Editing Lift (force) (section)

===Pressure differences===

[[Pressure]] is the [[Stress (mechanics)#Normal and shear stresses|normal force]] per unit area exerted by the air on itself and on surfaces that it touches. The lift force is transmitted through the pressure, which acts perpendicular to the surface of the airfoil. Thus, the net force manifests itself as pressure differences. The direction of the net force implies that the average pressure on the upper surface of the airfoil is lower than the average pressure on the underside.<ref>A uniform pressure surrounding a body does not create a net force. (See [[buoyancy]]). Therefore pressure differences are needed to exert a force on a body immersed in a fluid. For example, see: {{Citation|first=G.K.|last=Batchelor|author-link=George Batchelor|title=An Introduction to Fluid Dynamics|year=1967|publisher=Cambridge University Press|isbn=978-0-521-66396-0|pages=14–15}}</ref>

These pressure differences arise in conjunction with the curved airflow. When a fluid follows a curved path, there is a pressure [[gradient]] perpendicular to the flow direction with higher pressure on the outside of the curve and lower pressure on the inside.<ref>"''...if a streamline is curved, there must be a pressure gradient across the streamline...''" {{citation|journal=Physics Education|first=Holger|last=Babinsky|date=November 2003|title=How do wings work?|doi=10.1088/0031-9120/38/6/001|bibcode=2003PhyEd..38..497B|volume=38|issue=6|page=497|s2cid=1657792 }}</ref> This direct relationship between curved streamlines and pressure differences, sometimes called the [[Euler equations (fluid dynamics)#Streamline curvature theorem|streamline curvature theorem]], was derived from Newton's second law by [[Leonhard Euler]] in 1754:

:<math>\frac{\operatorname{d}p}{\operatorname{d}R}= \rho \frac{v^2}{R} </math>

The left side of this equation represents the pressure difference perpendicular to the fluid flow. On the right side of the equation, ρ is the density, v is the velocity, and R is the radius of curvature. This formula shows that higher velocities and tighter curvatures create larger pressure differentials and that for straight flow (R → ∞), the pressure difference is zero.<ref>Thus a distribution of the pressure is created which is given in Euler's equation. The physical reason is the aerofoil which forces the streamline to follow its curved surface. The low pressure at the upper side of the aerofoil is a consequence of the curved surface." ''A comparison of explanations of the aerodynamic lifting force'' Klaus Weltner Am. J. Phys. Vol.55 No.January 1, 1987, p. 53 [http://aapt.scitation.org/doi/pdf/10.1119/1.14960] {{Webarchive|url=https://web.archive.org/web/20210428194849/https://aapt.scitation.org/doi/pdf/10.1119/1.14960|date=April 28, 2021}}</ref>