Bernoulli equation Definition and 49 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. jdgotts

    I Help Regarding Application of Bernoulli in a Boundary Layer

    Hey all, I recently took an aerodynamics exam that included the question "Please Explain how the Bernoulli Equation can be Applied Inside a Boundary Layer". Now, it is my belief that the Bernoulli equation, defined by my textbook as P+0.5ρV2=ℂ, requires inviscid flow to be properly applied...
  2. F

    Fluid flow inside a cylindrical tank

    Hi, I´m quite lost and would appreciate guidance I have solved for 2 tubes using Bernoulli´s equation before, but now how does it change? Is it really going to rise water level inside? Why?
  3. Dario56

    I Work - Energy Principle Application to Fluid Flow

    In classical and continuum mechanics if we want to find equation of motion of the body we draw force diagram and apply Newton's 2nd law. In continuum mechanics, equation of motion actually refers to a special point of the body known as center of mass (COM) which can be proven by definition of...
  4. Dario56

    I Work - Energy Principle Application to Fluid Flow

    Work - energy principle states that work done by net force acting on the body equals change in kinetic energy of the body. We are talking about continuum mechanics. This principle is usually introduced in mechanics of solid bodies. For us to describe the motion of the body, it is enough to know...
  5. Dario56

    I Bernoull's Equation Derivation From Work - Energy Principle

    Work - Energy principle states that work of resultant force or sum of work of all forces acting on some system equals change in kinetic energy of the system. For inviscid fluid flowing in a pipe such theorem can be used to derive Bernoulli's equation because as fluid flows it is subjected to...
  6. S

    How pressure increases in centrifugal compressor?

    I have read this in my textbook about principles of operation of centrifugal compressor: "The impeller is rotated at high speed by the turbine and air is continuously induced into the center of the impeller. Centrifugal action causes it to flow radially outwards along the vanes to the impeller...
  7. lahanadar

    Cause-effect relation between pressure & velocity

    For a steady, non-viscous and incompressible flow, one can apply both Bernoulli's principle (no potentials) as $$p+\frac{\rho v^2}{2} = p_t$$ where ##p##, ##\rho,##, ##v##, and ##p_t## are static pressure, density, flow velocity, and total pressure, respectively, and continuitiy principle as...
  8. bjarke

    Hagen-Poiseuille vs Bernoulli

    I am trying to calculate the exit velocity of a nozzle (0,25Ø) that is connected to a high pressure syringe (10 BAR), however I cannot understand why Hagen-Poiseuille will have a higher exit velocity then Bernoulli when HP take viscosity into account.
  9. F

    Sorry, another Bernoulli equation paradox

    Let's start with a horizontal tube with a constant diameter. I'm not sure if it's important, but let's assume it's frictionless. I will have some fluid flowing in this tube and if it's important, we can make the fluid incompressible, inviscid, irrotational, etc. To create a flow in the tube...
  10. curiously new

    I Laminar-turbulent transition and Reynolds number

    Hi all! This is my first post here, so hopefully I am not in violation of any rules or etiquette. I'm looking to derive an equation for Reynolds number as function of pressure for a pinhole leak in a pressurized gas line. The line is regular air and is pressurized to 1 atmosphere and is leaking...
  11. Tymofei

    About deriving the Bernoulli equation from a particle model and its pressure

    Here i added a page from my fluid dynamics book where it shows particle model for deriving the equation. My question is why pressure is more at stream side aka 'positive "s" direction'.I would expected more pressure on the other side because for example when you trying to push a rigid object or...
  12. A

    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...
  13. rjomega

    Effect on Volume of a Change in the Pressure of Compressible Gas

    Will the available Volume of oxygen gas for use of patients increase when the pressure decreases from 12.4 MPa to 500 KPa? Is using boyle's law the right way to calculate the available volume?
  14. nohappy

    The Venturi Effect cannot explain how an eductor works

    I've google all the keywords like "eductor", "ejector", "Venturi pump". All of them are the mechanism or applications that utilize "Venturi Effect". All of them are trying to explain the "suction effect" by saying that restriction area makes the fluid's velocity higher and hence lower pressure...
  15. nuclearfireball_42

    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 ?
  16. Q

    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?
  17. S

    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...
  18. Logic hunter

    Contradicting values of pressure for liquid cross-sections

    Consider a fixed horizontal tube of uniform cross section with pressure being 1atm at one of it's end and 5atm at the other (former due to 'open to atmosphere' and latter due to force on a piston), then liquid would flow towards low pressure end. By equation of continuity all cross sections will...
  19. S

    Bernoulli lift

    Homework Statement We have a plane which lift area is A=500 m2 and weight of plane is m= 540 000 kg. That the plane can stay in air the pressure difference between top and bottom of the lifts have to be: B>G --> Δp= (m*g)/A = 10 594.8 Pa. Homework Equations Bernoulli equation The Attempt at...
  20. JTC

    Bernoulli to explain Magnus Effect

    Hi I have been reading some internet articles that state the Bernoulli equation does NOT explain the Magnus Effect. The articles state that the effect is due to circulation (Bernoulli requires inviscid flows) Could someone explain the cause of the Magnus effect without reference Bernoulli's...
  21. J

    Filling and emptying a barrel -- Bernoulli

    << Mentor Note -- thread moved from the schoolwork forums to ME for better views >> 1. Homework Statement Not really homework as this is for work. We are washing a cylindrical container (15.5 cm) with only one (1.8" dia) hole at the bottom the rest is sealed. Our machine sticks a .5" pipe with...
  22. E

    How do we use Bernoulli's Principle in this situation?

    Homework Statement A large water tank, open at the top, has a small hole in the bottom. When the water level is ## 30## ##m## above the bottom of the tank, the speed of the water leaking from the hole: A. is ##2.5## ##m/s## B. is ##24## ##m/s## C. is ##44## ##m/s## D. cannot be calculated...
  23. 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...
  24. TheRumpus

    Does air flowing through a tube cause a pressure change?

    I have been involved in a fairly furious debate about a fluid dynamics experiment regarding the pressure of air passing through a tube attached to a moving car. It is similar in concept to the issue of whether or not an opening in an aeroplane fuselage would suck people out, where the opening...
  25. M

    Fan Selection for air cooling of a computer

    I 90% understand the question and solution but unfortunately I have some hesitations about the formal solution in the book. First of all, when calculating P4-P3, choosing lines and points on them, why do book assume velocities of 3 and 4 to be the same...
  26. T

    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...
  27. RMalt

    Evaluate Problems using fluid dynamics

    I have a diagram similar to the following. Water entering the larger end is at 20degreesC. The larger end has a diameter of 8cm and Area 50.26cm2. The small side has a diameter of 3cm and Area 7.0685cm2. The water jet exerts a force of 87N on a flat plate at an unknown distance. Assuming no...
  28. T

    Bathtub Vortex Pressure difference

    The problem statement Example 1 A single particle I have a particle being forced by a radial centripetal force onto a smaller radius $$l_1=m_c.v_1.r_1$$ $$L_1=L_2$$ $$L_2=m_c.v_2.r_2$$ $$m_c.v_1.r_1=m_c.v_2.r_2$$ $$v_1.r_1=v_2.r_2$$ $$v_2=v_1\frac{r_2}{r_1}$$ Its increase from ##v_1## to...
  29. G

    When will the house be flooded?

    Homework Statement [/B] A simplified schematic of the rain drainage system for a barn is shown in the figure. Rain falling on the slanted roof runs off into gutters around the roof edge; it then drains through several downspouts (only one is shown) into a main drainage pipe M below the...
  30. W

    Bernoulli equation exercise from Fanning and Moody

    Homework Statement I would really appreciate it if you could give me a hand with this exercise, not sure on what I've done. Data: Moody: L=55*10-3m D=10-1m k=0.0002m Homework Equations ##Re=\frac{D*u*ρ} {μ}## ##Re=\frac{4*m} {pi*D*μ}## Relative roughness ##ξ=\frac k D## where k=rougness...
  31. W

    Bernoulli equation with losses dilemma

    Homework Statement Hello,Could you please lend me a hand with this problem?I would really appreciate it. Question:[/B] Their Answer: KL=2*0.75(2 elbows) D=5*10^-2 m ;f=0.001(fanning friction factor) Assumptions I made: Point 1 which is at the top of the liquid in the tank: h1=23 m...
  32. surfwavesfreak

    A Boundary condition question

    Hello everyone, The boundary condition : P=0, z=ζ is very common when studying irrotational flows. When cast with the Bernoulli equation, it gives rise to the famous dynamic boundary conditionn, which is much more convenient : ∂tφ+½(∇φ)2+gζ=0, z=ζ But what happens if the motion is rotational ...
  33. Houeto

    What kind of ODE is this?

    consider ODE : Show that the solution to this ODE is: Can someone tell what kind of ODE is it?I thought,it's on the form of Bernoulli ODE with P(x)=0.Is it possible to still solve it by using Bernoulli Methodology?I mean by substituting u=y^1-a with a=2? Thanks
  34. MexChemE

    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...
  35. S

    Use of Bernoulli equation for pumps

    I have a doubt on the use of Bernoulli equation for pumps. Consider the situation in the picture. I marked different points: ##1## on the surface of first tank, ##2## in the exit from first tank, ##3## just before the pump, ##4## just after the pump and ##5## entering the second tank. Now...
  36. S

    Bernoulli equation related to minimum pressure inside train

    Homework Statement The train is travelling in a gallery at ##v=70 km/h##. Outside air is not moving and pressure is ##p_0=101325 Pa## and temperature ##T=283 K ##. The area of the face of the train is ##A=9 m^2##. Determine the minimum area of the section of the gallery such that the pressure...
  37. J

    Fluid Mechanics-Bernoulli and Energy Equations

    Homework Statement A well-fitting piston with 4 small holes in a sealed water-filled cylinder,shown in Fig is pushed to the right at a constant speed of 4 mm /s while the pressure in the right compartment remains constant at 50 kPa gage. Disregarding the frictional effects,determine the force...
  38. faradayscat

    U-shaped tube fluid dynamics

    Homework Statement [/B] Suppose we have a u-shaped tube filled with water, with oil added at one end which disturbs the equilibrium. Now say one end is blocked off and the other is exposed to air flow which reduces the pressure above the water and causes the water to climb back to equilibrium...
  39. Q

    Instantaneous Bernoullis

    Hi I am having trouble understanding how Bernoulli’s pressure gradients develop I know the equation in a steady flow state (after the developed state ‘flushes’ thru) but I want to understand what mechanism lowers the pressure and increases the speed as the flow develops The continuity...
  40. UMath1

    B Mass Continuity and Torricelli's Law

    Based on the law of mass continuity, when a pipe narrows then the speed of the fluid increases. Then why is it that when draining a tank the speed of the fluid only depends on the height of water above and not on the size of the hole? Wouldn't a narrower hole mean that that the speed must be...
  41. 0

    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...
  42. Alettix

    Using Bernoulli's Equation

    Hello! I have encountered some trouble with choosing the right reference points when using Bernoulli's equation and I would be glad if you could help me sort it out with this made up example. :) 1. Homework Statement There is a large, open, cylindrical water tank with a cross section area of...
  43. vincekillics

    Question for Syringe

    Homework Statement It is time for aged physics lecturers to have their flu shots but even that can be interesting. Assume the density of the vaccine in the syringe is the same as the density of water. The diameter of the syringe is 6mm, the length of the needle is 3cm and by reading the packet...
  44. evan-e-cent

    At what pressure does steam exiting a nozzle reach mach 1

    I am planning to build a Hero's steam engine or Aeolipile. This is the steam engine invented by the Greeks about 2000 years ago. It is attributed to Hero or Heron (10-70 AD) but was also referred to in 15 BC. My grandfather build one for a show. I have inherited it but rather than try to repair...
  45. O

    Why Does Bernoulli's Equation Apply to Wind Tunnels?

    I hope this question doesn't have too obvious of an answer. Basically, I still cannot grasp why Bernoulli's equation applies for wind tunnels and pitot-static probes. According to my textbook ("Introduction to Flight" by Anderson), Bernoulli's equation holds only when comparing two points...
  46. F

    Bernoulli's equation and (mostly) sealed containers.

    (This is more of a conceptual question than a real homework question; thank you all so much for your help though!) :D 1. Homework Statement Let's say that I have a large soda bottle. I drill a small hole through the side of it, put my finger over it to seal the hole, and fill the bottle up...
  47. E

    Laminar Flow Due To Gravity

    < Mentor Note -- please remember to use the Homework Help Template when posting schoolwork questions > Consider the fully developed laminar flow due to gravity of water in a vertical circular tube. Assume atmospheric pressure at inlet and outlet. Show that the relationship between diameter and...
  48. Soumalya

    Flow work and kinetic energy in fluid flow

    Considering horizontal flow in a pipe (under the action of pressure forces only) if we focus on a fluid element we have a pressure gradient across the same i.e, a net pressure force will always act over the element that essentially maintains the flow.Now, in a region in the flow field if the...
  49. Soumalya

    Accelerating and Non-accelerating Coordinates - Fluid flow

    Referring to the problem in the attachment, the author mentions that if we consider the coordinate system attached to the bicycle and the bicycle accelerates or decelerates, the flow past the bicycle becomes unsteady. For an unsteady flow, we know that nothing changes at a given location on a...