R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 25
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(7) transform into^=(9)aiU+aV2y'^d=β,ϋ+ A7,Vwhere αϊ , a ,2βι , and β are known functions of £ and 77. Either U or V2can be eliminated from this system. If we differentiate the first equationwith respect to η, and substitute for dV/dη its value from the second equation and for V its value from the first equation, we obtaindUΛdη d£\2^, 1 da \dU2a δηdη // d£22,dU+dηon —.hI οίφι-if α does not vanish. If a22+-—α2) U,ση /= 0, the first equation (9) is already an equa2tion for U, and U will also satisfy(10a)αφση\d *^2dη d£=^ dU _j_ da±dηδη^126II. G E N E R A L T H E O R E M SSimilarly one can obtain a linear second-order partial differential equationof this type for V.Under the hypotheses of theorem A the values of u and ν are given alongan arc ABin the x,y-pl&ne which is nowhere tangent to a characteristic.Under the transformation (6) the arc AB is mapped into an arc AB in the£,77-plane, which can never have a horizontal or vertical tangent (see Fig.48).
T h e values of u and υ given on A Β are the values of u and ν on AB,and these determine the values ofU and V along AB,by(8). Thevalues of U along AB determine the derivative of U in the direction tangent to the curve, which, by hypothesis, is never the ξ-direetion; the firstof Eqs. (9) gives dU/θξ along AB.From the derivatives in t w o differentdirections, the derivative in any direction can be found, and in particulardU/θη. Thus the problem of finding U may be stated as follows: U satisfiesa differential equation of the form(11)£(U)m£%+ a % + b f . +σξ σησζσηcU-0,where a, b, and c are functions of ξ and η [not to be confused with thea, b in Eqs.
(1)], and the values of U, dU/θξ, and dU/θη are given at allpoints of an arc AB which at no point has its tangent parallel to either ofthe axes. T h e problem is the same for V. T h e problem is solved if it can beshown how to find the value of U at a point C which has the same abscissa as Β and the same ordinate as AACBD(or vice versa). T h e rectanglein the £,TJ-plane corresponds to the curvilinear quadrangleACBDin the x,y-p\&ne.
I t then follows that U may be computed at any point Ρinterior to this rectangle, using the given data on an appropriate segmentofAB.T h e solution of this problem was given in 1860 by Bernhard Riemann,who laid the foundation for the analytic theory of supersonic flow.F I G . 48. Cauchy problem in characteristic plane.10.55. Riemann's solutionRIEMANN'S127SOLUTION47T h e solution given by Riemann for Eq.
(11), subject to the conditionsgiven along AB, is based on the use of a second function Ω(£,?7), whichsatisfies another differential equation, the so-called adjoint equation, namely,(12)ΞβίΠΓν ' θϊ - T + ( ~ * ~ 6η) - °'αbcΩvwhere α, 6, and c are the same as in Eq. (11). I t may be noted that theadjoint of (12) is exactly (11).For any two differentiable functions t/(£,*?) and Ώ(ξ,η) we consider thefollowing expressions:(13)'-ssWW-Ks-*)-Differentiating these expressions, adding and subtracting the productand collecting terms, we find thatσξcUil.σηwhere £(U) has been written for the left-hand member of Eq. (11) and9ΪΙ(Ω) for the left-hand member of Eq.
(12). N o w if U satisfies (11) andΩ satisfies (12), the right-hand member of Eq. (14) vanishes. L e t acurve G bound an area A in the £,rj-plane in which both equations aresatisfied. If Stokes' Theorem (Sec. 6.1) is applied to this area for the vector ( - F , X , 0 ) , we find(15)' f & d , - Y d &= //(f+ ^)<M=0with the line integral taken around 6. T h e same result can be obtained byspecializing the Divergence Theorem (2.27) to two dimensions and applying it to the vector(X,Y).This formula (15) will be applied to the circuit ABC (Fig. 48). First,however, we shall prescribe boundary conditions for Ω, which so far issubject only to the differential equation (12). If (ξι ,ηι) are the coordinatesof C, we shall require that(16)= 6Ωalong AC.= αΩalongO(C) = O f e , i h ) . = 1,θηCE.128II.
GENERALTHEOREMST h e function Ω depending on the four variables £, η, £ι, r?i, which satisfiesEqs. (12) and (16), is known as the Riemann function of the problem (11).I t depends only on the coefficients of the differential equation (11) and isindependent of prescribed boundary data for U. I t may therefore be determined once and for all for a given equation £(U)= 0. On account of(16) the expressions for X and Y are considerably simpler on the straightline parts of 6 ; along CA, for example, we have Υ = £θ([/Ω)/θξ. For thissame line, άη = 0 so that the part of the integral in (15) along CA is2 [JcA(-Y)άξ =Γ£ (UQ) dH = U(C)σξJA-U(A)V(A).Similarly, the part of the integral along EC is2 [°Χάη=JBΓJB- (UQ)θηdv= U(C)-U(B)Q(E).When these values are inserted in (15) and the equation solved forU(C),there results(17)U(C)= I [U(A)U(A)Δ+U(B)Q(B)]-Γ(Χ άη — Υάξ)?J Αwhere the last integral is a line integral along the curve AB.
If the solutionΩ of E q . (12), under the boundary conditions (16), has been found, thenformula (17) gives an expression for U, using the values of Ω and itsderivatives as well as the prescribed values of U and its derivates alongAB.Assuming the existence of Ω, it is possible (though lengthy) to verifythat the formula (17) actually does satisfy Eq. (11) and takes the requiredvalues along AB.49T h e proof is now complete except for the construction of the Riemannfunction Ω, corresponding to Eq.
(11). T h e boundary conditions for Ω arerelatively simple. In some simple cases Ω can be given explicitly, and ingeneral, Ω can be constructed by the method of successive approximations.In the same way the function V can also be found anywhere in the rectangle ABCD.From U and V the solutions u and ν are found in termsof ξ and η, by means of ( 8 ) , everywhere in ABCD or, in terms of χ and y,everywhere in the quadrangle bounded by the characteristics through Aand B. Thus Riemann's formula (17) leads to a complete proof of theorem A [which establishes the existence and uniqueness of the solution ofthe system (1) in the characteristic quadrangle for given values of u andν along AB] provided that the system (1) is linear.5010.6I N T E R C H A N G E OF129VARIABLES6.
Interchange of variablesIf the right-hand members of the two equations (1) are zero, a v e r yefficient transformation may be applied, one which will be used extensivelylater on. T h e essential idea of the transformation is interchanging the rolesof the dependent and independent variables.If two variables u and ν depend on two independent variables χ and y,e.g., by Eqs. (1) and appropriate boundary conditions, the relationshipmay be considered as a correspondence between a point P(x y)in thex,?/-plane and a point Q(u,v) in the w,i>-plane.
(See Fig. 49.) A displacement of Ρ in the x,?/-plane causes a corresponding displacement of Q. T h esame relationship can be considered in reverse, as a correspondence between Q and P ; then χ and y are considered functions of u and v. A n yfunction Φ of χ and y may equally well be considered a function of u and v.T h e differential of Φ, for corresponding displacements of Ρ and Q, may bewritten in the two formsydΦdΦdΦdΦάΦ = — dx + — dy = — du + — dv.dxdydudv(18)In additiondx, dx ,dx — — du + — dv.dudv77dy, dydy = — du + — dv.dudv77ΊNext, these expressions for dx and dy are substituted in (18), and two particular displacements, one with dv = 0 and the other with du = 0, areconsidered. W e see that this leads to the equations(19)^ ^ _|_ ^ ^2/ — ^dx dudy duduX1dxdx dvdΦ dy _ dΦdy dvdv 'T h e pair of equations (19) can be solved for dΦ/dx and dΦ/dy wheneverthe determinant of coefficientsj_dx dy _ dx dydu dvdv dudoes not vanish, yielding(20)- (—^— =dxJ \du dvdv du/Jdy= IdΦ dxf'J\ dv dudΦ dx^du dv )I t should be noted that J is independent of Φ.In the particular cases Φ = u and Φ = v, respectively, these solutions(20) lead to the formulas130, .^II.
G E N E R A Ldu _ 1 dydx ~ JdvTHEOREMSdu _ _]_dxdy ~Tdv'1<3y __ J _ θ|/dx"J du'dv_ _ \^dx~dy~~ 7du'When these substitutions are made in Eqs. (1) for the case where the righthand members are zero, the factor 1 /J may be divided out, and we obtainαϊ(22)dydx, dybi-dxΛα - — j -α —, dxιdy j.o — +du7 dxo — =0.du02dvdyα 2dvdvdv3du34du= U,4This is again a planar system, with x,y as unknowns and u,v as independentvariables. This interchange of variables in (1) is possible only if the righthand members of Eqs. (1) vanish.T h e importance of this transformation in fluid mechanics lies in the factthat in actual applications the coefficients a\, · · · , 6 in Eqs.
(1) are oftenfunctions of u and ν only, and do not involve χ and y. These same coefficientsoccur in Eqs. (22), which is then a linear system in the new independentvariables u and v. T h e advantages of this situation are obvious.I n obtaining Eqs. (20) to (22) it was assumed that the determinant:4(23)jwas different from zero.= ^ i ^ - ^ ^=dudvdv dud(u,v)51From (21) and (23) it follows also that:· _ dudv^dx dydudvdy dx_ d(u,v)d(x,y)_1J'T h e geometrical significance of these Jacobian determinants, or Jacobians,is well known: when Ρ is displaced so as to cover a rectangle of area dx dy,the corresponding point Q covers a curvilinear '"parallelogram" of areajdxdy;conversely, a rectangular area dudv about Q corresponds to anarea J du dv about P.
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