Daniel Bernoulli

Daniel Bernoulli (Groningen, 8 February 1700 – Basel, 8 March 1782) was aDutch-Swiss mathematician and was one of the many prominent mathematicians in theBernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability andstatistics. Bernoulli’s work is still studied at length by many schools of science throughout the world. In Physics :- He is the earliest writer who attempted to formulate a kinetic theory of gases, and he applied the idea to explain Boyle’s law. 2] He worked with Euler on elasticity and the development of the Euler-Bernoulli beam equation. [9] Bernoulli’s principle is of critical use inaerodynamics. [4] Daniel Bernoulli, an eighteenth-century Swiss scientist, discovered that as the velocity of a fluid increases, its pressure decreases The relationship between the velocity and pressure exerted by a moving liquid is described by the Bernoulli’s principle: as the velocity of a fluid increases, the pressure exerted by that fluid decreases.
Airplanes get a part of their lift by taking advantage of Bernoulli’s principle. Race cars employ Bernoulli’s principle to keep their rear wheels on the ground while traveling at high speeds. The Continuity Equation relates the speed of a fluid moving through a pipe to the cross sectional area of the pipe. It says that as a radius of the pipe decreases the speed of fluid flow must increase and visa-versa. This interactive tool lets you explore this principle of fluids.
You can change the diameter of the red section of the pipe by dragging the top red edge up or down. Principle In fluid dynamics, Bernoulli’s principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid’s potential energy. [1][2] Bernoulli’s principle is named after the Dutch-Swiss mathematician Daniel Bernoulliwho published his principle in his book Hydrodynamica in 1738. 3] Bernoulli’s principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli’s equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli’s principle is valid for incompressible flows (e. g. most liquid flows) and also for compressible flows (e. g. gases) moving at low Mach numbers. More advanced forms may in some cases 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 mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure andpotential 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 mass (the sum of pressure and gravitational potential ? g h) is the same everywhere. [4] Bernoulli’s principle can also be derived directly from Newton’s 2nd law. 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. [5][6] 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. ————————————————- Incompressible flow equation
In most flows of liquids, and of gases at low Mach number, the mass density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. For this reason the fluid in such flows can be considered to be incompressible and these flows can be described as incompressible flow. Bernoulli performed his experiments on liquids and his equation in its original form is valid only for incompressible flow. A common form of Bernoulli’s equation, valid at any arbitrary point along a streamline where gravity is constant, is: |  | |  |  | | | | | | | where: is the fluid flow speed at a point on a streamline, is the acceleration due to gravity, is the elevation of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration,  is the pressure at the chosen point, and is the density of the fluid at all points in the fluid. For conservative force fields, Bernoulli’s equation can be generalized as:[7] where ? is the force potential at the point considered on the streamline. E. g. for the Earth’s gravity ?  gz. The following two assumptions must be met for this Bernoulli equation to apply:[7] * the fluid must be incompressible – even though pressure varies, the density must remain constant along a streamline; * friction by viscous forces has to be negligible. By multiplying with the fluid density ? , equation (A) can be rewritten as: or: where: is dynamic pressure, is the piezometric head or hydraulic head (the sum of the elevation z and the pressure head)[8][9] and  is the total pressure (the sum of the static pressure p and dynamic pressure q). 10] The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head H: The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli’s equation ceases to be valid before zero pressure is reached. In liquids – when the pressure becomes too low – cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure.
At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid Simplified form In many applications of Bernoulli’s equation, the change in the ? g z term along the streamline is so small compared with the other terms it can be ignored. For example, in the case of aircraft in flight, the change in height z along a streamline is so small the ? g z term can be omitted. This allows the above equation to be presented in the following simplified form: where p0 is called total pressure, and q is dynamic pressure. 11] Many authors refer to the pressure p as static pressure to distinguish it from total pressure p0 and dynamic pressure q. In Aerodynamics, L. J. Clancy writes: “To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure. “[12] The simplified form of Bernoulli’s equation can be summarized in the following memorable word equation: static pressure + dynamic pressure = total pressure[12]
Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure p and dynamic pressure q. Their sum p + q is defined to be the total pressure p0. The significance of Bernoulli’s principle can now be summarized as total pressure is constant along a streamline. If the fluid flow is irrotational, the total pressure on every streamline is the same and Bernoulli’s principle can be summarized as total pressure is constant everywhere in the fluid flow. 13] It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight, and ships moving in open bodies of water. However, it is important to remember that Bernoulli’s principle does not apply in the boundary layer or in fluid flow through long pipes. If the fluid flow at some point along a stream line is brought to rest, this point is called a stagnation point, and at this point the total pressure is equal to the stagnation pressure.
Applicability of incompressible flow equation to flow of gases Bernoulli’s equation is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli’s equation – in its incompressible flow form – can not be assumed to be valid. However if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas, (so the simple energy balance is not upset).
According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature, however this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle, or in an individualisentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density.
Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0. 3 is generally considered to be slow enough. [edit]Unsteady potential flow The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics. For an irrotational flow, the flow velocity can be described as the gradient ?? f a velocity potential ?. In that case, and for a constant density? , the momentum equations of the Euler equations can be integrated to:[14] which is a Bernoulli equation valid also for unsteady – or time dependent – flows. Here ?? /? t denotes the partial derivative of the velocity potential ? with respect to time t, and v = |?? | is the flow speed. The function f(t) depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment t does not only apply along a certain streamline, but in the whole fluid domain.
This is also true for the special case of a steady irrotational flow, in which case f is a constant. [14] Further f(t) can be made equal to zero by incorporating it into the velocity potential using the transformation Note that the relation of the potential to the flow velocity is unaffected by this transformation: ?? = ??. The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke’s variational principle, a variational description of free-surface flows using the Lagrangian (not to be confused with Lagrangian coordinates). ————————————————- edit]Compressible flow equation Bernoulli developed his principle from his observations on liquids, and his equation is applicable only to incompressible fluids, and compressible fluids at very low speeds (perhaps up to 1/3 of the sound speed in the fluid). It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli’s equation and all rely on nothing more than the fundamental principles of physics such as Newton’s laws of motion or the first law of thermodynamics.
Compressible flow in fluid dynamics For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces, [15]   (constant along a streamline) where: p is the pressure ? is the density v is the flow speed ? is the potential associated with the conservative force field, often the gravitational potential In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation becomes [16]   (constant along a streamline) here, in addition to the terms listed above: ? is the ratio of the specific heats of the fluid g is the acceleration due to gravity z is the elevation of the point above a reference plane In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term gz can be omitted. A very useful form of the equation is then: where: p0 is the total pressure ?0 is the total density [edit]Compressible flow in thermodynamics Another useful form of the equation, suitable for use in thermodynamics, is: [17]
Here w is the enthalpy per unit mass, which is also often written as h (not to be confused with “head” or “height”). Note that  where ? is the thermodynamic energy per unit mass, also known as the specific internal energy. The constant on the right hand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of energy, b is constant along any given streamline. More generally, when b may vary along streamlines, it still proves a useful parameter, related to the “head” of the fluid (see below).
When the change in ? can be ignored, a very useful form of this equation is: where w0 is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly proportional to the temperature, and this leads to the concept of the total (or stagnation) temperature. When shock waves are present, in a reference frame in which the shock is stationary and the flow is steady, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected.
An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy. ————————————————- Real-world application Condensation visible over the upper surface of a wing caused by the fall in temperature accompanying the fall in pressure, both due to acceleration of the air. In modern everyday life there are many observations that can be successfully explained by application of Bernoulli’s principle, even though no real fluid is entirely inviscid [21] and a small viscosity often has a large effect on the flow. Bernoulli’s principle can be used to calculate the lift force on an airfoil if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli’s principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lift force. nb 1][22] Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli’s equations[23] – established by Bernoulli over a century before the first man-made wings were used for the purpose of flight. Bernoulli’s principle does not explain why the air flows faster past the top of the wing and slower past the underside. To understand why, it is helpful to understand circulation, the Kutta condition, and the Kutta–Joukowski theorem. The carburetor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburetor and mix it thoroughly with the incoming air. The low pressure in the throat of a venturi can be explained by Bernoulli’s principle; in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure. * The Pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft. These two devices are connected to theairspeed indicator which determines the dynamic pressure of the airflow past the aircraft.
Dynamic pressure is the difference betweenstagnation pressure and static pressure. Bernoulli’s principle is used to calibrate the airspeed indicator so that it displays the indicated airspeed appropriate to the dynamic pressure. [24] * The flow speed of a fluid can be measured using a device such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed.
Subsequently Bernoulli’s principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect. * The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli’s equation, and is found to be proportional to the square root of the height of the fluid in the tank. This is Torricelli’s law, showing that Torricelli’s law is compatible with Bernoulli’s principle. Viscosity lowers this drain rate. This is reflected in the discharge coefficient, which is a function of the Reynolds number and the shape of the orifice. 25] * In open-channel hydraulics, a detailed analysis of the Bernoulli theorem and its extension were recently (2009) developed. [26] It was proved that the depth-averaged specific energy reaches a minimum in converging accelerating free-surface flow over weirs and flumes (also[27][28]). Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics. * The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper. [edit]

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