R. von Mises - Mathematical theory of compressible fluid flow, страница 2

In particular, if I be taken as the direction of q, the directioncosines may be expressed in terms of q, t o givedd,d,dwhere s is used in place of I to designate the direction of the line of flow; forthis direction ds =q dt. B y d/dt in Eqs. (1) and (2) is meant, not partialdifferentiation with respect to t at constant x, y, z, but rather differentiationfor a given particle, whose position changes according to E q . ( 3 ) :d(4)^tdx_d_i_dyd^,dzd=dtdtdt dxd=dt.+dd~xqxdt dy,+dq" d ydt dz.+d*dzq= dtd+,ddi'qT h e acceleration vector dq/dt is the time rate of change of the velocityvector q for a definite material particle which moves in the direction of qat the rate q = ds/dt. T h e operation d/dt may be termed particle differen­tiation, or material differentiation, as distinguished from partial differentia-1.23NEWTON'S EQUATION FOR A N IN VISCID F L U I Dtion with respect to time at a fixed position.

An alternative form of Eq. (4)is(θ&~h +d- -(q grad)Here, grad is considered as a symbolic vector with the components d/dx,d/dy, and d/dz in accordance with the well-known notation of grad / forthe vector with components df/dx, df/dy, and df/dz, which is called thegradient of f, and the scalar product q*grad means the product q times thecomponent of grad in the q-direction, i.e., q d/ds = q d/dx + q d/dy +xyq d/dz. Equations (4) or (4') will be referred to as the Euler rule of differ­zentiation?When conditions are such that the partial derivative with respect totime, d/dt, vanishes for each variable, the phenomenon is called steady, asin steadyflow,steady motion. Particle differentiation then reduces toq d/ds.2.

Newton's equation for an inviscid fluidT h e above equation (1) holds for any continuously distributed mass inwhich the density ρ is defined at each point and at each moment of time.Different types of continua may be characterized by the form of the termwhich represents the forces arising from interaction of neighboring ele­ments.

A n inviscid fluid is one in which it is assumed that the force actingon any surface element d§>, at which two elements of the fluid are in con­tact, acts in a direction normal to the surface element. I t can be shown (see,for example, [16],* p. 2) that at each point Ρ the stress, or force per unitarea, is independent of the orientation (direction of the normal) of dS. T h evalue of this stress is called the hydraulic pressure, or briefly pressure, p,at the point P. T h e word "pressure" indicates that the force is a thrust,directed toward the fluid element on which it acts: negative values of ρare not admitted.For a small rectangular cell of fluid of volume dV = dx dy dz (see F i g . l ) ,two pressure forces act in the z-direction, on surface elements of area dy dz.Taking the x-axis toward the right, the left-hand face experiences a forceρ dy dz directed toward the right, while the right-hand face experiences aforce (p +dp) dy dz directed toward the left; here dp =(dp/dx) dx.

T h eresultant force in the rc-direction is thus— dp dy dz =yυ—^-dxdydz=dx*—^dxdV.Therefore the internal force per unit volume, appearing in Eq. ( 1 ) , has* Numbers in brackets refer to the bibliography of standard works, p. 502 if.4I.INTRODUCTIONζ~^1Γ(P+dp)dz|- —dxF I G . 1. Rectangular volume element with pressure forces in x-direction.x-component — dp/dx; similarly, the remaining components are found tobe — dp/dy and — dp/dz. Hence(I)P~= pg ~ grad ρexpresses part ( b ) of Newton's Principle.

This equation was first given byLeonhard Euler (1755), but is usually known as Newton's equation?T h e vector Eq. ( I ) is equivalent to the three scalar equations(ΐ')dqxdpdtdxdqdpypl t=pg*dqdpzEquations ( I ) and ( Γ ) are valid only for inviscid fluids. If viscosity ispresent, additional terms must be included in the expression for the internalforce per unit volume, thus generalizing E q . ( I ) .

This will be discussed later(Sec. 3.2).T h e motion of the fluid may also be described by following the positionof each single particle of the fluid for all values of t. From this point of viewthe so-called Lagrangian equations of motion are obtained, which determinethe position (x, y, z) of each particle as a function of t and of three parame­ters identifying the particle, e.g., the position of the particle at t = to .Except in certain cases of one-dimensional flow, the Eulerian equations aremore manageable.In either case the following basic concepts apply. T h e space curve de­scribed by a moving particle is called its path or trajectory. T h e family ofsuch trajectories is defined by the differential equation dr/dt = q(r, t).

Onthe other hand, at each fixed time t = t , there is a (two-parameter) familyοϊ streamlines: for t = U each streamline is tangent at each point to thevelocity vector at this point, its differential equation being dx X q(r, t ) = 0,4001.3or dx:dy:dzE Q U A T I O N OF C O N T I N U I T Y= q :q :q ,xywhere qz= q (x,xx5y, z, to), etc. Thus in general theof streamlines varies in time. Clearly the streamline through r atfamilyt = to, is tangent to the trajectory of the particle passing through r at thatinstant.If, in particular, the motion is steady, i.e., at each point Ρthe samething happens at all times, then there is only one family of streamlines,which then necessarily coincides with the trajectories.In the general case it is convenient to consider, in addition, the particlelines (or " world l i n e s " ) .

T h e y are defined in four-dimensional x, y, z, i-space,each line corresponding to one particle. T h e y appear in Chapter I I I as thex, Mines. A particle path is the projection onto x, y, 2-space of the respectiveparticle line.3. Equation of continuityIn order to express part (a) of Newton's Principle—conservation of mass— i n the form of a differential equation, the differentiation indicated in E q .(2) could be carried out by transforming the integral suitably (see below,Sec.

2.6). I t is simpler, however, to consider again the rectangular cell ofFig. 1. Fluid mass flows into the cell through the left-hand face at the rateof pq dy dz units of mass per second, and out of the right-hand face at ratex[pq + d(pq )]xxdy dz, where d(pq ) is the product of dx and the rate of changexof pq in the ^-direction, or [d(pq )/dx]xxdx. Thus the net increase in theamount of mass present in this volume element caused by flow across thesetwo faces is given b y — [d(pq )/dx]xdx dy dz, with analogous expressions forthe other pairs of faces. If we use the expression div v, divergence of thevector v, for the sum (dv /dx)x+(dv /dy)y+(dv /dz), the total change inzmass per unit time is —div (pq) dV. N o w if the mass of each moving par­ticle is invariant in time, the difference between the mass entering the celland that leaving the cell must be balanced by a change in the density of themass present in the cell.

A t first, the mass in the cell is given by ρ dV, andafter time dt by (p + dp) dV, where dp = (dp/dt)dt. Thus the rate of changeof mass in the cell, per unit time, is given by (dp/dt) dV, so thatdiv(pq) =(ID- ^ .This relation, valid for any type of continuously distributed mass, is knownas the equation ofcontinuity.5A slightly different form of ( I I ) is obtained by carrying out the differ­entiation of pq, givingI.6INTRODUCTIONor, using the Euler rule (4)(ΙΙ')Either form, ( I I ) or (II')> will be used in the sequel. In the case of a steadyflow, Eq.

( I I ) reduces todiv (pq) = 0,while in the case ρ = constant, that of an incompressible fluid, E q . ( I F )givesdiv q = 0,whether or not the flow is steady. T h e equation of continuity remains un­altered when viscosity is admitted.4. Specifying equationT h e equations ( I ) and ( I I ) , which express Newton's Principle for the mo­tion of an inviscid fluid and are usually referred to as Euler's equations, in­clude one vector equation and one scalar equation, or four scalar equations.There are, however, five unknowns: q , q , q , p, and p, in these fourequations.

I t follows that one more equation is needed in order that a solu­tion of the system of equations be uniquely determined for given "boundaryconditions \ Boundary conditions, in a general sense, are equations involv­ing the same variables, holding, however, not in the four-dimensional x, y, z,ί-space, but only in certain subspaces, as at some surface Φ(χ, y, z) = 0,for all t (boundary conditions in the narrower sense), or at some timet = to, for all x, y, ζ (initial conditions).xyz,There exists no general physical principle which would supply a fifthequation to hold in all cases of motion of an inviscid fluid, as do Eqs. ( I )and ( I I ) .