R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 26
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T h e case J — oo (i.e., j = 0) means that an areaaround Ρ is mapped into a single arc through Q, while if J vanishes, an areaabout Q corresponds to an arc at P. ( I n either case the arc may degenerateto a single point.) I n both cases, therefore, the transformation degeneratesand the two systems (1) and (22) are no longer equivalent.
T h e geometricalsignificance of these exceptional cases will be discussed in the followingsection, while the physical meaning of the transformation in general willappear later (e.g., Sec. 12.3).7. Geometrical interpretationWhen the correspondence between the x,i/-plane and the w,i;-plane isdefined by differentiable functions u(x,y), v(x,y), or x(u,v), y(u,v), then10.7GEOMETRICAL INTERPRETATION131in the infinitesimal neighborhoods of a pair of corresponding points Ρ andQ the mapping can be considered a linear (affine) transformation:du = a n dx+ai2dy(25)dx = βηdudy2Χdu + β+β dvΧ2ordv = a i dx + a 2 dy22β=22dv.Here the coefficients a and β are, of course, partial derivatives:,.dw(26)α π = - ,α12ax= - ,•··,ax£π = - ,ft,= - ,· · ·,and the determinant J of (23) becomesJ = ftife - Α Λ =(27)«n«22 — «ι α ι22Either set a or β determines the other set, from the formulas (21). Thus,the specifying of a set of four independent parameters determines the affinetransformation.
T h e system (1) of t w o planar differential equations, whichcan now be writtenαι<2ιι +α2αΐ2 +α^α ι +aabiaba+6 a i +6 a2A22=a,=6,(28)n+2i232422serves to reduce the number of independent parameters α by t w o ; i.e., thedifferential equations single out a subset of qo transformations from the2set of o o possible transformations. T h e equations (28) are therefore not suf4ficient to determine completely the mapping of the neighborhood of Ρ ontothe neighborhood of Q. ( I t is different in the case of one-dimensional problems, where there is only one parameter t o be determined and one differentialdetermines α inequation to determine i t : du/dx = f(x,u)a=a dx as= /.) T w o out of the four parameters can still be chosen.
This may bedone by supposing that t w o neighboring pairs of correspondingP(x,y),Q(u,v)pointsand P\(x\ ,2/1), Q\{u\ ,vi) are known (see Fig. 4 9 ) ; this isequivalent to supposing that for one special set of values dx\ , dyi (notboth zero), the corresponding values du\ , dv\ are given, or(29)a n dxi +ai2dyi = dui ,a2 idxi + a2 2dyi = dv .xTogether, Eqs. (28) and (29) give four linear equations for the four parameters a. One must then consider whether this system determines the parameters in question.
A unique solution is obtained if and only if the determinant of coefficients is different from zero. Taking φ as the angle betweenthe x-direction and Ρ Pi , we set dx\ = ds cos φ and dyi = ds sin φ. Thenthe determinant of coefficients of our system (28), (29), namely,II. GENERAL132THEOREMSαϊabibdxidyi000dxi22a3b30,40is exactly the right-hand member of Eq.
( 2 ' ) except for a factor — (ds) ,2with the a's and b's evaluated for the given values of the coordinates ofP(x,y)andQ(u,v).YνPIo,of•uF I G . 49. Increments in corresponding planes.Thus, in the elliptic case, when Eq. ( 2 ' ) has no real roots, the equations(28) and (29) always determine the parameters a for arbitrary values ofdux and dvi, i.e., PPi and QQi may be chosen arbitrarily. In the hyperboliccase, however, there are two characteristic directions through P , determined by Eq.
( 2 ' ) , for the given correspondence P , Q. If P P i is not one ofthese directions, the equations can be solved, as above, for arbitrary Qi .But if P P i is a characteristic, say a σ , the four equations (28) and ( 2 9 ) ,whose determinant vanishes, are in general (i.e., for arbitrary right sides)not consistent. T h e y will be consistent only for particular right sidessuch that the four equations become linearly dependent. In this case adefinite relation must exist between dui and dth . I t is obvious that thisrelation must be the compatibility condition. T o show it formally write inEqs. (28) and (29), for brevity, X\ , x , Xz, X\ instead of a n , an , «21 , a .Then, exactly as on p.
119, multiply the first Eq. (28) by Bi (with φ = φ ) ,the second by —Α , and add. In view of the identities (4a) and (4c) thisgives immediately+2 22+ΛΔι (χι2cos φ++ x sin φ ) + K(x+23cos φThen multiply the first Eq. (29) by ΔΧ 2++ x sin φ ) = Β α+4— Aib.λ, the second by K, and add; theresult is:Δι (χι cos φ2++x2sin φ )++K(x3cos φ++z4sin φ )+= ^ Δ ι2+Κ.10.7GEOMETRICALINTERPRETATION133These equations are consistent only if(30)f LdaAu± Lda++= BKia-A.b,+and this is the first Eq. (5) for the σ ^ ΐ Γ β ^ ί ο η . Hence, if PP\ has the σ direction, Eq.
(30) must hold for the increments dui and dvi of QQi, inorder to make the equations consistent. In this case there exists a oneparameter family of solutions a .++ikT h e geometric interpretation can be followed up further in the casewhere the right sides of the given equations vanish: a = b = 0. Then thecondition (30) leads to±/->i\(31)tadviΔ1ιn*=^=2- X 'where ψ is the angle between the ^-direction and QQi . On using the secondof the expressions in (4c) for K, Eq. (31) may be written(32)Δcot φ— Δ1 2 cot φ =24Δη ,where φ corresponds to ψ* (see Fig.
49).Furthermore, in this case (a = b = 0) the interchange of variablestransforms (1) into the system (22), which may be written as±αφη — αφη — αφ ι + αφ2Mil-&2&2 -&3&1 +22= 0,Μ22 =0,with the fiik as in (26). For this system the equation analogous to (2') is(33)Δ12 cos ψ +2(Δι4— Δ ) cos ψ sin φ +2 3Δ34sin ψ = 0,2which is satisfied by cot ψ as given in Eq. (32) whenever cot φ satisfies Eq.(2'). Therefore, if PP\ is a characteristic direction for the system ( 1 ) , with= b = 0, the equations (28) and (29) are consistent only when the corresponding direction QQi is also characteristic for the transformed system; toeach characteristic direction φ or φ~ in the x,?/-plane, there corresponds adefinite characteristic direction, ψ or ψ~, in the ^,^-plane, given by (32).Thus we can state: A system of two planar equations between x,y and u,vwith vanishing right sides determines for each pair of corresponding pointsP(x,y) and Q(u,v) a set of q o affine transformations, each of which mapstwo directions φ and φ~ through Ρ into two directions ψ and ψ~ through Q.These two pair of directions (real only in the hyperbolic case) are the characteristic directions of the system (1) and of the transformed system (22),respectively.a++2++T h e directions φ and φ~ are determined from Eq.
(2') as soon as Ρ and Qare given; the directions ψ and ψ~ are then determined from Eq. (32). A++134II. GENERAL THEOREMSparticular affine transformation can be singled out by giving the t w o quotients QQi /PPi++and QQi~/PP{~, where P i , P{~, Qi , and φ Γ are points++on the corresponding characteristics. ( T h a t these quotients, and not thelengths themselves, fix the transformation follows from the fact that multiplying dx\, dyi , du\, and dvi by the same constant leaves E q s .
(29)unaltered.) T h e transformation is determined, except for a scale factor, bythe ratio of these quotients. I n particular, if the first quotient is zero, Eqs.(25) show that 1 /D must vanish [see E q . (27)]; the area around Ρ is mappedinto a segment of the line QQ , at least if only first-order terms are con2sidered. Thus, if D =oo (or D = 0 ) , the area around Ρ (or around Q)is mapped into a curve element in the other plane whose directionis oneof the two characteristic directions in that plane.
This phenomenon, knownas that of the limit line of the x,i/-plane in the case D = 0, and that ofthe edge in the u,y-plane in the case D =o c , appears repeatedly in connection with applications. A general discussion will be given in A r t . 19.CHAPTER IIIONE-DIMENSIONAL FLOWArticle 11S t e a d y Flow with Viscosity and Heat Conduction1. General equations for parallel nonsteady flowA one-dimensional, or parallel flow is one in which (a) the velocityvector q is always parallel to a fixed direction, and (b) all derivatives indirections perpendicular to this direction vanish.
Taking the x-axis in thedirection of motion, we have(1)Qx = u,qyA =A == q = 0,zdy0.dzIn this case the divergence of q reduces to du/dx, and the equation ofcontinuity (1 . I I ) becomes(2)A( u)P+| = 0 .dtdxIn Newton's equation (1.1), when gravity is omitted, only the first component remains, givingPdu _dpd t ~~dxin the case of inviscid flow.In the more general case of a viscous fluid it is clear that for any volumeelement dx dy dz (Fig.
6) no shearing stresses due to viscosity (friction) canexist on the four faces parallel to the x-direction, since the particles adjacent to those faces move along with the same speed. Consequently, all. shearing components vanish, and the only viscosity effect in the x-directionis the normal tensile stress σ' (positive when directed outward) on the twofaces perpendicular to this direction. Thus the quantity ρ used in the inviscid case is to be replaced by ρ — σ' , leading toχχ135III.
ONE-DIMENSIONAL136FLOWin agreement with E q . (3.8). T h e relation of σdiscussed presently.to the velocity u will beζA s specifying condition we suppose that the motion is adiabatic exceptfor heat conduction, i.e., simply adiabatic (see Sec. 1.5). T h e equationspecifying simply adiabatic conditions, E q . (3.23), was discussed at theend of Sec. 3.5; when Q' is replaced by the left-hand member of the energy equation (3.19), with gravity omitted, we have [see Eq.
(3.24)]|(|-+ t 7)^ . i+divftgrad„in the general case of simply adiabatic flow, and!(£+ϋ)+^ . ιέ( . 0 )in the one-dimensional case. Here w is defined by Eq. (2.5) and w' by Eq.(3.10); in the one-dimensional case these equations givew=^ ,' = _^,w+w'w*-d\=d[u(dxdxdxIf we assume, further, that the fluid is a perfect gas, then the relations (1.6)and (1.9) hold:(5)andρ = gRpTThen U [see Eq. (2.13)] is given by c Tvcv=°Ry -= gRT/(y-Γ1).
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