find the (wrong) result that the velocity at position 2 is ##v_2 = 0## also! (We assume that ##A_2## is small but finite) Hence, to answer the question...
14. ### Bernoulli's Equation and water rockets

So I'm playing around with some water rockets and I'm trying to figure out how fast the exhaust velocity of the water is. I've had an experimental approach using high fps camera to record and analyse (using tracker) the exhaust velocity. I'm using a 0,5 l soda bottle with 0,085 L ; 0,135 ; L...
15. ### A question regarding Bernoulli's equation

P1 + ρgh1 + 0.5ρv21 = P2 + ρgh2 + 0.5ρv22 In the derivation of this equation from the theorem of Work-Kinetic Energy, pressures ( P1 and P2) represent are derived from F = PA, forces affected by other portions of fluid upon the fluid in the middle (which is our concern) at 2 different points. So...
16. ### Is Bernoulli's Equation related to the Conservation of Mechanical Energy?

So the Bernoulli's Equation.. My question : Are the terms on the left hand side equal to the total mechanical energy? So can I rewrite this equation as ?
17. ### Pumps and Bernoulli's equation

Hello, My understanding is that pumps (whatever type) add energy to the fluid and cause the fluid to move. The fluid can be either brought to a higher elevation or not. I am unclear on how pumps "provide a larger pressure". Do they? If so, in what sense? Are pumps simply speeding up, i.e...
18. ### Deriving Bernoulli's Equation Using Euler's Law of Motion

Hi all, I have attached an image of a page out of the book I am using for context. The blue arrow in Figure 12-3 describes the motion of the particle. I figured the net force would need to be in the same direction, but apparently the net force opposes the motion. So, in Figure 12-3 the pressure...
19. ### Bernoulli's Equation Pressure at Blockage

Would it be because I'm comparing two different streams as opposed to two portions of the same stream?
20. ### Bernoulli's equation applicability question....

Hello, I just want to make sure I am on the right track: the three terms in Bernoulli's equation add to the same exact constant for any two points along the same streamline if the fluid is: stationary incompressible inviscid However, if the fluid is also irrotational, the the three terms add...
21. ### Help with Bernoulli's equation for a central heating system

so far I have found the velocity 1 and 2 by dividing the volume flow rate over the area which I got from pi x dia squared/4 my v1 = 1.01859m/s and v2= 2.82942m/s i have then figured out a pressure for the 15mm pipe which i got an answer of 2.71 bar however i am stuck on the rest of the question...
22. ### Demystifying the Often Misunderstood Bernoulli's Equation - Comments

Greg Bernhardt submitted a new blog post Demystifying the Often Misunderstood Bernoulli's Equation Continue reading the Original Blog Post.
23. ### Fluid mechanics: Bernoulli's equation and conservation of mass

Homework Statement A mechanical servo-mechanism comprising of a movable piston-cylinder within a vertical cylinder operates based on a venturi contraction in a horizontal 350mm diameter pipe that delivers a fluid of relative density 0.95. The upper end of the 100mm diameter vertical cylinder is...
24. ### When to apply Pascal's Law vs Bernoulli's equation

Hello, I was solving a problem regarding pressure at different elevations. The question regarded water flowing through a pipe which travels up 5 meters. I used Pascal's Law (p = p(initial) + rho*g*h : rho is density of fluid, g is gravity and h is the height) and came up with an answer...
25. ### Different answers from equation of continuity and Bernoulli

Hi there, So I was doing the dishes this morning using a sink wand hat can toggle between different flow speeds. The way that I've always thought of this working is using the equation of continuity: Volume flow rate: = Area*velocity Pressing a button on the wand decreases the cross-sectional...
26. ### Bernoulli's equation, different streamlines, and wings

Hello Everyone, Bernoulli's equations expresses the conservation of mechanical energy for a particular fluid parcel moving inside a time-independent flow. The parcel is restricted to move and remain along a particular streamlines. The sum of the trinomial is equal to a constant on every...
27. ### Does Bernoulli's equation apply to a fountain?

Can I use Bernoulli's equation to calculate certain measures of a fountain, or does it only apply to fluids in pipes? Also if so, how could I calculate the pressure inside a tube used in a fountain?
28. ### Bernoulli's Equation, Find gauge pressure at 2nd point

Homework Statement At a certain point in a pipeline, the velocity is 1 m/s and the gauge pressure is 3 x 105 N/m2. Find the gauge pressure at a second point in the line 20 m lower than the first if the cross-section at the second point is one half that at the first. The liquid in the pipe is...
29. S

### Confusion about Bernoulli's Equation & Airplane Wing

Hi, I'm a little confused about the theory behind this problem related to fluids/Bernoulli's equation: "An airplane wing is designed so that the speed of the air across the top of the wing is 251 m/s when the speed of the air below the wing is 225 m/s. The density of the air is 1.29 kg/m3...
30. ### Bernoulli's equation, Venturi effect, hydraulic head, nozzle

Hi, I have this problem: I have a vertical tube 1 meter D, in the bottom end there is a nozzle with 0,5 m D. The tube is full of water. the tube length is 10 meters and the nozzle length is 2 meters. I need to calculate the power of this by this equation: W = Q * g * h * p where W is watt, Q...
31. ### Trouble understanding energy aspect of Bernoulli's Equation

So the Bernoulli's EQ comes from conservation of energy. From the figure, I see that if Force 1 is greater than Force 2, the water will move to the right. The distance Force 1 travels gives work of F1d1, and the distance Force 2 is pushed back gives work F2d2, and net work on the system is F1d1...
32. ### Using Bernoulli's Equation to find Power Requirement

Homework Statement Calculate head required for the pump and then its power requirement assuming 70% efficiency. The lower storage vessel is vented to atmosphere (assume 1 bar pressure) . I have the following given information: Pipe Area = 0.00636m^3. Flow(Q)= 0.01m^3/s Average Velocity =...
33. ### Would the velocity of a fluid increase right after exiting a pipe into open air?

If you have an non-viscous incompressible fluid flowing in a pipe whose static pressure is higher than atmospheric pressure, then after exits the pipe will the dynamic pressure increase? The static pressure of the fluid right after exiting should decrease because it should be equal to the...
34. ### Find velocity of air using Bernoulli principle

Homework Statement A horizontal stream of air is blown just above the open end of the hares apparatus with an initial speed "V". The density of air is 1.2 kgm-3.Then the water column rises to 6 cm in the respective limb.Density of water is 103 kgm-3 Using Bernoulli`s principle find "V"...
35. M

### How to show Bernoulli's equation applies on same streamline?

When deriving Bernoulli's equation from Navier Stokes, how do we know it is only valid along a streamline? At the very end of my derivation, assuming Newtonian, incompressible, inviscid, irrotational flow I have ##\nabla(\partial_t \phi + |\vec{u}|^2/2+p/\rho + g z) = \vec{0} \implies \partial_t...
36. ### Why is Bernoulli's Equation Isentropic

I have trouble understanding why we classify an inviscid adiabatic incompressible flow along a streamline as isentropic I understand this from a Thermodynamic definition/explanation $$dS = dQ/T$$ Adiabatic Invsicid $$dQ =0= dS$$ So no heat added or lost no change in entropy I'm fine with that...
37. ### Bernoulli's Eq: P1=P2 & Torricelli's Eq

When using Bernoulli's equation to describe fluid that is coming out from a spigot, why is it that P1 = P2 are the same? This cancellation will eventually lead to the Torricelli's equation.
38. ### Bernoulli's Equation and pressure differences

Homework Statement Problem in attached image Homework Equations $$P_1+\frac{\rho v_1^2}{2}=P_2+\frac{\rho v_2^2}{2}$$ The Attempt at a Solution I understand everything in the solution except why $$P_A-P_B=h(\rho_{Hg}-\rho)g$$ Why do we have to subtract the density of water from that of...
39. ### Different forms of Bernoulli's equation

I was reading some textbooks on doppler echo for medicine and came across this version of the bernoulli equation for blood flow in the aorta. P_{1} - P_{2}= 1/2 \rho (v{_{2}}^{2}- v{_{1}}^{2}) + \rho \int_{1}^{2} \frac{\overrightarrow{dv}}{dt}\cdot \overrightarrow{ds} +...
40. ### Bernoulli's equation and water exit speed from tank opening

Homework Statement Use Bernoulli’s equation to calculate how fast the water emerges from the open tap (at position 2) in the figure(a). You may assume that the water at position 1 moves negligibly slowly (b) The tap is rotated to create a fountain as shown in (b) Calculate the maximum height h...
41. ### How can Bernoulli's equation be applied in this problem?

Homework Statement Homework EquationsThe Attempt at a Solution First I would like to mention that I have solved this problem using force approach and got correct result . My problem is that when I saw the official solution , I was quite surprised to see the application of Bernoulli's...
42. ### Modeling ideal gas flow using Bernoulli's equation

Hello, PF! I'm currently brushing up my fluid mechanics and came across some questions while studying the compressible flow of an ideal gas using Bernoulli's equation. First, consider incompressible flow in the following system Neglecting any changes in elevation, the Bernoulli equation for...
43. ### The pressure term in Bernoulli's equation

Seeing as P(F/A) + KE + PE = Constant, if the fluid's flowing through a constriction (So the area's decreasing(Which would net an overall larger pressure)), How does the pressure term decrease? Does the Force decrease even more greatly than the Area, to net an overall lower pressure? Assuming...
44. ### Velocity of liquid from bottom pinhole of closed container

What I know: Below link is about Torricelli's law. Velocity of liquid coming out of bottom of the tank i.e comes after using Bernoulli's equation square root of (2*g*h*) where "h" is height of fluid in the container and "g" is acceleration due to gravity...
45. ### How water flows even after adverse pressure gradient?

In nature, gradient is always required for flow; whether it is temperature gradient for heat transfer or pressure difference for fluid flow. There is a case of Venturimeter in which we have throat section. After throat there is a divergent section. How could flow even happen in that adverse...
46. ### Bernoulli's equation to find the flow rate

Homework Statement " A horizontal water pipe has a radius of 10 cm and a pressure of 8*10^4 Pa at one end. At the other end the radius is 5 cm and the pressure is 6*10^4 Pa. What is the water flow rate through this pipe? Homework Equations P1 + 0.5 * ρ * v1^2 + h1ρg = P2 + 0.5 * ρ * v2^2 +...
47. ### An Open Cylindrical Tank of Acid....

Homework Statement An open cylindrical tank of acid rests at the edge of a table 2.20\cdot 10^0\ m above the floor of the chemistry lab. If this tank springs a small hole in the side at its base, how far from the foot of the table will the acid hit the floor if the acid in the tank is...
48. ### What is the Optimum Height for Drinking a Milkshake with a Vertical Straw?

You and a friend of yours went for a drink of chocolate milkshake(ρ=1200 kg/m3). The waiter brings your drink in two glasses 200 mm tall with a straw 8 mm in diameter and 300 mm long. Given that human lungs capacity can develop approximately 3000 Pa of vacuum pressure and assuming that the straw...
49. ### How is Bernoulli's equation related to saxon bowls?

Homework Statement This is a lab that we have to design and carry out that goes above and beyond our course. I am investigating how the diameter of a hole in a bowl affects its sink time when put in a larger bowl of water (saxon bowl, or sinking bowl as they are known.) I will have to plot them...
50. ### Calculate pressure using Bernoulli's equation

Homework Statement You have been given a milkshake (ρ = 1200kg/m^3). The glass is 200mm tall a straw 8mm diameter and 300mm long. Show that human lungs would be unable to drink the milkshake through a vertical straw. (answer should be around 3000Pa) I have no idea what to do as I don't know...