R. von Mises - Mathematical theory of compressible fluid flow, страница 7

This time, however, the statement (1.8) of theFirst L a w must be modified to allow for the rate of dissipation of energy,which is θ dV for a volume element dV. T h e modified equation, for a unitmass, readsno\n>_LdTθ.d/Awhere Q', as before, denotes the heat delivered to the particle per unit oftime and mass by radiation or conduction from surrounding fluid elements,etc.

When Eqs. (12) and (13) are combined, the energy equation reads( 1 4 )' ( i+(M.fcr)+u ± * _ < ,30I. I N T R O D U C T I O Nif c is considered to be a constant, as for a perfect gas. The difference be­tween Eqs. (14) and (2.7) lies only in the additional term w' / ρ representingthe work done per unit mass and time against the viscous stresses at thesurface of the element.If the flow is considered as being strictly adiabatic, the condition Q' = 0supplies the specifying equationv<(i(i»+fM.&r)i ± ^ _ o .+This equation does not lead to the simple condition dS/dt = 0 given in(1.12) as the specifying equation for strictly adiabatic flow in the case ofan inviscid perfect gas; for, instead of (1.8), it follows from (13) thatn,,dT=C" M+Vd /l\Θd t K p ) - p -If Τ and c are replaced by the expressions given in (1.6) and (1.9), theresult that replaces (1.12) isv(16)Q'T § at=6- ,ρwhere the definition (1.7) of the entropy S is used. Thus (15) is equivalentto(17)T§atd= -,ρθand either (15) or (17) may be taken as the specifying equation for thestrictly adiabatic flow of a viscous perfect gas.Incidentally, it follows from E q .

(17) that, since θ is nonnegative,dS/dt ^ 0 for strictly adiabatic flow, as compared with dS/dt = 0 in thecase of an inviscid fluid, i.e., the entropy of a particle of a viscous perfectgas can never decrease if no heat input or output occurs. This is an ex­pression of the Second L a w of Thermodynamics.T h e result (14) holds only if the fluid is such that c is constant. Fora viscous fluid for which an equation of state of the form (2.9) holds, func­tions U(p, p) and S(p, p) can be found, as shown in Sec. 2.3, satisfyingv' S - ' + i - T + ' i G )and subject to the unchanged restriction (2.11). T h e energy equation isthen(ie<(i+lM.

c )+- ± « ! _ t fdS_ θdtρT3.5HEAT31CONDUCTIONrather than Eq. (2.14) as for an inviscid fluid. T h e specifying equation forstrictly adiabatic flow is again (17), or (15) with c T replaced by U.vT h e integration of the energy relation over a finite volume can be car­ried out as in Sec. 2.7. One new term appears: W, the integral of w' dV. I tcan be shown by a formal transformation, or inferred from the physicalmeaning of w', that W is the work done per unit time against the viscousstresses on the surface of the volume under consideration.

Thus if t' isthe viscous stress on a surface element dS with outward normal n, we haven(20)W'=ίw dV = -f t' .q dS.n18T h e integrated energy equation will then read( )21f+w+' =f > =f ( Tf -e)avWvPQdVv Ptin the case of a viscous perfect gas, as compared to E q . (2.36); the functionΕ is the same as that in (2.34).5. Heat conductionAll relations so far discussed in this article are valid regardless of whetheror not there is heat conduction within the fluid. Heat conduction can be afactor in the mechanics of a fluid only if it occurs explicitly or implicitlyin the specifying condition.

For example, the specifying condition maybe that a particle experiences no heat input or output other than theexchange of heat with surrounding particles by means of conduction. Inorder to set up the specifying equation corresponding to this case, it isnecessary first to discuss the mechanism of heat flow.I t is usual to assume that at each point of a continuously distributedmass the flow of heat in any direction is proportional to the derivative ofthe temperature function in that direction, and that the flow takes placefrom higher values of Τ toward lower values. Consider again the rectangu­lar cell of Fig.

6. Supposing dT/dx is positive, heat will flow through the leftface of the cell in the negative x-direction at the rate of k(dT/dx) dy dzper unit time, where k is the coefficient of (internal) thermal conductivityfor the substance. T h e flux through the opposite face is-[k — +(k ^ ) dx dy dz.dxdx \ dx/J *'so that the net gain of heat resulting from flow across these two faces isk ( HOk'd x dydz32I.INTRODUCTIONand the net gain of heat per unit time and volume, resulting from flowacross all faces, isHere k may be a given constant, or a given function of T.If a fluid motion occurs under simply adiabatic conditions (Sec. 1.5), i.e.,there is no external heat exchange (by radiation, etc.), the total heatinput per unit time and volume, pQ', must be given by Eq.

(22), or(23)Q' = - div (k gradT),pas compared with the relation Q' = 0 corresponding to strictly adiabaticflow. T h e specifying condition (23) may be expressed also by substitutingfor Q' from (23) in the energy equations (14) or ( 1 9 ) :(24)d_ ( i+ +u) += 1 divgh(Jb grad T)or(24')Τ 4 atd- = - div (k gradρρT).These equations differ from those for strictly adiabatic flow in having(1/p) div (k grad T) in place of zero on the right.6. General form of specifying equation19For some purposes it seems useful to have a general form of specifyingequation that is not so restrictive as form ( 1 .

I l i a ) , yet more explicit thanEq. ( 1 . I l l ) , and which includes both viscous and inviscid fluids, with orwithout heat conduction. Such a form of specifying equation is(Illb)tA+ tB=>cand the various cases are obtained by specializing, in appropriate ways,the coefficients A, B, and C.Take first C = 0 and A and Β as the partial derivatives with respect toρ and ρ of a function F of ρ and p.

Then Eq. ( I l l b ) expresses the fact thatF(p, p) has a constant value for each particle. Combined with the boundarycondition that at some time this value is the same for all particles, weobtain the specifying equation of an elastic fluid. In particular, if thefunction F is the entropy S(p, p) then ( I l l b ) is the specifying equationfor the adiabatic flow of an inviscid fluid.3.6G E N E R A L F O R M OF S P E C I F Y I N GEQUATION33A general class of conditions is expressed by ( I l l b ) Ίϊ A, B, and C areassumed to be arbitrary functions of x, y, z, t, and of the flow variables p,p, q.

T h e corresponding wide class of fluid motions is characterized by com­mon properties. This class does not include the case of viscous or heatconducting fluids and may be designated as the class of ideal fluid motions.( W e shall consider a particular case in Sec. 9.6.)T h e specifying equation for the case of viscosity and heat conduction isincluded in ( I l l b ) if the term C on the right is allowed to depend also onderivatives of p, p, and q. In fact, if T(p, p) and S(p, p) are given, and wetake(25)A = Tf,dpΒ =Τ —,dp1then the left-hand side of ( I l l b ) is T(dS/dt) and according to (16) mustequal Q + θ/ρ. Here θ is defined by (11) as a given function of the viscousstresses σ'', τ and of the spatial derivatives of q; and the σ ' , τ are in generalassumed to be given functions of these spatial derivatives of q with co­efficients depending on ρ and p.

In the strictly adiabatic case Q' = 0,while in the simply adiabatic case when heat conduction is admitted withno other heat input or output, Q is determined by (23) as a function ofthe spatial derivatives of T(p, p). T o cover more general cases, one wouldhave to add to the right-hand side of Eq. (23) an appropriate functionexpressing the heat production per unit of time and mass.ffIf the total heat input [and, therefore, the quantity C in Eq. ( I l l b ) ] isgiven as a function of x, y, z, t, p, p, q, and the spatial derivatives of p, p, q,then the system of equations that governs the motion consists of the fol­lowing: Eq. ( I a ) , which generalizes Eq.

(1.1), Eqs. ( l . I I ' ) and ( I l l b ) . T h esystem can then be written in the form:-77 = g - - grad ρ + atρρ(26)^ = - p d i v qdpd t, CΒ=ApdlVq+A >where v, as in ( I a ) , is the resultant viscosity force per unit volume. Sinceν depends on the spatial derivatives of q and possibly on the flow variables(through the viscosity coefficients) the Eqs. (26) can be interpreted asfollows: At each moment and for each particle the material derivatives of thefive unknowns q, ρ, p are determined as functions of the instantaneous valuesof p, p, ({ at the point x, ?/, ζ and in a neighborhood of the point.34I. I N T R O D U C T I O NArticle 4Sound Velocity.

W a v e Equation1. The problemSuppose that the fluid mass extends indefinitely in all directions, so thatno boundaries are encountered, and that the fluid is inviscid and elastic.T h e latter assumption means, as in Sec. 1.4, that a universal and one-toone relation between ρ and ρ exists.

This (p, p)-relation is assumed to bedifferentiable, and such that density increases with pressure, and viceversa. Then dp/dp is nonnegative, and we may writeif- = a\(1)dpT h e dimensions of the left-hand member are given by(ilf/L ) =3L /T\2Thus a has the dimensions L/T,(ML/T L )/22a velocity. W i t h thenotation of Eq. ( 1 ) , any derivative of ρ can be expressed in terms of thecorresponding derivative of p, e.g., dp/dx = adp/dx, and, in particular,grad ρ — a grad p.(2)A state of rest, q = 0, with uniform pressure and density, ρ = p , ρ = po,0is certainly compatible with the equation of motion (1.1), and the equationof continuity ( l .

I I ) , if gravity is neglected, for all the other terms in theseequations are derivatives.20T h e values ρ =p,ρ =0po are assumed tosatisfy the given (p, p)-relation.N o w we consider a small perturbationinitial disturbance:of the state of rest, caused b y anto each (x, y, z, t) there will correspond small valuesof q, ρ — po, and ρ — p . T h e qualification small means that any product0of two or more of these quantities or their derivatives will be supposednegligible compared to first-order terms. T h e particle derivative, d/dt, ofany of these quantities may then be replaced by the partial derivatived/dt, since the additional terms are all of second order.

Moreover, on theleft-hand side of Eq. (1.1), ρ dq/dt may be replaced by p dq/dt, the dif­0ference (p — po) dq/dt being of second order. On the right-hand side,grad ρ becomes, by Eq. ( 2 ) , a grad p, for which one has to substitute2a02(a2grad p, where a is the value of α for ρ = p ,002Eq. (1.1) becomes(3)ρ = p , the difference0— ao ) grad ρ being small of second order. Thus, with gravity neglectedpo^7 =dt-a20grad p.4.2D'ALEMBERT'SSOLUTION35Similarly, the equation of continuity ( l . I I ' ) has to be replaced bypodiv q = -(4)|;.dtThese two equations, (3) and ( 4 ) , are the determining equations for asmall perturbation in an inviscid elastic fluid originally at rest. There isone scalar and one vector equation, with one scalar unknown, p, and onevector unknown, q.