R. von Mises - Mathematical theory of compressible fluid flow, страница 13

24.Here it follows from (6) that the circulation must have a common valuefor all circuits encircling the same tube. In fact, in computing the circulationΓ along6 2 one can choose for($ 2 the surface consisting of the part of themantle between G and (B together with any surface & spanning 6. A s above,the surface integral is zero on the mantle, so that the integral over d hasexactly the same value as the integral over Q, giving Γ = Γ.* This commonvalue of the circulation is also called the vorticity of the tube; it is a scalarquantity, not to be confused with the magnitude of the vortex vector.

Inthe case of a vortex filament, the vorticity dT is given by the product ofthe length of the vortex vector by the normal cross section of the tube.22227If a vortex tube be divided into several tubes of finite cross section (orinto an infinite number of vortex filaments), the vorticity of the wholetube is the sum (integral) of the individual vorticities. This follows from theadditivity of circulation [or from Eq. (6)].A vortex tube cannot begin or end in the interior of the fluid, but must eitherbe a closed tube (like a torus or doughnut) or else (provided it does not meeta boundary) must extend indefinitely in either direction.

For at an end, ifthere were one, a continuous transition would be possible along the mantlefrom curves of type6 1 to those of type 62 , which is inconsistent with thefact that Γι = 0, while Γ = constant 9^ 0.23 . Kelvin's theoremSo far in this article only pure kinematics has been discussed: the con­cepts of circulation and mean rotation, as well as the relation between them,are valid for any type of continuously distributed material.

W e shall nowspecialize to the case of an inviscid elastic fluid, so that the basic equationsof Arts. 1 and 2 hold.T h e fluid particles lying on any closed circuit at some moment willstill form a closed circuit at a later time, since for reasons of continuity noseparation of particles can occur; a preliminary question, and one whichcan be given a very simple and decisive answer, is the following: how doesthe circulation change during this transition?* When there is no surface spanning G (in the fluid) the result follows from anargument similar to that at the end of the preceding section.62II.

GENERALTHEOREMSIn Fig. 25 the solid line represents a circuit 6,'and the dotted line thecircuit G' formed by the same material particles after time dt. T h e circula­tions along these two circuits are given byandq»dl(8)?' =jq-dl',where the integrals are evaluated along 6 and 6 ' respectively. L e t Ρ bethe position of an arbitrary particle of C and Q that of a particle of Q at adistance dl away, and let P' and Q' be the positions of these particles aftertime dt.

ThenPP'= qdtandQQwhere d/dl denotes the directional derivative in the direction of the tangentto 6 at P. T h e corresponding element of arc dV on G' can be computed fromthe vector equation ~PP' + P'Q' = PQ + QQ', giving(9)dX = P'Q'= PQ + QQ' -PP'dq= dl + ^ dl dt,dtwhile the value of q corresponding to P' is given by(10)Then, using Eqs. ( 8 ) , ( 9 ) , and (10), and omitting a term of higher order,we findr ' - r - / [ , . | Sw£ .w£ . ^iW). ](11)F I G . 25.

T w o circuits formed by the same particle.6.4 H E L M H O L T Z ' V O R T E X63THEOREMSwhere the integral is to be extended along C Equation (11) is still truefor any continuously distributed mass. For an inviscid fluid, however, thevalue of the acceleration vector dq/dt may be taken from the equation ofmotion (1.1)8=~ ρggradV jwhere in the notation of Sec. 2.1, g = — g grad h. Moreover, for an elasticfluid the notion of pressure head P/g (Sec. 2.5) can be used, giving- grad ρ = grad P.ΡThus the expression for dq/dt takes the form(12)^_(g+- g r a d (gh + P)*d\==gradhP)>and=- |(gh + P)dl.When this is inserted in (11), there results(13)Γ' -Γ = dt j> d[j - gh -p]dlT h e quantity within the bracket is a single-valued function of positionand time.

Thus, since at a given time the integral is extended around theclosed circuit β its value is zero. T h u s E q . (13) gives Γ' — Γ = 0 or: In aninviscid elastic fluid, the circulation around any circuit does not change as theparticles forming the circuit move along. This is Kelvin's theorem. T h etheorem depends essentially on the fact that the equation of motion canbe expressed in the form (12), i.e., on the fact that in an inviscid elasticfluid the acceleration vector is a gradient, or (since the curl of a gradient van­ishes identically) that the curl of the acceleration is zero.9104.

Helmholtz' vortex theoremsStarting from Kelvin's theorem, it is easy to derive two theorems onvortex motion which had been proved earlier by Helmholtz (although by adifferent method; see Sec. 6 ) . Consider a vortex tube 5C of infinitesimal crosssection (Fig. 26). T h e particles of JC after time dt still form a tube 3C',because no separation of particles can occur; we first wish to determinewhether 3C' is still a vortex tube. In Sec. 2 we found that the circulationmust vanish along any circuit Ci lying on the mantle of a vortex tube, but64II. GENERALTHEOREMSnot encircling it.

From Kelvin's theorem it follows that the circulationalong the new position Ci of this circuit must also vanish. Also, c i lies onthe surface of 3C'. For an infinitesimal circuit β ( , according to Stokes'Theorem, the circulation is the product of the area enclosed within thecircuit by the component of the vortex vector normal to that area. Sincethis product is zero, the vortex vector must be tangent to the area elementat @ί , and the circuit e j must lie on the mantle of some vortex tube. Thisis true for any infinitesimal circuit Gi on 3C and the corresponding c { on 3C',so that 3C' is also a vortex tube of infinitesimal cross section.

Thus wearrive at this statement: Particles lying on a vortex line at some momentmove in such a way that they form a vortex line at every moment. A shorterexpression of this first vortex theorem is: The vortex lines are material lines,in the sense that they always consist of the same particles or materialpoints.11Each vortex tube has a certain vorticity, equal to the circulation alongany circuit encircling the tube, such as <E in Fig.

26. B y Kelvin's theorem,the circulation has the same value along the corresponding circuit C on3C', so that the vorticity of the vortex tube 3C' is the same as that of 3C.Thus we can state Helmholtz' second theorem: The vorticity of a vortextube does not change as its particles move along.22In Sec. 2 it was seen that vortex tubes cannot come to an end in the in­terior of the fluid, but must either meet a boundary, extend indefinitely, orbe closed.

Tubes of the latter type can be observed in air as smoke rings,produced by imparting a rotational motion to the smoke particles. A c ­tually, the smoke rings do not persist indefinitely, in apparent contradictionof the vortex theorems. This is due to the presence of viscosity effects,which are disregarded in the theory of inviscid fluids. T h e vortex theoremsfollow from the fact that the acceleration vector is a gradient (and there­fore curl-free). T o arrive at this statement (12) it was necessary to neglectall stress components other than the pressure ρ (all shearing stresses),F I G .

26. Vortex filaments formed by same particles.6.5MEAN ROTATION AND BERNOULLIFUNCTION65and to assume the existence of a relation between ρ and ρ in order to makepossible the definition of P.5. Mean rotation and the Bernoulli functionIn Sec. 2.5 we introduced the total headπΡΗ = %- + h + - ,2tf9which was shown to be constant along each streamline during steady flow(Bernoulli equation). Let us consider the relation of the Bernoulli functionΗ to the mean rotation of the fluid or to curl q.Starting from the equation of motion for an inviscid elastic fluid in theform (12), we subtract grad (q /2) from both sides and use Eq.

(14), to ob­tain2(14)2(15)^-grad ( 0=-grad(gH).In order to interpret the vector on the left, we compute its x-component.Using the Euler rule of differentiation and q = q + q + q , and denot­ing briefly curl q by λ , we see that the x-component of the left-hand side is22xy2zdq:dtdx\2/dt^ \dzdx)qz\dxdy)qv(16)^+rtt(λ^-Kq )y=^ + ( U q ) ,at,and from Eq. (15)(17)a£ +dtq(curl q X q ) =-grad(gH)uThis (vector) equation (which is a form of Newton's equation for an in­viscid fluid), includes the (scalar) Bernoulli equation and more.

In fact, forsteady flow dq/dt = 0, so that(18)grad Η =- - (curl q X q ) .lSince a vector product is perpendicular to each of its factors, Eq. (18)shows that the vector grad Η is perpendicular to q. Hence the directionalderivative of Η along a streamline is zero, and Η must be constant alongthe streamline. Moreover, grad Η has no component in the direction ofcurl q, the direction of the vortex lines. Thus: In the steady flow of an inviscidelastic fluid the surfaces on which the Bernoulli function has constant valuesare composed of streamlines and vortex lines.66II. GENERALTHEOREMST h e most important consequence of ( 1 8 ) is the following.