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A fluid flowing past the surface of a body exerts a force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction. If the fluid is air, the force is called an aerodynamic force.

Lift is the force generated by propellers and wings to propel aircraft and keep them in the air. Birds, bats, insects, fish, flying reptiles, and even falling plant seeds have usefully exploited lift for millions of years.

Overview

Forces on an airfoil

Lift is most commonly associated with the wing of a fixed-wing aircraft, although lift is also generated by propellers, kites, helicopter rotors, rudders, sails and keels on sailboats, hydrofoils, wings on auto racing cars, wind turbines and other streamlined objects. Lift is also exploited in the animal world, and even in the plant world by the seeds of certain trees. While the common meaning of the word "lift" assumes that lift opposes weight, lift in its technical sense can be in any direction since it is defined with respect to the direction of flow rather than to the direction of gravity. When an aircraft is flying straight and level (cruise) most of the lift opposes gravity. However, when an aircraft is climbing, descending, or banking in a turn the lift is tilted with respect to the vertical. Lift may also be entirely downwards in some aerobatic manoeuvres, or on the wing on a racing car. In this last case, the term downforce is often used. Lift may also be largely horizontal, for instance on a sail on a sailboat.

An airfoil is a streamlined shape that is capable of generating significantly more lift than drag. Non-streamlined objects such as bluff bodies and flat plates may also generate lift when moving relative to the fluid, but will have a higher drag coefficient dominated by pressure drag.

Description of lift on an airfoil

There are several ways to explain how an airfoil generates lift. Some are more complicated or more mathematically rigorous than others; some have been shown to be incorrect. For example, there are explanations based directly on Newton’s laws of motion and explanations based on Bernoulli’s principle. Either can be used to explain lift. This article will start with a simple explanation based on Newton's laws; more complicated and alternative explanations will follow.

Newton's laws: lift and the deflection of the flow

Deflection

Streamlines around an airfoil in a wind tunnel. Note the curved streamlines above and below the foil, and the overall downward deflection of the air.

Lift is a reaction force—an airfoil deflects the air as it passes the airfoil. Since the foil must exert a force on the air to change its direction, the air must exert a force of equal magnitude but opposite direction on the foil. In the case of an airplane wing, the wing exerts a downward force on the air and the air exerts an upward force on the wing.

This follows from the second and third of Newton's laws of motion: The net force on an object is equal to its rate of momentum change, and: To every action there is an equal and opposite reaction.

The air changes direction as it passes the airfoil and follows a path that is curved. Whenever airflow changes direction, a reaction force is generated opposite to the directional change.

Pressure differences

Lift may also be described in terms of air pressure: pressure is the normal force per unit area. Wherever there is net force there is also a pressure difference, thus deflection/flow turning indicates the presence of a net force and therefore a pressure difference. The direction of the net force implies that the average pressure on the upper surface of the wing is lower than the average pressure on the underside.

Whenever a fluid follows a curved path, there is a pressure gradient perpendicular to the flow direction. This direct relationship between curved streamlines and pressure differences was derived from Newton's second law by Leonhard Euler in 1754:

${\frac {\operatorname {d} p}{\operatorname {d} R}}=\rho {\frac {v^{2}}{R}}$

where R is the radius of curvature, p is the pressure, ρ is the density, and v is the velocity. This formula shows that higher velocities and tighter curvatures create larger pressure differentials and that for straight flow (R → ∞) the pressure difference is zero.

Flow on both sides of the wing

File:NasaPressureDiagram.jpg

Variation of the pressure around an airfoil as obtained by a solution of the Euler equations.

In the picture above, observe that the air is turned both above and below the wing so both the upper and lower surface contribute to the flow turning and therefore the lift. In fact, for typical airfoils at subsonic speeds the top surface contributes more flow turning than the bottom surface, and the pressure deviation along the top is significantly larger than along the bottom. A common explanation describes lift as merely the result of the air molecules bouncing off the lower surface of the wing, but since this ignores the airflow around the top of the wing it usually leads to incorrect results. However, at hypersonic speeds, this model becomes applicable.

Angle of attack

Airflow around an airfoil. The angle α is the angle of attack.

The angle of attack is the angle between an airfoil and the oncoming air. A symmetrical airfoil will generate zero lift at zero angle of attack. But as the angle of attack increases, the air is deflected through a larger angle and the vertical component of the airstream velocity increases, resulting in more lift. For small angles a symmetrical airfoil will generate a lift force roughly proportional to the angle of attack.

As the angle of attack grows larger, the lift reaches a maximum at some angle; increasing the angle of attack beyond this critical angle of attack causes the air to become turbulent and separate from the wing; there is less deflection downward so the airfoil generates less lift. The airfoil is said to be stalled.

Cambered airfoils will generate lift at zero angle of attack. When the chordline is horizontal, the trailing edge has a downward direction and since the air follows the trailing edge it is deflected downward. When a cambered airfoil is upside down, the angle of attack can be adjusted so that the lift force is upwards. This explains how a plane can fly upside down.

Limitations of deflection/turning

• While the theory correctly reasons that deflection implies that there must be a force on the wing, it does not explain why the air is deflected. Intuitively, one can say that the air follows the curve of the foil, but this is not very rigorous or precise.
• The theory, while correct in as far as it goes, is not sufficiently detailed to support the precise calculations required for engineering. Thus, textbooks on aerodynamics use more complex models to provide a full description of lift.

Bernoulli's principle: lift, pressure, and speed

Bernoulli's principle states that within an airflow of constant energy, when the air flows through a region of lower pressure it speeds up and vice versa. Thus, there is a direct mathematical relationship between the pressure and the speed, so if one knows the speed at all points within the airflow one can calculate the pressure, and vice versa. For any airfoil generating lift, there must be a pressure imbalance, i.e. lower average air pressure on the top than on the bottom. Bernoulli's principle states that this pressure difference must be accompanied by a speed difference.

Bernoulli's principle does not explain why the air flows faster over the top of the wing; to explain that requires some other physical reasoning. It is in providing that additional reasoning where some explanations oversimplify things.

Conservation of mass

Streamlines around an airfoil in a wind tunnel. Note the narrower streamtubes above and the wider streamtubes below the foil.

If one takes the experimentally observed flow around an airfoil as a starting point, then lift can be explained in terms of pressures using Bernoulli's principle and conservation of mass.

Returning to the picture from the previous section, the flow approaching an airfoil can be divided into streamtubes, which are defined based on the area between two streamlines. By definition, fluid never crosses a streamline in a steady flow. Assuming that the air is incompressible, the rate of flow (e.g. liters or gallons per minute) must be constant within each streamtube since matter is not created or destroyed. If a streamtube becomes narrower, the flow speed must increase in the narrower region to maintain the constant flow rate. This idea is called "conservation of mass", and for incompressible flow mass is conserved within each streamtube.

The picture shows that the upper stream tubes constrict as they flow up and around the airfoil. Conservation of mass says that the flow speed must increase as the stream tube area decreases. Similarly, the lower stream tubes expand and the flow slows down.

From Bernoulli's principle, the pressure on the upper surface where the flow is moving faster is lower than the pressure on the lower surface where it is moving slower. The pressure difference thus creates a net aerodynamic force, pointing upward and downstream to the flow direction. The component of the force perpendicular to the free stream is lift; the component parallel to the freestream is drag. In conjunction with this force by the air on the airfoil, the airfoil imparts an equal-and-opposite force on the surrounding air that creates the downwash, in accordance with Newton's third law. Measuring the momentum transferred to the downwash is another way to determine the amount of lift on the airfoil.

Limitations of explanations based on Bernoulli's principle

• The explanation above does not explain why the streamtubes change size. To see why the air flows the way it does requires more sophisticated analysis.

• Sometimes a geometrical argument is offered to demonstrate why the streamtubes change size: it is asserted that the top "obstructs" or "constricts" the air more than the bottom, hence narrower streamtubes. For conventional wings that are flat on the bottom and curved on top this makes some intuitive sense. But it does not explain how flat plates, symmetric airfoils, sailboat sails, or conventional airfoils flying upside down can generate lift, and attempts to calculate lift based on the amount of constriction do not predict experimental results.

• In deriving Bernoulli's principle, assumptions may be made (such as constant energy or incompressible flow) that are not applicable to real-world airfoils. For instance, a sailboat that is accelerating is removing energy from the flow while an airplane in level flight is adding energy to the flow, so energy is not constant. For high speed aircraft moving at transonic speeds the effects of compressibility can't be neglected.

• A common explanation using Bernoulli's principle asserts that the air must traverse both the top and bottom in the same amount of time and that this explains the increased speed on the (longer) top side of the wing. But this assertion is false; it is typically the case that the air parcels traveling over the upper surface will reach the trailing edge before those traveling over the bottom.

A more detailed physical description

Flow around an airfoil: the dots move with the flow. Note that the velocities are much higher at the upper surface than at the lower surface. The black dots are on timelines, which split into two – an upper and lower part – at the leading edge. The part of a timeline below the airfoil does not catch up with the one above. Colors of the dots indicate streamlines. The airfoil is a Kármán–Trefftz airfoil, with parameters μ_x = –0.08, μ_y = +0.08 and n = 1.94. The angle of attack is 8°, and the flow is a potential flow.

Lift is generated in accordance with the fundamental principles of physics. The most relevant physics reduce to three principles:

• Conservation of Momentum, which is a direct consequence of Newton's laws of motion, especially Newton's second law which relates the net force on an element of air to its rate of momentum change,
• Conservation of Mass, including the common assumption that the airfoil's surface is impermeable for the air flowing around, and
• Conservation of Energy, which says that energy is neither created nor destroyed.

In addition, one needs an expression relating the fluid stresses (consisting of pressure and shear stress components) to the properties of the flow. The pressure depends on the other flow properties, such as its mass density, through the (thermodynamic) equation of state, while the shear stresses are related to the flow through the air's viscosity.

Navier–Stokes and Euler equations

The above conditions can be expressed as partial differential equations that constrain the fluid flow. Unlike the physics of rigid bodies where the equations of motion describe the physical location of the rigid body, fluid dynamics describes the flow as a velocity field, i.e. a solution associates each point in space with a vector that represents the speed and direction of the flow at that point. Once the velocity field is solved for, other quantities of interest may be computed from the velocity field. For instance, Bernoulli's equation can be used to determine the pressure at each point on the airfoil from the speed of the velocity field. The pressures can then be "added up" (integrated) to determine the net aerodynamic force.

Application of the viscous shear stresses to Newton's second law for an airflow results in the Navier–Stokes equations. These equations are notoriously difficult to solve, but in many instances approximations suffice for a good description of lifting airfoils. In large parts of the flow viscosity may be neglected; such an inviscid flow can be described mathematically through the Euler equations, resulting from the Navier-Stokes equations when the viscosity is neglected. Neither the Navier-Stokes equations nor the Euler equations lend themselves to exact analytic solutions; usually engineers have to resort to numerical solutions to solve them, but Euler's equation can be solved by making further simplifying assumptions.

Potential flow, the Kutta condition, and circulation

File:Diagram showing superposition of uniform and vortex flow.jpg

Uniform flow plus vortex flow (circulation) gives the total flow below.

One further simplifying approximation is to assume that the flow is irrotational, i.e. that the flow does not rotate around itself. Mathematically, this is expressed by saying that the curl of the flow field is everywhere equal to zero. Irrotational flows have two nice properties: 1) an irrotational flow can be expressed as the gradient of a scalar function called a potential (the flow is then called potential flow) 2) solutions to complicated flows can be expressed as the sum of simpler flow fields. These two properties make the calculations tractable.

In particular, solutions can be expressed as the sum of a uniform flow (i.e. a steady flow equal to the free stream velocity) plus a free vortex flow, i.e. a circular flow around the airfoil with the speed inversely proportional to the radius. (Vortex flow is irrotational, except at the center of its circulation; since the center is inside the airfoil, the flow itself remains irrotational.) A vortex flow of any strength may be added to this uniform flow and the equation is solved, thus there are many flows that solve the Euler equations.

In order to arrive at a unique (physical) solution, one can apply the Kutta condition, which says that for steady flow the rear stagnation point is coincident with the trailing edge of the airfoil. The magnitude of the vortex flow is adjusted so that the Kutta condition is met. Another way to say this is that the airflow around the airfoil develops enough vorticity to satisfy the Kutta condition. Vorticity is measured by a number called the circulation, so an airfoil in a steady flow will develop sufficient circulation to satisfy the Kutta condition.

Streamlines and streamtubes around a NACA 0012 airfoil at moderate angle of attack. The stagnation line is the boundary between the two shaded streamtubes.

The image to the right shows the streamlines over a NACA 0012 airfoil computed using potential flow theory and applying the Kutta condition as a boundary condition. The stagnation streamline is the streamline that intersects the airfoil. In the front it typically intersects the arifoil on the lower surface near the leading edge. The stagnation streamline leaves the airfoil at the sharp trailing edge, according to the Kutta condition.

The physical interpretation of circulation is that the vortex flow adds to the uniform flow above the wing and subtracts from it below the wing. Thus, the speed of the air above the wing is higher than below it, as can be seen in the narrower streamtube. By Bernoulli's principle, this implies a pressure difference and therefore lift. The amount of lift is directly proportional to the circulation.

Flowfield formation

Consider the case of an airfoil accelerating from rest in a viscous flow. When there is no flow, there is no lift and the aerodynamic forces acting on the airfoil are zero. At the instant when the flow is “turned on”, the flow is undeflected downstream of the airfoil and there are two stagnation points on the airfoil (where the flow velocity is zero): one near the leading edge on the bottom surface, and another on the upper surface near the trailing edge. The dividing line between the upper and lower streamtubes intersects the body at the stagnation points. Since the flow speed is zero at these points, by Bernoulli's principle the static pressure at these points is at a maximum. As long as the second stagnation point is at its initial location on the upper surface of the wing, the circulation around the airfoil is zero and, in accordance with the Kutta–Joukowski theorem, there is no lift. The net pressure difference between the upper and lower surfaces is zero.

The effects of viscosity are contained within a thin layer of fluid called the boundary layer, close to the body. As flow over the airfoil commences, the flow along the lower surface turns at the sharp trailing edge and flows along the upper surface towards the upper stagnation point. The flow in the vicinity of the sharp trailing edge is very fast and the resulting viscous forces cause the boundary layer to accumulate into a vortex on the upper side of the airfoil between the trailing edge and the upper stagnation point. This is called the starting vortex. The starting vortex and the bound vortex around the surface of the wing are two halves of a closed loop. As the starting vortex increases in strength the bound vortex also strengthens, causing the flow over the upper surface of the airfoil to accelerate and drive the upper stagnation point towards the sharp trailing edge. As this happens, the starting vortex is shed into the wake, and is a necessary condition to produce lift on an airfoil. If the flow were stopped, there would be a corresponding "stopping vortex". Despite being an idealization of the real world, the “vortex system” set up around a wing is both real and observable; the trailing vortex sheet most noticeably rolls up into wing-tip vortices.

The upper stagnation point continues moving downstream until it is coincident with the sharp trailing edge (as stated by the Kutta condition). The flow downstream of the airfoil is deflected downward from the free-stream direction and, from the reasoning above in the basic explanation, there is now a net pressure difference between the upper and lower surfaces and an aerodynamic force is generated.

Other alternative explanations for the generation of lift

Many other alternative explanations for the generation of lift by an airfoil have been put forward, a few of which are presented here. Most of them are intended to explain the phenomenon of lift to a general audience. Although the explanations may share features in common with the explanations above, additional assumptions and simplifications may be introduced. This can reduce the validity of an alternative explanation to a limited sub-class of lift generating conditions, or might not allow a quantitative analysis. Several theories introduce assumptions which proved to be wrong, like the equal transit-time theory.

"Popular" explanation based on equal transit-time

An illustration of the (incorrect) equal transit-time explanation of airfoil lift.

An explanation of lift frequently encountered in basic or popular sources is the equal transit-time theory. Equal transit-time states that because of the longer path of the upper surface of an airfoil, the air going over the top must go faster in order to catch up with the air flowing around the bottom, i.e. the parcels of air that are divided at the leading edge and travel above and below an airfoil must rejoin when they reach the trailing edge. Bernoulli's Principle is then cited to conclude that since the air moves faster on the top of the wing the air pressure must be lower. This pressure difference pushes the wing up.

However, the hypothesis of equal transit time has no basis in theory and is universally contradicted by experiment. Although it is true that the air moving over the top of a wing generating lift does move faster, there is no requirement for equal transit time. In fact the air moving over the top of an airfoil generating lift moves much faster than the equal transit theory would imply.

The assertion that the air must arrive simultaneously at the trailing edge is sometimes referred to as the "Equal Transit-Time Fallacy".

Coandă effect

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In a limited sense, the Coandă effect refers to the tendency of a fluid jet to stay attached to an adjacent surface that curves away from the flow, and the resultant entrainment of ambient air into the flow. The effect is named for Henri Coandă, the Romanian aerodynamicist who exploited it in many of his patents.

More broadly, some consider the effect to include the tendency of any fluid boundary layer to adhere to a curved surface, not just the boundary layer accompanying a fluid jet. It is in this broader sense that the Coandă effect is used by some to explain why the air flow remains attached to the top side of an airfoil. Jef Raskin, for example, describes a simple demonstration, using a straw to blow over the upper surface of a wing. The wing deflects upwards, thus demonstrating that the Coandă effect creates lift. This demonstration correctly demonstrates the Coandă effect as a fluid jet (the exhaust from a straw) adhering to a curved surface (the wing). However, the upper surface in this flow is a complicated, vortex-laden mixing layer, while on the lower surface the flow is quiescent. The physics of this demonstration are very different from that of the general flow over the wing. The usage in this sense is encountered in some popular references on aerodynamics. In the aerodynamics field, the Coandă effect is commonly defined in the more limited sense above and viscosity is used to explain why the boundary layer attaches to the surface of a wing.

Methods to determine lift on an airfoil

Lift coefficient

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If the lift coefficient for a wing at a specified angle of attack is known (or estimated using a method such as thin airfoil theory), then the lift produced for specific flow conditions can be determined using the following equation:

$L={\tfrac {1}{2}}\rho v^{2}AC_{L}$

where

• L is lift force,
• ρ is air density
• v is true airspeed,
• A is planform area, and
• $C_{L}$ is the lift coefficient at the desired angle of attack, Mach number, and Reynolds number

Kutta–Joukowski theorem

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Lift can be calculated using potential flow theory by imposing a circulation. It is often used by practising aerodynamicists as a convenient quantity in calculations, for example thin-airfoil theory and lifting-line theory.

The circulation $\Gamma$ is the contour integral of the tangential velocity of the air on a closed loop (also called a 'circuit') around the boundary of an airfoil. It can be understood as the total amount of "spinning" (or vorticity) of air around the airfoil. The section lift/span $L'$ can be calculated using the Kutta–Joukowski theorem:

$L'=\rho v\Gamma \,$

where $\rho$ is the air density, $v$ is the free-stream airspeed. Kelvin's circulation theorem states that circulation is conserved for any circuit moving with the fluid. When an aircraft is at rest, circulation is zero.

The challenge when using the Kutta–Joukowski theorem to determine lift is to determine the appropriate circulation for a particular airfoil. In practice, this is done by applying the Kutta condition, which uniquely prescribes the circulation for a given geometry and free-stream velocity.

A physical understanding of the theorem can be observed in the Magnus effect, which is a lift force generated by a spinning cylinder in a freestream. Here the necessary circulation is induced by the mechanical rotation acting on the boundary layer, causing it to induce a faster flow around one side of the cylinder and a slower flow around the other. The asymmetric distribution of airspeed around the cylinder then produces a circulation in the outer inviscid flow.

Pressure integration

The force on the wing can be examined in terms of the pressure differences above and below the wing, which can be related to velocity changes by Bernoulli's principle.

The total lift force is the integral of vertical pressure forces over the entire wetted surface area of the wing:

$L=\oint p\mathbf {n} \cdot \mathbf {k} \;\mathrm {d} A,$

where:

• L is the lift,
• A is the wing surface area
• p is the value of the pressure,
• n is the normal unit vector pointing into the wing, and
• k is the vertical unit vector, normal to the freestream direction.

The above lift equation neglects the skin friction forces, which typically have a negligible contribution to the lift compared to the pressure forces. By using the streamwise vector i parallel to the freestream in place of k in the integral, we obtain an expression for the pressure drag D_p (which includes induced drag in a 3D wing). If we use the spanwise vector j, we obtain the side force Y.

${\begin{aligned}D_{p}&=\oint p\mathbf {n} \cdot \mathbf {i} \;\mathrm {d} A,\\[1.2ex]Y&=\oint p\mathbf {n} \cdot \mathbf {j} \;\mathrm {d} A.\end{aligned}}$

One method for calculating the pressure is Bernoulli's equation, which is the mathematical expression of Bernoulli's principle. This method ignores the effects of viscosity, which can be important in the boundary layer and to predict friction drag, which is the other component of the total drag in addition to D_p.

The Bernoulli principle states that the sum total of energy within a parcel of fluid remains constant as long as no energy is added or removed. It is a statement of the principle of the conservation of energy applied to flowing fluids.

A substantial simplification proposes that other forms of energy changes are inconsequential during the flow of air around a wing and that energy transfer in/out of the air is not significant, so the sum of pressure energy and speed energy for any particular parcel of air must be constant. Consequently, an increase in speed must be accompanied by a decrease in pressure and vice-versa. It is named for the Dutch-Swiss mathematician and scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others.

Bernoulli's principle provides an explanation of pressure difference in the absence of air density and temperature variation (a common approximation for low-speed aircraft). If the air density and temperature are the same above and below a wing, a naive application of the ideal gas law requires that the pressure also be the same. Bernoulli's principle, by including air velocity, explains this pressure difference. The principle does not, however, specify the air velocity. This must come from another source, e.g., experimental data.

In order to solve for the velocity of inviscid flow around a wing, the Kutta condition must be applied to simulate the effects of viscosity. The Kutta condition allows for the correct choice among an infinite number of flow solutions that otherwise obey the laws of conservation of mass and conservation of momentum.

Lift forces on bluff bodies

Template:Further2 The flow around bluff bodies – i.e. without a streamlined shape, or stalling airfoils – may also generate lift, besides a strong drag force. This lift may be steady, or it may oscillate due to vortex shedding. Interaction of the object's flexibility with the vortex shedding may enhance the effects of fluctuating lift and cause vortex-induced vibrations. For instance, the flow around a circular cylinder generates a Kármán vortex street: vortices being shed in an alternating fashion from each side of the cylinder. The oscillatory nature of the flow is reflected in the fluctuating lift force on the cylinder, whereas the mean lift force is negligible. The lift force frequency is characterised by the dimensionless Strouhal number, which depends (among others) on the Reynolds number of the flow.

For a flexible structure, this oscillatory lift force may induce vortex-induced vibrations. Under certain conditions – for instance resonance or strong spanwise correlation of the lift force – the resulting motion of the structure due to the lift fluctuations may be strongly enhanced. Such vibrations may pose problems and threaten collapse in tall man-made structures like industrial chimneys.

See also

Organisational Psychologist Alfonzo Lester from Timmins, enjoys pinochle, property developers in new launch singapore property and textiles. Gets motivation through travel and just spent 7 days at Alejandro de Humboldt National Park.

• Aerodynamic force
• Banked turn
• Circulation control wing
• Downforce
• Drag (physics)
• Foil (fluid mechanics)
• Küssner effect
• Kutta condition
• Kutta–Joukowski theorem
• Lift-induced drag
• Lift-to-drag ratio
• Lifting-line theory

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References and notes

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Further reading

• Introduction to Flight, John D. Anderson, Jr., McGraw-Hill, ISBN 0-07-299071-6 – The author is the Curator of Aerodynamics at the Smithsonian Institution's National Air & Space Museum and Professor Emeritus at the University of Maryland.
• Understanding Flight, by David Anderson and Scott Eberhardt, McGraw-Hill, ISBN 0-07-136377-7 – The authors are a physicist and an aeronautical engineer. They explain flight in non-technical terms and specifically address the equal-transit-time myth. Turning of the flow around the wing is attributed to the Coanda effect, which is quite controversial.
• Aerodynamics, Clancy, L.J. (1975), Section 4.8, Pitman Publishing Limited, London ISBN 0-273-01120-0.
• Aerodynamics, Aeronautics, and Flight Mechanics, McCormick, Barnes W., (1979), Chapter 3, John Wiley & Sons, Inc., New York ISBN 0-471-03032-5.
• Fundamentals of Flight, Richard S. Shevell, Prentice-Hall International Editions, ISBN 0-13-332917-8 – This book is primarily intended as a text for a one semester undergraduate course in mechanical or aeronautical engineering, although its sections on theory of flight are understandable with a passing knowledge of calculus and physics.
• "Observation of Perfect Potential Flow in Superfluid", Paul P. Craig and John R. Pellam (1957) Physical Review 108(5), pp. 1109–1112, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park. – Experiments under superfluidity conditions, resulting in the vanishing of lift in inviscid flow since the Kutta condition no longer is satisfied.
• "Aerodynamics at the Particle Level", Charles A. Crummer (2005, revised 2012) – A treatment of aerodynamics emphasizing the particle nature of air, as opposed to the fluid approximation commonly used. http://arxiv.org/pdf/nlin/0507032.pdf
• "Flight without Bernoulli" Chris Waltham Vol. 36, Nov. 1998 THE PHYSICS TEACHER – using a physical model relying only on Newton’s second law, the author presents a rigorous fluid dynamical treatment of flight. http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf
• Bernoulli, Newton, and Dynamic Lift Norman F. Smith School Science and Mathematics vol 73 Part I: http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb09040.x/pdf Part II http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb08998.x/pdf

External links

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I such as you to be, am a brand new projects specialist. You've got the conception that new tasks personnel should be showflat certain. Should you're eager, let me train you the right way to master the entire show flats island vast as a substitute of getting to stay just at 1 place. Is that attainable you may ask, well, I've achieved it in 6 months, you can too. Which company is well-recognized and is actually dedicated for developing rookie within the industry in venture sales market with success? Can a rookie join the company's core group from day one? I wish to propose a third class, which I have been grooming my agents in the direction of, and that is as a Huttons agent, you will be able to market and have knowledge of ALL Huttons projects, and if essential, projects exterior of Huttons as properly.

GPS has assembled a high workforce of personnel who are additionally well-known figures in the native actual property scene to pioneer this up-and-coming organization. At GPS Alliance, WE LEAD THE WAY! Many people have asked me how I managed to earn S$114,000 from my sales job (my third job) at age 24. The reply is easy. After graduation from NUS with a Historical past diploma, my first job was in actual estate. Within the ultimate part of this series, I interview one of the top agents in ERA Horizon Group and share with you the secrets to his success! Learn it RIGHT HERE

Notice that the application must be submitted by the appointed Key Government Officer (KEO) such as the CEO, COO, or MD. Once the KEO has submitted the mandatory paperwork and assuming all documents are in order, an email notification shall be sent stating that the applying is permitted. No hardcopy of the license might be issued. A delicate-copy could be downloaded and printed by logging into the CEA website. It takes roughly four-6 weeks to course of an utility.

• Discussion of the apparent "conflict" between the various explanations of lift
• NASA tutorial, with animation, describing lift
• NASA FoilSim II 1.5 beta. Lift simulator.
• Explanation of Lift with animation of fluid flow around an airfoil
• A treatment of why and how wings generate lift that focuses on pressure.
• Physics of Flight – reviewed. Online paper by Prof. Dr. Klaus Weltner.
• How do Wings Work? – Holger Babinsky
• Plane and Pilot Magazine Bernoulli Or Newton: Who's Right About Lift?
• One Minute Physics How Does a Wing actually work? You Tube video