Here, is the speed of sound, determined by the equation of state of the fluid and conservation of energy. For polytropic gases, we have , where is the specific heat ratio and is the stagnation enthalpy, in which case the Chaplygin's equation reduces to
The Bernoulli equation (see the derivation below) states that maximum velocity occurs when specific enthalpy is at the smallest value possible; one can take the specific enthalpy to be zero corresponding to absolute zero temperature as the reference value, in which case is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.[2][3]
with the equation of state acting as third equation. Here is the stagnation enthalpy, is the magnitude of the velocity vector and is the entropy. For isentropic flow, density can be expressed as a function only of enthalpy , which in turn using Bernoulli's equation can be written as .
Since the flow is irrotational, a velocity potential exists and its differential is simply . Instead of treating and as dependent variables, we use a coordinate transform such that and become new dependent variables. Similarly the velocity potential is replaced by a new function (Legendre transformation)[4]
such then its differential is , therefore
Introducing another coordinate transformation for the independent variables from to according to the relation and , where is the magnitude of the velocity vector and is the angle that the velocity vector makes with the -axis, the dependent variables become
The continuity equation in the new coordinates become
For isentropic flow, , where is the speed of sound. Using the Bernoulli's equation we find
chaplygin, equation, dynamics, named, after, sergei, alekseevich, chaplygin, 1902, partial, differential, equation, useful, study, transonic, flow, displaystyle, frac, partial, partial, theta, frac, frac, partial, partial, frac, partial, partial, here, display. In gas dynamics Chaplygin s equation named after Sergei Alekseevich Chaplygin 1902 is a partial differential equation useful in the study of transonic flow 1 It is 2 F 8 2 v 2 1 v 2 c 2 2 F v 2 v F v 0 displaystyle frac partial 2 Phi partial theta 2 frac v 2 1 v 2 c 2 frac partial 2 Phi partial v 2 v frac partial Phi partial v 0 Here c c v displaystyle c c v is the speed of sound determined by the equation of state of the fluid and conservation of energy For polytropic gases we have c 2 g 1 h 0 v 2 2 displaystyle c 2 gamma 1 h 0 v 2 2 where g displaystyle gamma is the specific heat ratio and h 0 displaystyle h 0 is the stagnation enthalpy in which case the Chaplygin s equation reduces to 2 F 8 2 v 2 2 h 0 v 2 2 h 0 g 1 v 2 g 1 2 F v 2 v F v 0 displaystyle frac partial 2 Phi partial theta 2 v 2 frac 2h 0 v 2 2h 0 gamma 1 v 2 gamma 1 frac partial 2 Phi partial v 2 v frac partial Phi partial v 0 The Bernoulli equation see the derivation below states that maximum velocity occurs when specific enthalpy is at the smallest value possible one can take the specific enthalpy to be zero corresponding to absolute zero temperature as the reference value in which case 2 h 0 displaystyle 2h 0 is the maximum attainable velocity The particular integrals of above equation can be expressed in terms of hypergeometric functions 2 3 Derivation EditFor two dimensional potential flow the continuity equation and the Euler equations in fact the compressible Bernoulli s equation due to irrotationality in Cartesian coordinates x y displaystyle x y involving the variables fluid velocity v x v y displaystyle v x v y specific enthalpy h displaystyle h and density r displaystyle rho are x r v x y r v y 0 h 1 2 v 2 h o displaystyle begin aligned frac partial partial x rho v x frac partial partial y rho v y amp 0 h frac 1 2 v 2 amp h o end aligned with the equation of state r r s h displaystyle rho rho s h acting as third equation Here h o displaystyle h o is the stagnation enthalpy v 2 v x 2 v y 2 displaystyle v 2 v x 2 v y 2 is the magnitude of the velocity vector and s displaystyle s is the entropy For isentropic flow density can be expressed as a function only of enthalpy r r h displaystyle rho rho h which in turn using Bernoulli s equation can be written as r r v displaystyle rho rho v Since the flow is irrotational a velocity potential ϕ displaystyle phi exists and its differential is simply d ϕ v x d x v y d y displaystyle d phi v x dx v y dy Instead of treating v x v x x y displaystyle v x v x x y and v y v y x y displaystyle v y v y x y as dependent variables we use a coordinate transform such that x x v x v y displaystyle x x v x v y and y y v x v y displaystyle y y v x v y become new dependent variables Similarly the velocity potential is replaced by a new function Legendre transformation 4 F x v x y v y ϕ displaystyle Phi xv x yv y phi such then its differential is d F x d v x y d v y displaystyle d Phi xdv x ydv y therefore x F v x y F v y displaystyle x frac partial Phi partial v x quad y frac partial Phi partial v y Introducing another coordinate transformation for the independent variables from v x v y displaystyle v x v y to v 8 displaystyle v theta according to the relation v x v cos 8 displaystyle v x v cos theta and v y v sin 8 displaystyle v y v sin theta where v displaystyle v is the magnitude of the velocity vector and 8 displaystyle theta is the angle that the velocity vector makes with the v x displaystyle v x axis the dependent variables become x cos 8 F v sin 8 v F 8 y sin 8 F v cos 8 v F 8 ϕ F v F v displaystyle begin aligned x amp cos theta frac partial Phi partial v frac sin theta v frac partial Phi partial theta y amp sin theta frac partial Phi partial v frac cos theta v frac partial Phi partial theta phi amp Phi v frac partial Phi partial v end aligned The continuity equation in the new coordinates become d r v d v F v 1 v 2 F 8 2 r v 2 F v 2 0 displaystyle frac d rho v dv left frac partial Phi partial v frac 1 v frac partial 2 Phi partial theta 2 right rho v frac partial 2 Phi partial v 2 0 For isentropic flow d h r 1 c 2 d r displaystyle dh rho 1 c 2 d rho where c displaystyle c is the speed of sound Using the Bernoulli s equation we find d r v d v r 1 v 2 c 2 displaystyle frac d rho v dv rho left 1 frac v 2 c 2 right where c c v displaystyle c c v Hence we have 2 F 8 2 v 2 1 v 2 c 2 2 F v 2 v F v 0 displaystyle frac partial 2 Phi partial theta 2 frac v 2 1 frac v 2 c 2 frac partial 2 Phi partial v 2 v frac partial Phi partial v 0 See also EditEuler Tricomi equationReferences Edit Wikimedia Commons has media related to Chaplygin equation Chaplygin S A 1902 On gas streams Complete collection of works Russian Izd Akad Nauk SSSR 2 Sedov L I 1965 Two dimensional problems in hydrodynamics and aerodynamics Chapter X Von Mises R Geiringer H amp Ludford G S S 2004 Mathematical theory of compressible fluid flow Courier Corporation Landau L D Lifshitz E M 1982 Fluid Mechanics 2 ed Pergamon Press p 432 Retrieved from https en wikipedia org w index php title Chaplygin 27s equation amp oldid 1054968613, wikipedia, wiki, book, books, library,