R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 20
Текст из файла (страница 20)
Thus, if Adx(χι ,y)Φ1does not vanish anywhere on χ = x , the data can be extended to the0line χ = xi . Then one can proceed to compute the values of the derivativeswith respect to y on χ = χι and, if A does not vanish on χ = χι , the valuesof d $/dx on χ = xi and the values of Φ and dΦ/dx on χ = x222= X\ + dx.Thus, so long as A is different from zero, one can carry out an approximateintegration of the differential equation in the half-plane χ ^ x0(and, ofcourse, in the same way in the other half-plane), and the smaller the increment dx the better the approximation. I t may be noted that the transitionfrom (.T ,2/0) to (xo + dx, y ) does not require a knowledge of the values on00the entire linethe point(x0but only on a small interval of this line containing,2/0),or even on an arc of some curve tangent to χ = x at0,2/o).(xoFor the particular case of Eq.
(1) we have(4)A = 1 -^ .N o w , if the motion under consideration is subsonic (q < a), then a fortioriqx2/a2 < 1 and A remains positive everywhere. In this case no difficultycan arise in extending the solution. T h e situation is different in a region ofsupersonic flow (q > a). In this case it can happen that q , the x-component of q, has exactly the value of the local sound velocity a. A t such apoint P, the angle a between the ?/-direction, and the velocity vector q(Fig. 42) satisfiesx(5)SIN aq _ a_ 1~q ~ q ~ TVxthus a is the Mach angle defined in Sec. 5.2, and the ?/-direction coincidesF I G . 42.
^/-direction as characteristic direction.102II. GENERALTHEOREMSwith one of the Mach directions at P. I t appears that the integration processcannot be extended across an element dy if the angle between dy and q isthe M a c h angle.Since the x- and ^-directions in (1) are entirely arbitrary, this meansthat the process of step-by-step integration fails whenever one must cross aline element which makes the angle a with the velocity vector, i.e., wheneverthe component q normal to the line element is a.
T h e curves which crossstreamlines at the angle a have been called Mach lines (Sec. 5.4); througheach point Ρ pass two Mach lines C and C~ (Fig. 43), with q = a alongeach line. T h e significance of the Mach lines in connection with the differential equation (1) may now be seen. Even if Φ is known everywhere in theregion R between C and C~, the differential equation leads to no information about the values of Φ on the other sides of the Mach lines.
If a solutionof Eq. (1) can be found in the region R' with the same values of ΘΦ/dx andΘΦ/dy on C as the solution in R, then the t w o solutions combined determine a flow pattern in R + R with continuous velocity (and pressure)values, which satisfies the differential equation everywhere, even on thev+v++fF I G . 43. Characteristics as separation lines.F I G .
44. T w o different solutions patched together along straight characteristic.9.2G E N E R A L T H E O R Y OF103CHARACTERISTICSboundary line C . (If, e.g., the line element on C is dy, then the possiblydiscontinuous quantity θ Φ/θχ occurs with coefficient A = 0.) These twosolutions can be quite different analytically, in the sense that there is noTaylor series which converges to the combined solution in any region whichis partly in R and partly in R'. For example, it can happen that a uniformparallel flow changes into a curvilinear flow along a Mach line (which inthis case must be a straight line, as shown in Fig.
44). Such a change isnot possible in subsonic flow; there the solutions are everywhere analytic,and there are no curves beyond which a solution cannot be extended.++22Curves (or surfaces, in the case of more dimensions) 'along which analytically different solutions of a differential equation or a system of such equations can be patched together (with certain restrictions following from theequations themselves) are called characteristics of the equation or system.Insofar as the preceding argument is accepted as rigorous, it has beenshown that the Mach lines are characteristics of the two-dimensional potential equation ( 1 ) .2. General theory28In most branches of physics the mathematical problem can be formulated as follows: A set of k unknowns U\ , u , · · · , u , which are functionsof η independent variables Χι , x , · · · , x , is subject to k first-order differential equations which are linear in the derivatives du /dx .
Since the coefficients may depend on the unknowns u , as well as on the independentvariables, the differential equations are, in general, nonlinear. W e shall usethe term planar to indicate that they are nevertheless linear in the derivatives. In the case of fluid flow, there are k = 5 unknowns: q , q , q ,p, and p; and η = 4 independent variables: x, y, z, and /; the system ofequations consists of Newton's equation ( I ) , the equation of continuity( I I ) , and the specifying equation ( I I I ) .22knKyKxyzA n expression of the formdudu ..dua- H· ·· +a —dxidxdxcan always be considered as the (η-dimensional) scalar product of a vectora, with components a\ , · · · , a , and the vector grad u, which also has ηcomponents du/dx .
Since there are k unknowns u and k equations, itrequires k n-dimensional vectors a , where both L and κ run from 1 to k,to express the system of planar differential equations in the formai-H,2n2nnyK2l(tk(6)Σa„-grad uK=b,(t = 1, 2, · · · , k).Writing the system in vector form has the advantage of its being inde-104II.G E N E R A LT H E O R E M Spendent of the choice of coordinate axes. In discussing any particularphysical fact in connection with Eqs. ( 6 ) , one may choose the coordinateaxes so as to make the components of the vectors describing the situationas simple as possible.27T h e fc gradients have nk components. Suppose that for some choice ofthe axes, the components of the gradients are known in η — 1 of thecoordinate directions at P, i.e., (n — l)fc values are given arbitrarily; thenthe k equations (6) serve, in general, to determine the remaining k components, in the direction normal to the " p l a n e " of the η — 1 given directions at P.
In the case η = 3, there are given 2k gradient components,parallel to some plane (a plane in the usual sense of the w o r d ) , and Eqs. (6)serve to compute the remaining fc components. I t can happen, however,that for some plane Ε and (n — l)k given components of grad u parallelto E, Eqs.
(6) fail to determine the remaining k components. A necessaryand sufficient condition for this is that the determinant of coefficients ofthese k components in Eqs. (6) vanishes or, what is equivalent, that thereexists a linear combination of the Eqs. (6) that does not contain any ofthese components. A plane Ε for which this happens will be termed exceptional for the system (6) at P . Our problem is to find at each point whichplanes, if any, are exceptional, or, equivalently, to find the direction of thenormal λ to any such exceptional plane.KW e consider the linear combination of the equations of the system (6)obtained by multiplying by factors c*i , « 2 ,respectively andsumming:k(7)Σkk«ι Σa c*grad ul(K=Σ<*A >orι= 1κ=1i=l(7')kwhereΣA *grad uKK=B,kA< =(8)Σ1U =ι, · · ·, fc),=1Equation (7') is a necessary consequence of (6) and must be satisfied bysolutions u of (6) for arbitrary choices of the factors a .
T h e left-handi= lmember of Eq. (7') is a sum of scalar products, each of the form Α * · grad u ,with A a linear combination of the vectors a , ι = 1, · · · , / < ; ; eachscalar product is also equal to the product of A by the component ofLKKKl(CK9.2105G E N E R A L T H E O R Y OF C H A R A C T E R I S T I C Sgrad u in the direction of A* . Suppose that for some choice of the factorsKa(not all zero) the vectors Atare all parallel to a common plane E; thenKthis plane is surely exceptional since there exists a linear combination ofEqs. ( 6 ) that does not contain gradient components normal t o E.
For thissame reason the (n — l)k gradient components parallel to Ε are not arbitrary since there exists a linear relation between them. N o w if the vectorsAare parallel to some plane Ε and if λ is the normal to E, then λ satisfiesKλ·Αλ · Α ι = 0,(9)When the AK= 0,2λ · Α * = 0.are replaced by the expressions from ( 8 ) , we obtain fromEqs. (9) k linear homogeneous equations t o be satisfied by the k factors<*i(^«an) + α ( λ · Ε ι )++a (X*8i )«i(^«ai2) + α ( λ ^++a * ( * « a * ) = 0,2(90222 2)+ a ^«a fc) +Oii(X»Siik)22k=kl0,2· · · + a (X»a. )kkk=0.A nontrivial solution « ι , · · · , a*- of these equations exists if the determinant of coefficients vanishes, i.e., ifλ^ιΧ·2ίη(10)λ-au2λ ·a iλ · a22λ-a^λ-ajfciλ-a^^•a k2kEq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
