R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 21
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(10) is therefore a sufficient condition that λ be normal to an exceptional plane; by reversing the steps in the proof it is easily verified thatEq. (10) is also a necessary condition on λ .T o write the determinant (10) one has simply to arrange the original kequations ( 6 ) in such a way that the terms involving U\ form the first column, the terms with u the second, etc., and to replace each ^-derivative2by λ] , each ^ - d e r i v a t i v e by λ , etc. If, using the so-called summation2convention, we write the Eqs.
( 6 ) as(6')duKα,ίΚthe determinant equation (10) readsdo')α λίΚμμII = 0μ =1,ι, κ =1,T h e fact that condition (10) is necessary, can also be seen as follows. W ewill show that if λ does not satisfy Eq. (10), then λ cannot be the normal106II.
G E N E R A LTHEOREMSto an exceptional plane. Thus suppose λ is such that the determinant in (10)does not vanish, and that λ has length 1. T a k e the xi-axis parallel to λ ;then λ has components (1,0, · · · , 0 ) and the elements of the determinant in(10) are the xi-components of the k vectors a « . If the components of thek gradients are given in the plane normal to λ , the unknowns in Eqs. (6)2are the ^-components of the gradients, viz., dui/dxi,du^/dxi,··· ,dUk/dxi.Then (6) is a system of k linear equations for the k unknowns dUi/dxi whosedeterminant consists of the avcomponents of the vectors a « and does notvanish, by hypothesis.
Thus it is possible to compute the unknown components from the given ones, which means, we recall, that the plane normalto λ is not exceptional.If E q . (10) is expressed in terms of the components of λ , the result is ahomogeneous algebraic equation of degree k in these components, withcoefficients depending on the coefficients of the given equations at P. Thusthe endpoints of the vectors λ satisfying condition (10) lie on a " c o n e " oforder k in η-space with vertex at P. T h e cone need not be real and may bedegenerate, but in general E q .
(10) defines a cone of directions at P . Eachplane through Ρ normal to a direction λ is exceptional, and any (real) surface tangent to such a plane at each of its points is a characteristic surface,according t o the definition given at the end of Sec. I .303. Compatibility relationsW e have not quite completed the basic theory. Equation (10) defines theexceptional directions λ * . When such a direction λ = λ * has been found, wehave still t o determine the previously mentioned combinations ( 7 ' )of the original equations, where the a* are such that all A ? are parallel t oone and the same exceptional " p l a n e " E* which is perpendicular t o λ * .In other words, once a λ * with components λ? , \ * , · · · , λ£ has been foundas a solution of (10)—a generator of the cone mentioned above—we mustsubstitute these λ? , λ * , · · · , λ ί in ( 9 ' ) and determine a corresponding setof multipliers < * ? , « ? , · · · , < * ? ; the set of vectors A ? , A ? , · · · , A f definedby (8) and formed with these a ? is then parallel t o the exceptional planeE*, and substitution of these A* into ( 7 ' ) yields the desired combination:k(11)ΣkΑ.* grad uK= Σaffe.
= B*.On the left side of (11) there are only differentiations in directionsA* , i.e. parallel to E* and perpendicular t o λ * . W e call (11) a compatibility relation. This name is justified b y the fact that (11) restrictsthe arbitrariness of the derivatives of the u "along £"*", or parallel t o E*.In the particular case η = 2, an exceptional plane Ε is simply a line of characteristic direction (since η — 1 = 1) and differentiation along Ε beK9.4FIRST EXAMPLES107comes the well-known directional differentiation.
Corresponding to one X*there may be more than one compatibility relation, i.e., combinations ofthe original equations which do not contain any differentiation in the λ * direction. (See Sec. 6.)Thus we can state that with respect to any λ * the original set of k equations can be so combined as to split into t w o parts: one part, consisting ofr equations (k — r being the rank of the coefficient matrix in (9') for λ = λ * ) ,includes derivatives of the u only in directions perpendicular to λ * . TheseKequations are the compatibility relations.
Each of the additional (k — r)equations, forming the other part, includes at least one derivative of anunknown in the λ * ^ ΐ Γ β ^ ί ο η .3 1These rather brief indications must suffice. T h e y will become more concrete when we consider various examples in the next sections and, particularly, the case of general fluid motion where η = 4, k = 5 in Sec. 6. In A r t . 10the case η = k = 2 will be dealt with in detail, and we shall see that evenin this case the general and detailed discussion of the compatibility relations (Sec. 10.2) is not too simple.
In Chapters I I I , I V , and V we shall dealwith the problems of nonsteady parallel flow and steady plane flow respectively. In both cases the basic equations present comparatively simple examples of the Eqs. (10.1). A s far as the compatibility relations are concerned,the main simplification consists in the fact that the right sides of the basicequations are zero.4. First examplesSuppose that a function Φ(χ,ν)equationoftheform(2)is subject t o a second-order differentialwhere A,B,C, and Fdependon x,y,ΘΦ/θχ, ΘΦ/dy, but not explicitly on Φ. T o express this problem in the formof system ( 6 ) , we take the first derivatives of Φ as unknowns:ΘΦThen Eq. (2) and the condition that U\ and u are derivatives of the same2function may be written as+(12)BidUidu \\dydx)dUi~dy2du2~dx* If the coefficients A,B,C,and F also depend on Φ, a third equation, ΘΦ/dx = u ,is added to Eqp.
(12) so as to form a system of three planar equations for u\ , w , and Φ.The characteristics are still given by (13).x2108II. GENERALTHEOREMSHere k = η = 2. When this system is put in the form ( 6 ) , the four vectorsatKaream(A,B)a :(S,C)a i:(0,1)a :(-1,0).21 22 2Thus, condition (10), determining λ, becomesCX 1A\i + Βλ+λ— λι2(13)22= -[ΑΛΊ +2 £ λ ι λ + CX ] = 0.2222In this two-dimensional case, the " c o n e " of directions at Ρ consists ofdirections λ in the x,y-plane with components satisfying Eq. (13);''exceptional planes'' are straight lines each normal to a λ and inz,2/-plane and the "surface elements'' normal to these directions λ arements of arc.
N o w λ /λι , determined from (13), gives the slopes oftwo λ-directions; hence the slopes of the two characteristic directionsgiven by2(130- r = τ [Β ± VB*λ2-twothetheeletheareAC].ΛFor the particular case of steady potential flow, governed by Eq. ( 1 ) ,the coefficients A, B, and C may be given explicitly. Choosing the x-axisparallel to q at P, we have A = 1 — Μ , Β = 0, C = 1; then2where a is the Mach angle. Thus, if Μ < 1, no real characteristic directionsexist; for Μ ^ 1, the characteristic directions are the Mach directions atP, so that the characteristic lines turn out to be the Mach lines.
Moreover,since there are no other solutions of Eq. (13), the Mach lines are the onlycharacteristic lines in the case of steady plane potential flow, and, moregenerally, of Eq. (7.43).Let us now consider the compatibility relations corresponding to Eqs.(12). Call φ the angle a characteristic direction makes with the ar-axis. ThenEq.
(13) shows that there are two values of φ which we denote by φ andφ~. W i t h X i = — sin</>, λ = cos φ, Eq. (13) gives A sin φ — 2 ^ β ί η φ cos φ +C cos φ = 0, or+2222B -A tan φ -Ctan φ= 0.Denote by d/θσ differentiation along a characteristic. Then, multiplyingthe first Eq. (12) by cos φ, the second by A sin φ — Β cos φ (these being9.4FIRST109EXAMPLESappropriate multipliers a ), adding, and using the above relation, we obtainLimmediately(14)A ^da+ C cot φ —2da= F cos φas the desired compatibility relation.
I t consists of two equations, one forφ and da , the other for φ" and da~, or, in other words, one along eachcharacteristic.Equation (14) allows a useful modification: from the above quadraticequation for tan φ it is seen that++_tan φtan φ+C=—.Hence the two equations contained in (14) can be written(14')A(jg +tan ^= F cos φ*.In differential form, and using the abbreviation dx = cos φ da, we obtain(14")duf + tan φ * dut= γdotas the required compatibility relation.If our equations have come from one of the flow problems governed byEq.
(7.43) we know that the characteristic curves are the Mach lines sothatφ+= 0 +a,φ~ =θ -a,where θ is the angle between the x-axis and the flow direction and a theMach angle. In Eq. (7.43) with ν =F =0 for plane potential flow we have0 and (14 ) reduces torThis relation between the derivatives of the dependent variables along thecharacteristics, may be compared to (dy/dx)*For (7.43), with ν =a1, qx= Ui , q— U\ we obtain with φ * = 0 ±= tan φ*.= ih , Fy=—u a/y,2α:2(16)dwf + tan (θ =F a) duf=—RRy(a2-Mi )2dx .±and A=110II. GENERALTHEOREMSThis equation simplifies if polar variables g, 0 are used, since a.
is a functionof q alone. Introducing into (16)du\ = dq cos 0 — άθ q sin 0,du2= dq sin 0 + dd q cos 0,we obtain after some simple manipulationsin a sin 0 dx——-—- — ,cos (Θ + a) y.da,sin a sin 0 dxg tan a + a0 = cos (0y,j- — a)r —dq——2q tan α(17)ΊΛd0 =ΊΛ9,~+along a G ,~along a C .ΛThese relations, as well as Eqs. (15), will be better understood after thestudy of Sees. 10.6,7. If the same polar variables are introduced into (15)rather than (16) we obtain Eqs. (17) with their right sides replaced by zeros(see Sec. 16.3).5. Further examplesSome caution is needed in considering the three-dimensionalproblem ofsteady potential flow. Suppose the differential equation for Φ is of the formp(18) Alp +i + Atdx2a dyA2, ^ * + 2 Bdzl2^ - + 2 B ^+ 2B ^dy dzdz dx3= 0,dx dywhere, as before, the coefficients do not depend on Φ itself.
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