R. von Mises - Mathematical theory of compressible fluid flow, страница 5

Bernoulli equationIf we use Eqs. (2) and ( 3 ) , Eq. (1) can be written2| ( |+ ^ ) =-^q-gradp.W i t h the further assumption that the motion is steady, so that dp/dt = 0,the Euler rule (1.4) givesq . g r a dP= ^ ,18I.INTRODUCTIONand the above relation becomesUnder the hypothesis that a (p, p)-relation exists for the particle in ques­tion, ρ can be regarded as a function of p. A new function can be introduced,say P, whose derivative with respect to ρ is 1/p:(19)JThen dP/dt=(l/p)(dp/dt)ρand (18) becomes(20)which is known as the Bernoulliequation.Usually P/g (which has the dimension of length) is called the pressurehead, and q/2g the velocity head.

Equation (20) is equivalent to10where i f is a constant (which may be different for each particle), or inwords: During the steady motion of a fluid that is either elastic or subject to aparticle-wise (ρ, p)-relation the sum Η of the velocity head, the geometricalelevation, and the pressure head, has a constant value for each particle. Forall particles on the same streamline this constant is the same.

T h e quantityΗ is known as the total head or as the Bernoulli constant or the Bernoullifunction (since it may vary from one streamline to the n e x t ) .11T h e Bernoulli equation can also be given in differential form, by multi­plying through by dt in E q . ( 1 8 ) :(21)q dq + g dh +- dp = 0Ρor, neglecting gravity,(21')PQdq + dp = 0,where the differentials refer, of course, to the changes in q, h, and ρ for adefinite particle.T h e Bernoulli equation in differential form can also be derived directlyfrom Eq.

(1.1) as the component of Newton's equation in the direction ofq, where d/dt is to be identified with q(d/ds).For the four cases listed previously, the function Ρ is given b y2.5BERNOULLI EQUATION19( a ) Incompressible flow, or ρ = p :0Ρ = — +Poconstant;(b) Isothermal flow of a perfect gas, or — =ΡΡ = — log ρ +Po—:Poconstant;(22)(c) Polytropic flow, or - = — :p"Po*Ρ = — % ^p~κ — 1 po{K+l),K(d) Linearized condition, ρ =Ρ = 2P<P 2r>Ρ +2constant = — ?+κ — 1ρconstant;ρPo -constant.Again, the additive constant in Ρ is arbitrary.If the Bernoulli equation (20) is compared with the energy equation (16)for an elastic fluid, it is seen that Ρ includes both the strain energy and thesurface work w. In fact, the equalitydP _ de _j_ wdtρdtfollows for steady motion directly from ( 4 ) , using (15), ( 5 ) , and (19).

Thusthe pressure head P/g is not an energy, but the sum of an energy term anda work term.I n thermodynamics the sum of internal energy U plus the quotient p/pis known as enthalpy:(23)+pIn adiabatic flow, where Q' = 0, it follows from E q . (10) thatdldtdUdt+dt \pjμdt \p/dt\pjpdtdt'since each particle possesses a (p, p)-relation expressed by the constancyof its entropy. Thus, the quantity Ρ in the case of adiabatic flow differsfrom the enthalpy / only by an additive constant (which still may be dif­ferent for each particle). If the motion is isentroprc, Ρ as well as / is anI.20INTRODUCTIONover-all function of the state variables, and in this case they differ by anover-all constant.6. Two integral theoremsIn order to extend the energy equation (7) to a volume of finite dimen­sions, some integral transformation formulas are needed.

These will nowbe derived.(a) Let V be a volume bounded by a closed surface S. L e t F be a functionof x, y, ζ having first partial derivatives which, together with F, are con­tinuous in V and on S.T h e formula to be proved is(24)[ ^fFdV =cos (ft, z) ds,JgJy OZwhere (ft, z) denotes the angle between the z-axis and the outer normal to S.This formula transforms the volume integral over V into a surface integralover S.

The surface S may consist of a finite number of sections, on each ofwhich cos (ft, z) varies continuously.Suppose first that V is such that a parallel to the z-axis intersects S in atmost two points 1 and 2. T h e volume integral may then be evaluated as aniterated integral:L£(25)ddv= L I£dA d=dzL^- > >F)dAwhere the region of integration, A, for the double integral is the projectionof V onto the x, ?/-plane, while the integral with respect to ζ is taken fromthe point 1 where the horizontal cylinder with cross section dA first cuts V,to the point 2 where it leaves V (see Fig. 4 ) ; F\ and F denote the values ofF at these points. Let d§>i and d§> denote the areas of the parts of S cut offby this cylinder, and fti and n<i the directions normal to these surface ele­ments.

Then since dA, dSi , and dS are positive quantities,222dA =— cos (fti , z) d$i = cos (712, z) d&2.Thus, the last integral in (25) may be rewritten asJ(26)(F2-Fi) dA = jF cos (ft , z) d§> + j222F cos (fti, z) dSixU=/ F cos (ft, z) dS.Equations (25) and (26) give exactly (24). One can easily see that the re­striction to a surface with only two intersections with a parallel to thez-axis is unessential.2.6TWO INTEGRAL21THEOREMSFIG.

4. Integral transformation from volume to surface integral.Analogous formulas hold for the x- and ^/-directions. In particular,these formulas may be applied to the cases F = v , v , and v , where v ,v , and v are the components of a vector v, each formula being taken withdifferentiation in the direction of the component. When these results areadded, the integrand on the left in Eq. (24) is div v, while the integrand onthe right is v cos (n, x) + v cos (n, y) + v cos (n, z), which is exactlythe component v of ν in the direction n. Thusxyyzxzxyznf(27)div ν dV =fvnT h e right-hand member of Eq.

(27) may be called the flux of ν through thesurface $. T h e result (27) is known as Gauss Theorem, or as the Diver­gence Theorem.( b ) L e t / be a quantity associated with a fluid particle, i.e., a functionof some or all of the variables q , q , q , p, and p. Then the following for­mula holds:1xyzwhere ρ is the density and d(f pf dV)/dt means the rate of change of J pf dVas a certain portion of the fluid moves along.P y the Euler rule (1.4') and a computation similar to that used in deriv­ing Eq. (4) we haveyv(29)ρ l = ρId+ pq-grad / = ρ | + div (p/q) -/ div ( q ) .P22I.INTRODUCTIONt+dttF I G . 5.

Change of volume integral due to change and displacement of boundarysurface.From the equation of continuity ( l . I I ) , div (pq) = —dp/dt, so that(3o)IAdVIA i> L (£) -=+dVdV+fdVCombining the two integrands involving d/dt and applying (27) to the thirdintegral, we obtain(31)The left-hand side of Eq.

(28) expresses the total rate of change of / pf dV,due in part to the change within V of the integrand with respect to time(which produces the rate of change J d(pf)/dt dV), and in part to themotion of the boundary S which changes the volume over which the integralis extended. This flow takes place across each surface element dS; in timedt the surface element d& adds to the volume V a cylindrical element ofvolume d$-q dt (see Fig. 5). Then / pf dV evaluated only for the addi­tional volume equals dt /§ pfq d§> and the rate of change is thereforeSgpfQn dS. It is thus seen that the left-hand side of Eq. (28) is preciselyequal to the right-hand side of Eq.

(31), and the proof is complete. Ofcourse, with / a= 1, so that df/dt = 0, equation (28) reduces to theequation of continuity in its preliminary form (1.2).vnn}7. Energy equation for a finite massWith the aid of the foregoing integral theorems it is easy to derive fromEq. (7), which holds for a perfect gas, a relation between the energy, thework, and the heat input for a finite portion of the gas.Both sides of Eq.

(7) may be multiplied by ρ dV and the result integratedover V. With the abbreviation / = q/2 + gh + c T, and the value of wfrom (5), the result isv2.7E N E R G Y E Q U A T I O N FOR F I N I T E MASS23If we apply (28) to the first integral and (27) to the second, this becomes(33)jtjv PfJdV +pqd$ =nfvpQ' dV.N o w the total energy of the fluid mass enclosed by the boundary surface Sis given by(34)E = £P( |+ gh + c T^dVv=f^pfdV,while the work done, per unit time, against pressure on the surface S is(35)fW =pq dS,nsince the force has magnitude ρ dS, and the rate of motion in the directionopposite to the force is q .

Thus (33) givesn(36)^+W =jv PQ'dV,and the statement is the same for a finite portion of a perfect gas as for theinfinitesimal portion considered in Sec. 2: The total heat input per unit timeis equal to the time rate of change of the total energy (kinetic, potential, andinternal) plus the work done per unit time at the surface of the gas againstpressure.T h e statement (36) holds independently of the type of specifying equa­tion.

For adiabatic flow, Q' = 0, and (36) givesdE-f +(37)atW = 0.For an elastic fluid, the integral theorems may be applied as above,starting from E q . (16), to give(38)f+W = 0,where(39)E'=jpv(£+ gh + e^ dV.24I. I N T R O D U C T I O NArticle 3Influence of Viscosity. Heat Conduction1. Viscous stresses and hydraulic pressureFor an inviscid fluid the forces exerted on any fluid element by sur­rounding masses are normal to the surface element on which they act andhave the same intensity ρ whatever the orientation of the surface element.This intensity ρ (force per unit area) was called the hydraulic pressureat the point under consideration.