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Euler equations (fluid dynamics)
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==== Streamline curvature theorem ==== [[File:Streamlines around a NACA 0012.svg|frame|right| The "Streamline curvature theorem" states that the pressure at the upper surface of an airfoil is lower than the pressure far away and that the pressure at the lower surface is higher than the pressure far away; hence the pressure difference between the upper and lower surfaces of an airfoil generates a lift force. ]] Let <math>r</math> be the distance from the center of curvature of the streamline, then the second equation is written as follows: <math display="block"> \frac{\partial p}{\partial r} = \rho \frac{u^2}{r}~(>0), </math> where <math>{\partial / \partial r} = -{\partial /\partial n}.</math> This equation states:<blockquote> ''In a steady flow of an [[inviscid]] [[fluid]] without external forces, the [[center of curvature]] of the streamline lies in the direction of decreasing radial pressure.'' </blockquote> Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature.{{sfn|Babinsky|2003}} Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".{{sfn|Imai|1973|p=}} This "theorem" explains clearly why there are such low pressures in the centre of [[vortex|vortices]],{{sfn|Babinsky|2003}} which consist of concentric circles of streamlines. This also is a way to intuitively explain why airfoils generate [[lift (force)|lift forces]].{{sfn|Babinsky|2003}}
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