J.J. Stoker - Water waves. The mathematical theory with applications, страница 6

If 7for all curves C, as we assume, it follows easily by awell-known argument that the vector curl v vanishes everywhere:=* It should bemuchadded that the law of conservation of circulation holds underwider conditions than were assumed here (cf. [C.9], p. 19).BASIC HYDRODYNAMICS(1.2.4)curland the flowis=v(wyvzwxuz,uy )vx,9=0,then said to be irrotational. In other words, a motionin a nonviscous fluid which is irrotational at one instant alwaysremains irrotational. Throughout the rest of this book we shall assumeallflowsbe irrotational.toand Bernoulli's law1.3. Potential flowThe assumptionof irrotational flow results in anumberof sim-plifications in our theory which are of the greatest utility.

In thefirst place, the fact that curl v(cf. (1.2.4)) ensures, as is well=known, the existence of a single-valued velocity potential 0(x, y, 2; t)in any simply connected region, from which the velocity field can bederived by taking the gradient:in terms of theor,=grad(0 Xcomponents ofu(1.3.2)The=v(1.3.1)=velocity potentialis,X9v=Z ),y,,v:y9w =indeed, givenZ.by theline integral<e,v,z0(xThe vanishingisan exacty, 2; t)9=u dxI+ v dy + w dz.of curl v ensures that the expression to be integratedOnce it is known that the velocity com-differential.ponents are determined by (1.3.2), it follows from the continuityis a0, that the velocity potentialequation (1.1.6), i.e. div vsolution of the Laplace equation=V2(1.3.3)<Z>=XX++VVZZ=0,is thus a harmonic function. This factandagreat simplification, since the velocity field is derivablerepresentsfrom a single function satisfying a linear differential equation whichas one readily sees,has been very much studied and about which a great deal is known.Still another important consequence of the irrotational characterof a flow can be obtained from the equations of motion (1.1.4).

Bythat the equations ofmotion (1.1.4) can be written in the following vector form:making use ofgradt+(1.2.4), it is readily verified1~2use having beengrad (u2+v +w =22)pgrad -grad(gy),Qmadeof the fact that Q=constant. IntegrationWATER WAVES10of this relation leads to the important equation expressingcalled Bernoulli s law:whatis9(1.3.4)t+ I2(u*+ v* +te;2)+ ? + gy =C(t),Qmay depend on t, but not on the space variables.

Theretwo other forms of Bernoulli's law for the case of steady flows,one of which applies along stream lines even though the flow isnot irrotational, but since we make no use of these laws in this bookwe refrain from formulating them.The potential equation (1.3.3) together with Bernoulli's lawinwhich C(t)arecan be used to take the place of the equations of motionandthe continuity equation (1.1.6) as a means of determining(1.1.4)the velocity components u, u, 10, and the pressure p: in effect, u, vof (1.3.3), after which theand w are determined from the solutionpressure p can be obtained from (1.3.4). It is true that the pressureappears to be determined only within a function which is the same(1.8.4)9at each instant throughout the fluid.

On physical grounds it is,however, clear that a function of t alone added to the pressure phas no effect on the motion of the fluid since no pressure gradientsresult=from such an addition to the pressure. In0*-f fC(f ) df ,then0*isaharmonicfact,ifwefunctionsetwith=gradgrad 0* and the Bernoulli law with reference to it has ain (1.3.4)vanishing right hand side. Thus we may take C(t)without any essential loss of generality.While it is true that the Laplace equation is a linear differentialequation, it does not follow that we shall be able to escape all of thedifficulties arising from the nonlinear character of the basic differen-=tialequations of motion (1.1.4). As we shall sec, the problems ofremain essentially nonlinear because the Bernoulli lawinterest hereand another condition to be derivednext section, givenonlinear boundary conditions at free surfaces.

In the nextsection we take up the important question of the boundary conditions appropriate to various physical situations.(1.3.4),rise toin theBoundary conditionsWe assume the fluid under consideration1.4.have a boundarysurface S, fixed or moving, which separates it from some othermedium, and which has the property that any particle which is oncetoBASIC HYDRODYNAMICSon the surface remains on11Examples of such boundary surfacesit.*of importance for us are those in which S is the surface of a fixedthe bottom of the sea, forrigid body in contact with the fluidor the free surface of the water in contact with the air.such a surface S were given, for example, by an equationexampleIf(#,j/,z; t)(1.4.1)0, itfollows from (1.1.3) that the condition^ = u + <, +x+C =.dtwould hold on S. From (1.3.2) arid the fact that the vector ( X9 y Jis a normal vector to S it follows that the condition (1.4.1) can bewritten in the form,(,.4.2),<<=..v#+ ti+ c;a=.,which d/dn denotes differentiation in the direction of the normalS and v n means the common velocity of fluid and boundarysurface in the direction normal to the surface.In the important special case in which the boundary surface Sis fixed, i.e.

it isindependent of the time t, we have the conditionintod0(1.4.3)=on S.onis the appropriate boundary condition at the bottom of the sea,or at the walls of a tank containing water.ThisAnother extremely important special case is that in which S is afree surface of the liquid, i.e. a surface on which the pressure p isprescribed but the form of the surface is not prescribed a priori.We shall in general assume that such a free surface is given by theequationy(1.4.4)Onsuch a surface f=y=ri(x,z\t).forr)(a\ 3; t)anyparticle,and hence(1.4.1) yields the condition(1.4.5)xr, x-& +yZ *) Z+17,=on S.In addition, as remarked above, we assume that the pressure p isgiven on S; as a consequence the Bernoulli law (1.3.4) yields thecondition:*Actually, this propertyisa consequence of the basic assumption in con-tinuum mechanics that the motion of the fluid can be described mathematicallyas a topological deformation which depends continuously on the time t.WATER WAVES12(1.4.6)ff,+t+ I2+(012y+ 01) + P- =OR S.Q=in (1.3.4).)(As remarked earlier, we may take the quantity C(t)Thus the potential function must satisfy the two nonlinear boundaryconditions (1.4.5) and (1.4.6) on a free surface.

This is in sharp contrast to the single linear boundary condition (1.4.3) for a fixedboundary surface, but it is not strange that two conditions shouldbe prescribed in the case of the free surface since an additionalunknown function rj(x, z; t) the vertical displacement of the freesurface, is involved in the latter case.Later on we shall also be concerned with problems involving rigidbodies floating in the water and S will be the portion of the rigidbody in contact with the water.

In such cases the function r)(x, z\ t)will be determined by the motion of the rigid body, which in turnwill be fixed (through the dynamical laws of rigid body mechanics)by the pressure p between the body and the water in accordancewith (1.4.6). The detailed conditions for such cases will be workedout later on at an appropriate place.91.5.

Singularities of the velocity potentialIn our discussion up to now it has been tacitly assumed that allquantities such as the pressure, velocity potential, velocity components, etc. are regular functions of their arguments. It is, however,often useful to permit singularities of one kind or another to occuran idealization of, or an approximation to, certain physical situations.Perhaps the most useful such singularity is the point source or sinkwhich is given by the harmonic functionas(1.5.1)in three dimensions,and by=(1.5.2)2nlogr,r2=x*+ y*two dimensions. Both of these functions yield flows which areradially outward from the origin, and for which the flux per unitintime across a closed surface (for (1.5.1)) or a closed curve (for (1.5.2))surrounding the origin has the value c, as one readily verifies sinced0/dn= d0fdrfor r=constant.

That these functions represent atBASIC HYDRODYNAMICS13best idealizations of the physical situations implied in the wordsis clear from the fact that theyyield infinite velocitiesat r0. Nevertheless, it is very useful hereas in other branchessource and sink=of applied mathematicsto accept such infinities with the reservationthat the results obtained are not to be taken too literally in theimmediate vicinity of the singular point.We shall have occasion to deal with other singularities than sourcesor sinks, such as dipolesand multipoles, but thesewillbe introducedwhen needed.1.6.Notions concerning energy and energy fluxIn dealing with surface gravity waves in water it is importantin some detail the flow of energy in the fluidand useful to analyzepast a given surface S. LetRbe the region occupied by water andbounded by a "geometric" surface S which may, or may not, moveindependently of the liquid.

The energy E contained in R consistsof the kinetic energy of the water particles in R and their potentialenergy due to gravity; hence E is given by(1.6.1)or,E-alternatively,(1.6.2)R (0| + 01 + 01) + gy\Qdx dydz,byE= -JJJ(P+ e^t)dx dy dzR= 0.(1.3.4) with C(t)wish to calculate dE/dt, having in mind that the region Rnot necessarily fixed, but may depend on the time t. Quite generally,upon applying Bernoulli's lawWeisifE=f f f f(x, y, z;l)dx dy dz,Rit iswellknownthatSwhich v n denotes the normal velocity of the boundary S of Rtaken positive in the direction outward from R. In applying theformula for dE/dt we make use of the definition of the function /implied in (1.6.1) in the first term, but take / from (1.6.2) for theinsecond term.